diff --git a/-NA0T4oBgHgl3EQfPP8X/content/tmp_files/2301.02171v1.pdf.txt b/-NA0T4oBgHgl3EQfPP8X/content/tmp_files/2301.02171v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbc2869b3385bc47af0cde23ec33bdfd4c8ce784 --- /dev/null +++ b/-NA0T4oBgHgl3EQfPP8X/content/tmp_files/2301.02171v1.pdf.txt @@ -0,0 +1,2141 @@ +Strong Convergence of Peaks Over a Threshold +S. A. Padoan +Department of Decision Sciences, Bocconi University, Italy +and +S. Rizzelli +Department of Statistical Sciences, Catholic University, Italy +January 6, 2023 +Abstract +Extreme Value Theory plays an important role to provide approximation re- +sults for the extremes of a sequence of independent random variable when their +distribution is unknown. An important one is given by the Generalised Pareto dis- +tribution Hγ(x) as an approximation of the distribution Ft(s(t)x) of the excesses +over a threshold t, where s(t) is a suitable norming function. In this paper we +study the rate of convergence of Ft(s(t)·) to Hγ in variational and Hellinger dis- +tances and translate it into that regarding the Kullback-Leibler divergence between +the respective densities. We discuss the utility of these results in the statistical field +by showing that the derivation of consistency and rate of convergence of estimators +of the tail index or tail probabilities can be obtained thorough an alternative and +relatively simplified approach, if compared to usual asymptotic techniques. +Keywords: Contraction Rate, Consistency, Exceedances, Extreme Quantile, Gener- +alised Pareto, Tail Index. +2020 Mathematics Subject Classification: Primary 60G70; secondary 62F12, 62G20 +1 +Introduction +Extreme Value Theory (EVT) develops probabilistic models and methods for describ- +ing the random behaviour of extreme observations that rarely occur. These theoretical +foundations are very important for studying practical problems in environmental, cli- +mate, insurance and financial fields (e.g., Embrechts et al., 2013; Dey and Yan, 2016), +to name a few. +In the univariate setting, the most popular approaches for statistical analysis are the +so-called Block Maxima (BM) and Peaks Over Threshold (POT) (see e.g. B¨ucher and +Zhou, 2021, for a review). Let X1, . . . , Xn be independent and identically distributed +(i.i.d.) random variables according to a common distribution F. The first approach +concerns the modelling of k sample maxima derived over blocks of a certain size m, i.e. +Mm,i = max(X(i−1)m+1, . . . , Xim), i ∈ {1, . . . , k}. In this case, under some regularity +conditions (e.g. de Haan and Ferreira, 2006, Ch. 1), the weak limit theory establishes +that F m(amx+bm) converges pointwise to Gγ(x) as m → ∞, for every continuity point +x of Gγ, where Gγ is the Generalised Extreme Value (GEV) distribution, am > 0 and +bm are suitable norming constants for each m = 1, 2, . . . and γ ∈ R is the so-called +tail index, which describes the tail heaviness of F (e.g. de Haan and Ferreira, 2006, +Ch. 1). The second method concerns the modelling of k random variables out of the +n available that exceed a high threshold t, or, equivalently, of k threshold excesses Yj, +1 +arXiv:2301.02171v1 [math.PR] 5 Jan 2023 + +j = 1, . . . , k, which are i.i.d. copies of Y = X −t|X > t. In this context, the Generalised +Pareto (GP) distribution, say Hγ, appears as weak limit law of appropriately normalised +high threshold exceedances, i.e. for all x > 0, Ft(s(t)x) converges pointwise to Hγ(x) +as t → x∗, for all the continuity points x of Hγ(x), where Ft(x) = P(Y ≤ x) and +s(t) > 0 is a suitable scaling function for any t ≤ x∗, with x∗ = inf(x : F(x) < +∞). This result motivates the POT approach, which was introduced decades ago by +the seminal paper Balkema and de Haan (1974). Since then, few other convergence +results emerged. +For instance, the uniform convergence of Ft(s(t) · ) to Hγ and the +coresponding convergence rate have been derived by Pickands III (1975) and Raoult +and Worms (2003), respectively. Similar results but in Wasserstein distance have been +recently established by Bobbia et al. (2021). As for the GEV distribution, more results +are available. +In particular, there are sufficient conditions to ensure, in addition to +weak convergence, that F m(am · +bm) converges to Gγ for example uniformly and +in variational distance and the density of F m(am · +bm) converges pointwise, locally +uniformly and uniformly to that of Gγ (e.g. Falk et al., 2010, Ch. 2; Resnick, 2007, Ch. +2). +The main contribution of this article is to provide new convergence results that can +be useful in practical problems for the POT approach. Motivated by the utility in the +statistical field to asses the asymptotic accuracy of estimation procedures, we study +stronger forms of convergence than the pointwise one, as limt→x∗ D(Ft(s(t) · ), Hγ) = 0, +where D( · ; · ) is either the variational distance, the Hellinger distance or the Kullback- +Leibler divergence. In particular, we provide upper bounds for the rate of convergence +to zero of D(Ft(s(t) · ); Hγ) in the case that D( · ; · ) is the variational and Hellinger dis- +tance, and further translate them into bounds on Kullback-Leibler divergence between +the densities of Ft(s(t)·) and Hγ, respectively. +Estimators of the tail index γ (and other related quantities) are typically defined +as functionals of the random variables (Y1, . . . , Yk), as for instance the popular Hill +(Hill, 1975), Moment (Dekkers et al., 1989), Pickands (Pickands III, 1975), Maximum +Likelihood (ML, Jenkinson, 1969), Generalised Probability Weighted Moment (GPWM, +Hosking et al., 1985) estimators, to name a few. In real applications, the distribution +F is typically unknown and so is F(s(t) · ). Although, for large t, Hγ provides a model +approximation for Ft(s(t) · ), when one wants to derive asymptotic properties as the +consistency and especially the rate of convergence of the tail index estimators (or other +related quantities), still the fact that (after rescaling) the random variables (Y1, . . . , Yk) +are actually distributed according to Ft(s(t) · ) needs to be taken into account, which +makes asymptotic derivations quite burdensome. These are even more complicated if t +is determined on the basis of the (k + 1)-th largest order statistic of the original sample +X1, . . . , Xn, which is the most common situation in practical applications. In this case, +the threshold is in fact random and, up to rescaling, Ft(s(t) · ) only gives a conditional +model for the variables Yj given a fixed value t of the chosen statistic. Asymptotic +properties for POT methods have been studied in the last fifty years, see for example +Hall and Welsh (1984), Drees (1998), Dekkers and de Haan (1993) and the reference +therein. +Leveraging on our strong convergence results we can show that, for random sequences +(such as sequences of estimators) convergence results in probability that hold under the +limit model Hγ, are also valid for a rescaled sample of excesses over a large order statistic. +Precisely, we show that the distribution of the latter, up to rescaling and reordering, +is contiguous to that of an ordered i.i.d. sample from Hγ (e.g., van der Vaart, 2000, +Ch. 6.2). As a by product of this result, one can derive the consistency and rate of +convergence of a tail index estimator (or an estimator of a related quantity) by defining +it as a functional of the random sequence (Z1, . . . , Zk) which is distributed according the +2 + +limit model Hγ, and, if density of Ft(s(t)·) satisfies some regularity conditions, then the +same asymptotic results hold even when such estimator is defined through the sequence +of excesses. This approach simplifies a lot the computations as asymptotic properties +are easily derivable under the limit model. +The article is organised as follows, Section 2 of the paper provides a brief summary +of the probabilistic context on which our results are based. Section 3 provides our new +results on strong convergence to a Pareto model. Section 4 explains in what applications +concerning statistical estimation our results are useful. Section 5 provides the proofs of +the main results. +2 +Background +Let X be a random variable with a distribution function F that is in the domain of +attraction of the GEV distribution Gγ, shortly denoted as F ∈ D(Gγ). This means that +there are norming constants am > 0 and bm ∈ R for m = 1, 2, . . . such that +lim +m→∞ F m(amx + bm) = exp +� +− (1 + γx)−1/γ� +=: Gγ(x), +(2.1) +for all x ∈ R such that 1 + γx > 0, where γ ∈ R, and this is true if only if there is a +scaling function s(t) > 0 with t < x∗ such that +lim +t→x∗ Ft(s(t)x) = 1 − (1 + γx)−1/γ =: Hγ(x), +(2.2) +e.g., de Haan and Ferreira (2006, Theorem 1.1.6). The densities of Hγ and Gγ are +hγ(x) = (1 + γx)−(1/γ+1) +and +gγ(x) = Gγ(x)hγ(x), +respectively. Let U(v) := F ←(1 − 1/v), for v ≥ 1, where F ← is the left-continuous +inverse function of F and G←(exp(−1/x)) = (xγ − 1)/γ. +Then, we recall that the +first-order condition in formula (2.1) is equivalent to the limit result +lim +v→∞ +U(vx) − U(v) +a(v) += xγ − 1 +γ +, +(2.3) +for all x > 0, where a(v) > 0 is a suitable scaling function. In particular, we have that +s(t) = a(1/(1 − F(t))), see de Haan and Ferreira (2006, Ch. 1) for possible selections of +the function a. +A stronger convergence form than that in formula (2.2) is the uniform one, i.e. +sup +x∈[0, x∗−t +s(t) ) +|Ft(s(t)x) − Hγ(x)| → 0, +t → x∗. +To establish the speed at which Ft(s(t)x) converges uniformly to Hγ(x), Raoult and +Worms (2003) relied on a specific formulation of the well-known second-order condi- +tion. In its general form, the second order condition requires the existence of a posi- +tive function a and a positive or negative function A, named rate function, such that +limv→∞ |A(v)| = 0 and +lim +v→∞ +U(vx)−U(v) +a(v) +− xγ−1 +γ +A(v) += D(x), +x > 0, +3 + +where D is a non-null function which is not a multiple of (xγ − 1)/γ, see de Haan and +Ferreira (2006, Definition 2.3.1). The rate function A is necessarily regularly varying at +infinity with index ρ ≤ 0, named second-order parameter (de Haan and Ferreira, 2006, +Theorem 2.3.3). In the sequel, we use the same specific form of second order condition +of Raoult and Worms (2003) to obtain decay rates for stronger metrics than uniform +distance between distribution functions. +3 +Strong results for POT +In this section, we discuss strong forms of convergence for the distribution of rescaled +exceedances over a threshold. First, in Section 3.1, we discuss convergence to a GP +distribution in variational and Hellinger distance, drawing a connection with known +results for density convergence of normalized maxima. In Section 3.2 we quantify the +speed of convergence in variational and Hellinger distance. Finally, in Section 3.3, we +show how these can be used to also bound Kullback-Leibler divergences. Throughout, +for a twice differentiable function W(x) on R, we denote with W ′(x) = (∂/∂x)W(x) +and W ′′(x) = (∂2/∂x2)W(x) the first and second order derivatives, respectively. +3.1 +Strong convergence under classical assumptions +Let the distribution function F be twice differentiable. In the sequel, we denote f = F ′, +gm = (F m(am · +bm))′ and ft = F ′ +t. +Under the following classical von Mises-type +conditions +lim +x→∞ +xf(x) +1 − F(x) = 1 +γ , +γ > 0, +lim +x→x∗ +(x∗ − x)f(x) +1 − F(x) += −1 +γ , +γ < 0, +(3.1) +lim +x→x∗ +f(x) +� x∗ +x (1 − F(v)dv) +(1 − F(x))2 += 0, +γ = 0, +we know that the first-order condition in formula (2.3) is satisfied and it holds that +lim +v→∞ va(v)f(a(v)x + U(v)) = (1 + γx)−1/γ−1 +(3.2) +locally uniformly for (1 + γx) > 0. Since the equality gm(x) = F m−1(amx + bm)hm(x) +holds true, with bm = U(m), am = a(m) and hm(x) = mamf(amx + bm), and since +F m−1(amx+bm) converges to Gγ(x) locally uniformly as m → ∞, the convergence result +in formula (3.2) thus implies that gm(x) converges to gγ(x) locally uniformly (Resnick, +2007, Ch. 2.2). +On the other hand, the density pertaining to Ft(s(t)x) is +lt(x) := ft(s(t)x)s(t) = s(t)f(s(t)x + t) +1 − F(t) +and, setting v = 1/(1 − F(t)), we have a(v) = s(t) and v → ∞ as t → x∗. Therefore, +a further implication of the convergence result in formula (3.2) is that lt(x) converges +to hγ(x) locally uniformly for x > 0, if γ ≥ 0, or x ∈ (0, −1/γ), if γ < 0. In turn, by +Scheffe’s lemma we have +lim +t→x∗ V (Pt, P) = 0, +where +V (Pt, P) = sup +B∈B +|Pt(B) − P(B)| +4 + +is the total variation distance between the probability measures +Pt(B) := P +�X − t +s(t) +∈ B +����X > t +� +and P(B) := P(Z ∈ B), +and where Z is a random variable with distribution Hγ and B is a set in the Borel +σ-field of R, denoted by B. Let +H 2(lt; hγ) := +� �� +lt(x) − +� +hγ(x) +�2 +dx +be the square of the Hellinger distance. It is well know that the Hellinger and total +variation distances are related as +H 2(lt; hγ) ≤ 2V (Pt, P) ≤ 2H (lt; hγ), +(3.3) +see e.g. Ghosal and van der Vaart (2017, Appendix B). Therefore, the conditions in +formula (3.1) ultimately entail that also the Hellinger distance between the density of +rescaled peaks over a threshold lt and the GP density hγ converges to zero as t → x∗. +In the next subsection we introduce a stronger assumption, allowing us to also quantify +the speed of such convergence. +3.2 +Convergence rates +As in Raoult and Worms (2003) we rely on the following assumption, in order to derive +the convergence rate for the variational and Hellinger distance. +Condition 3.1. Assume that F is twice differentiable. Moreover, assume that there +exists ρ ≤ 0 such that +A(v) := vU′′(v) +U ′(v) + 1 − γ +defines a function of constant sign near infinity, whose absolute value |A(v)| is regularly +varying as v → ∞ with index of variation ρ. +When Condition 3.1 holds then the classical von-Mises conditions in formula (3.1) +are also satisfied for the cases where γ is positive, negative or equal to zero, respec- +tively. Furthermore, Condition 3.1 implies that an appropriate scaling function for the +exceedances of a high threshold t < x∗, which complies with the equivalent first-order +condition (2.2), is defined as +s(t) = (1 − F(t))/f(t). +With such a choice of the scaling function s, we establish the following results. +Theorem 3.2. Assume Condition 3.1 is satisfied with γ > −1/2. Then, there exist +constants ci > 0 with i = 1, 2, αj > 0 with j = 1, ..., 4, K > 0 and t0 < x∗ such that +H 2(lt; hγ) +K|A(v)|2 ≤ S(v) +(3.4) +for all t ≥ t0, where v = 1/(1 − F(t)) and +S(v) := +� +1 − |A(v)|α1 + 4 exp (c1|A(v)|α2) , +if γ ≥ 0 +1 − |A(v)|α3 + 4 exp (c2|A(v)|α4) , +if γ < 0 +. +5 + +Given the relationship between the total variation and Hellinger distances in (3.3), +the following result is a direct consequence of Theorem (3.2). +Corollary 3.3. Under the assumptions of Theorem 3.2, for all t ≥ t0 +V (Pt, P) ≤ |A(v)| +� +KS(v). +Theorem 3.2 implies that the Hellinger and variational distances of the probability +density and measure of rescaled exceedances from their GP distribution counterparts are +bounded from above by C|A(v)|, for a positive constant C, as the threshold t approaches +the end-point x∗. Since for a fixed x ∈ ∩t≥t0(0, x∗−t +s(t) ) it holds that +|Ft(s(t)x) − Hγ(x)| ≤ V (Pt, P) +and since Raoult and Worms (2003, Theorem 2(i)) implies that |Ft(s(t)x)−Hγ(x)|/|A(v)| +converges to a positive constant, there also exists c > 0 such that, for all large t, c|A(v)| +is a lower bound for variational and Hellinger distances. Therefore, since +c|A(v)| ≤ V (Pt, P) ≤ H (lt; hγ) ≤ C|A(v)|, +the decay rate of variational and Hellinger distances is precisely |A(v)| as t → x∗. +Differently from the result on uniform convergence in Raoult and Worms (2003), +our results on convergence rates in the stronger total variation and Hellinger topologies +are given for γ > −1/2. Although the bound in formula (3.4) remains mathematically +valid also for tail indices below −1/2, the restriction γ > −1/2 is imposed to guarantee +that constants α3, α4 in the definition of S(v) are positive, so that S(v) is positive and +bounded as t approaches x∗. Note that such a behaviour of S is essential to deduce +from the bound in formula (3.4) that the rate of convergence is |A(v)|. +3.3 +Kullback-Leibler divergences +A further implication of Theorem 3.2 concerns the speed of convergence to zero of the +Kullback-Leibler divergence +K (˜lt; hγ) := +� +ln +� +˜lt(x)/hγ(x) +� +˜lt(x)dx, +and the divergences of higher order p ≥ 2 +Dp(˜lt; hγ) := +� ���ln +� +˜lt(x)/hγ(x) +���� +p ˜lt(x)dx, +where ˜lt = (Ft(˜s(t) · ))′ and ˜s(t) is a scaling function possibly different from s(t), which +ensures that the support of the conditional distribution Ft(˜s(t)x) is contained in that +of the GP distribution Hγ when γ < 0, i.e. x ∈ R : x < (x∗ − t)/˜s(t) < −1/γ. We recall +indeed that, when γ is negative, the end-point (x∗ − t)/s(t) of lt converges to −1/γ as +t approaches x∗. Nevertheless, for t < x∗ it can be that (x∗ − t)/s(t) > −1/γ, entailing +that K (lt; hγ) = Dp(lt; hγ) = ∞. The introduction of a more flexible scaling function +˜s is thus meant to rule out this uninteresting situation. In order to exploit Theorem +3.2 to give bounds on Kullback-Leibler and higher order divergences, we first introduce +by the next two lemmas a uniform bound on density ratios and a Lipschitz continuity +result. +Lemma 3.4. Under the assumptions of Theorem 3.2, if ρ < 0 and γ ̸= 0, and if +˜s(t)/s(t) → 1 as t → x∗, then there exist a t1 < x∗ and a constant M ∈ (0, ∞) such +that +sup +t≥t1 +sup +0 −1/2. Then, there exists ϵ > 0 and L > 0 such that +H 2(hγ; hγ′(σ · )σ) < L2(|γ − γ′|2 + |1 − σ|2) +whenever |γ − γ′|2 + |1 − σ|2 < ϵ2. +Next, using the uniform bound on density ratio provided in Lemma 3.4 and the +Lipschitz continuity property established in Lemma 3.5, we are able to translate the +upper bounds on the squared Hellinger distance H 2(lt, hγ) into upper bounds on the +Kullback-Leibler divergence K (˜lt; hγ) and higher order divergences Dp(˜lt; hγ). +Corollary 3.6. Under the assumptions of Theorem 3.2 with in particular ρ < 0 and +γ ̸= 0, if there also exists B > 0 such that, for all large t < x∗, +|s(t)/˜s(t) − 1| ≤ B|A(v)|, +then there exists a t2 < x∗ such that, for all t ≥ t2 +(a) K (˜lt; hγ) ≤ 2M( +� +KS(v) + BL)2|A(v)|2 +(b) Dp(˜lt; hγ) ≤ 2p!M( +� +KS(v) + BL)2|A(v)|2, with p ≥ 2. +To extend the general results in Lemma 3.4 and Corollary 3.6 to the case of γ = 0 +seems to be technically over complicated. +Nevertheless, there are specific examples +where the properties listed in such lemmas are satisfied, such as the following one. +Example 3.7. Let F(x) = exp(− exp(−x)), x ∈ R, be the Gumbel distribution function. +In this case, Condition 3.1 is satisfied with γ = 0 and ρ = −1, so that Theorem 3.2 +applies to this example, and for an arbitrarily small ϵ > 0 we have +lt(x)/h0(x) ≤ exp(exp(−t)) < 1 + ϵ +for all x > 0 and suitably large t. Hence, the bounded density ratio property is satisfied +and it is still possible to conclude that Dp(lt; h0)/|A(v)|2 and K (lt; h0)/|A(v)|2 can be +bounded from above as in Corollary 3.6. +4 +Implications +From a statistical stand point, the results introduced in Sections 3 can be used to study +consistency and rate of contraction of estimators of the true value for a quantity of +interest relative to the distribution of threshold exceedances within a POT approach. +First, in Section 4.1, we illustrate an application to a density estimation problem. +Second, in Section 4.2, we discuss the problem of studying estimators’ asymptotic ac- +curacy in more general terms. A by product of our theory in Section 3 is that the +consistency of estimators of the GP distribution parameters or related quantities can +be easily derived by means of a contiguity result (e.g. van der Vaart, 2000, Ch. 6), +provided that appropriate regularity conditions are satisfied, avoiding complicated and +long calculations, typically required for example by popular estimators of the tail index +γ (Hall and Welsh, 1984; Drees, 1998; Dekkers and de Haan, 1993). +4.1 +Density estimation +Accurate density estimation for threshold excesses is a crucial problem for probabilis- +tic foresting of extremes, and, in particular, for the construction of reliable predictive +regions for future large observations. When a sample X1, . . . , Xn of i.i.d. random vari- +ables, with a common distribution F, is available, a simple method to estimate the +7 + +density ft of (approximately) a small fraction k/n of exceedances, with k ∈ N, over a +large quantile t = U(n/k), is as follows. Let X(n−k) < . . . < X(n) denote the k + 1 +largest order statistics of the sample. Then, for measurable functions Tk,i, i = 1, 2, let +�γk = Tk,1(X(n−k), ..., X(n)) +be a generic estimator of the tail index γ and +�sk = Tk,2(X(n−k), ..., X(n)) +be a generic estimator of the scaling function s(U(n/k)). Since under Condition 3.1 it +holds that +ft(x) ≈ hγ +� +x +s(U(n/k)) +� +1 +s(U(n/k)), +then a plug-in estimator of ft(x) exploiting its GP approximation is given by +�hk(x) := h�γk(x/�sk)(1/�sk). +By means of Theorem 3.2 the accuracy of the above estimator can be assessed by +quantifying its rate of contraction to the true density ft in Hellinger distance. This is +formally stated by the next result. +Proposition 4.1. Under the assumptions of Theorem 3.2 and assuming further that, +for t = U(n/k) and k ≡ k(n), the following conditions are satisfied as n → ∞: +(a) k → ∞ and k/n → 0, +(b) +√ +k|A(n/k)| → λ ∈ (0, ∞), +(c) |�γk − γ| = Op(1/ +√ +k) and |�sk/s(U(n/k)) − 1| = Op(1/ +√ +k), +it then holds that +H (ft;�hk) = Op(1/ +√ +k). +For some specific choices of the estimators �γk and �sk proposed in the literature +on POT methods (e.g. de Haan and Ferreira, 2006, Ch. +3–5), assumptions (a)–(b) +of Proposition 4.1 have been used along with the second order condition to establish +asymptotic normality of the sequence +√ +k +� +�γk − γ, +�sk +s(U(n/k)) − 1 +� +. +Such estimators thus comply with assumption (c) of Proposition 4.1, whose statement +allows to readily obtain the rate of contraction of �hk to ft in Hellinger distance. We +provide next two examples. +Example 4.2. Under the assumptions of Theorem 3.2 and conditions (a)–(b) of Propo- +sition 4.1, there exists a sequence of ML estimators of γ and s(U(n/k)) given by +(�γk, �sk) ∈ arg max +(γ,σ)∈D +k +� +i=1 +hγ +�X(n−k+i) − X(n−k) +σ +� 1 +σ +where D = (−1/2, ∞) × (0, ∞), satisfying condition (c) of Proposition 4.1, see Drees +et al. (2004) and Zhou (2009). +8 + +Example 4.3. The GPWM estimators of γ and s(U(n/k)) are defined as +�γk = 1 − +� Pk +2Qk +− 1 +�−1 +, +�sk = Pk +� Pk +2Qk +− 1 +�−1 +, +where +Pk = 1 +k +k−1 +� +i=0 +� +X(n−i) − X(n−k) +� +, +Qk = 1 +k +k−1 +� +i=0 +i +k +� +X(n−i) − X(n−k) +� +. +Under the assumptions of Theorem 3.2 and conditions (a)–(b) of Proposition 4.1, and +assuming further that γ < 1/2, such estimators satisfy condition (c) of Proposition 4.1, +see e.g. Theorem 3.6.1 in de Haan and Ferreira (2006). +4.2 +Estimation consistency +Popular estimators of the tail index γ as for example the Hill, Moment, Pickands, ML, +GPWM (Hill, 1975; Dekkers et al., 1989; Pickands III, 1975; Jenkinson, 1969; Hosking +et al., 1985), or estimators of other related quantities, are typically defined as suitable +functionals of peaks/excesses over a large order statistic X(n−k), defined though the k +larger statistics in a sample as +Yk := (X(n−k+1) − X(n−k), . . . , X(n) − X(n−k)). +Informally speaking, the random variable X(n−k) plays the role of a high threshold t +and the sequence (X(n−k+i) − X(n−k)) with i = 1, . . . , k (up to rescaling) is seen as +approximately distributed according to Hγ. +Let Z1, . . . , Zk be a sample of i.i.d. +random variables with GP distribution Hγ +and let Zk = (Z(1), . . . , Z(k)) be the corresponding order statistics. In this section we +establish the important statistical result that the distribution of the suitably rescaled +sequence Yk is contiguous to that of the sequence Zk. To this aim, we first recall the +notion of contiguity, see van der Vaart (e.g., 2000, Ch. 6.2) for more details. +Definition 4.4. Let Pk and Qk be two sequence of probability measures. Qk is said to +be contiguous with respect to Pk, in symbols Pk ▷Qk, if for all measurable set sequences +Ek for which Pk(Ek) = o(1) we also have Qk(Ek) = o(1). +As in Proposition 4.1, in the sequel we assume k ≡ k(n) and k → ∞ as n → ∞. +Proposition 4.5. Let Pk and Qk be the probability measures relative to the random +sequences Zk and Yk/˜s(X(n−k)), respectively. Then, under the assumptions of Corollary +3.6 and assumptions (a)–(b) of Proposition 4.1, we have that Pk ▷ Qk. +In statistical problems where the aim is to estimate a functional of the limiting +GP distribution, say θ := φ(Hγ), the contiguity result in Proposition 4.5 can be used +to show that a suitable estimator Tk(Yk) of the parameter θ is consistent, or formally +speaking D(Tk(Yk), θ) = op(1), for a suitable metric D of interest. The next result and +the subsequent discussion illustrate this point. +Corollary 4.6. Under the assumption of Proposition 4.5, if Tk is a scale invariant +measurable function on (0, ∞)k and Tk(Zk) is consistent estimator of θ as n → ∞, then +also Tk(Yk) is a consistent estimator of θ as n → ∞. +In real applications the distribution F of the original sample (X1, . . . , Xn) is typically +unknown and as a result also the distribution of Yk is unknown. +For this reason, +9 + +proving consistency of an estimator of the form Tk(Yk) for the parameter θ can be quite +burdensome, and this is especially true for the derivation of its rate of contraction. We +recall that quantifying the speed of convergence, or contraction rate, of an estimator +Tk(Yk) of a parameter θ concerns the derivation of a positive sequences ϵk such that +ϵk ↓ 0 and D(Tk(Yk), θ) = Op(ϵk) as k → ∞, for a suitable metric D. +On the contrary, to establish the consistency of an estimator of the form Tk(Zk) +for estimating θ and its contraction rate is much easier, and these preliminary results +can be readily extended to the more demanding estimator Tk(Yk) by our Corollary 4.6, +therefore establishing its consistency and the associated speed of convergence. +We conclude the section with the following remark. It should be noted that within +the POT approach it is common to use estimators defined on the basis of scale invariant +functionals Tk. This is the case for many estimators of the tail index γ as those afore- +mentioned. Nevertheless, the result of Proposition 4.5 extends also to estimators which +are not invariant to rescaling of the data, provided that the discrepancy D(Tk(Yk), θ) +can be suitably decomposed into several terms that depends on Yk/˜s(X(n−k)) up to an +op(1) reminder. +5 +Proofs +5.1 +Additional notation +For y > 0, we denote T(y) = U(ey) and, for t < x∗, we define the functions +pt(y) = +� T(y+T −1(t))−t +s(t) +− eγy−1 +γ +, +γ ̸= 0 +T(y+T −1(t))−t +s(t) +− y, +γ = 0 +, +with s(t) = (1 − F(t))/f(t), and +qt(y) = +� +1 +γ ln [1 + γe−γypt(y)] , +γ ̸= 0 +pt(y), +γ = 0 +. +Moreover, for x ∈ (0, x∗ − t), we let φt(x) = T −1(x + t) − T −1(t). Finally, for x ∈ R, +γ ∈ R, ρ ≤ 0 and σ > 0, we set +Iγ,ρ(x) = +� x +0 +eγs +� s +0 +eρzdzds +and ψx,γ = νx/σ(γ) +� +1/σ, with +νx(γ) = +�� +hγ(x), +1 + γx > 0 +0, +otherwise +. +5.2 +Auxiliary results +In this section we provide some results which are auxiliary to the proofs of the main ones, +presented in Section 3. Throughout, for Lemmas 5.1–5.6, Condition 3.1 is implicitly +assumed to hold true. +Lemma 5.1. For every ε > 0 and every α > 0, if γ ≥ 0, or α ∈ (0, −1/γ), if γ < 0, +there exist x1 < x∗ and κ1 > 0 such that, for all t ≥ x1 and y ∈ (0, −α ln |A(eT −1(t))|) +(a) if γ ≥ 0, then +eqt(y) ∈ +� +e±κ1|A(eT −1(t))|e2εy� +; +10 + +(b) if γ < 0, then +eqt(y) ∈ +� +e±κ1|A(eT −1(t))|e(γ−ε)y� +. +Proof. By Lemma 5 in Raoult and Worms (2003), for all ε > 0 there exists x0 such that +for all t ∈ (x0, x∗) and y > 0, +e−γx|pt(y)| ≤ (1 + ε)|A(eT −1(t))|Iγ,ρ(y)e(γ−ε)y. +Moreover, for a positive constant ϑ1 +Iγ,ρ(y)e(γ−ε)y ≤ +� +ϑ1e2εy, +γ ≥ 0 +ϑ1e(γ−ε)y, +γ < 0 +. +Combining these two inequalities, we deduce that +e−γy|pt(y)| ≤ +� +(1 + ε)|A(eT −1(t))|ϑ1e2εy, +γ ≥ 0 +(1 + ε)|A(eT −1(t))|ϑ1e(γ−ε)y, +γ < 0 +. +(5.1) +As a consequence, if γ ≥ 0, for any α > 0 there exists a constant ϑ2 such that +sup +y∈(0,−α ln |A(eT −1(t))|) +e−γy|pt(y)| ≤ ϑ2|A(eT −1(t))|1−2εα +(5.2) +while, if γ < 0, for any α ∈ (0, −1/γ) there exists a constant ϑ3 such that +sup +y∈(0,−α ln |A(eT −1(t))|) +e−γy|pt(y)| ≤ ϑ3|A(eT −1(t))|1−(ε−γ)α. +(5.3) +Therefore, choosing ε sufficiently small, e−γy|pt(y)| converges to zero uniformly over the +interval (0, −α ln |A(eT −1(t))|) as t → x∗. +It now follows that, if y ∈ (0, −α ln |A(eT −1(t))|) and t > x1 for a sufficiently large +value x1 < x∗, when γ ̸= 0 a first-order Taylor expansion of the logarithm at 1 yields +|qt(y)| = +���� +1 +γ +γe−γypt(y) +1 + ϑ(t, y)γe−γypt(y) +���� +≤ +� +ϑ4|A(eT −1(t))|e2εy, +γ > 0 +ϑ5|A(eT −1(t))|e(γ−ε)y, +γ < 0 +, +where ϑ(t, y) ∈ (0, 1) and ϑ4, ϑ5 are positive constants, while when γ = 0 it holds that +|qt(y)| = eγye−γy|pt(y)| +≤ ϑ6|A(eT −1(t))|e2εy, +where ϑ6 is a positive constant. The two results in the statement are a direct consequence +of the last two inequalities. +Lemma 5.2. For every ε > 0 and every α > 0, if γ ≥ 0, or α ∈ (0, −1/γ), if γ < 0, +there exist x2 < x∗ and κ2 > 0 such that, for all t ≥ x2 and y ∈ (0, −α ln |A(eT −1(t))|) +(a) if γ ≥ 0, then +1 + q′ +t(y) ∈ +� +e±κ2|A(eT −1(t))|e2εy� +; +11 + +(b) if γ < 0, then +1 + q′ +t(y) ∈ +� +e±κ2|A(eT −1(t))|e(γ−ε)y� +. +Proof. If γ ̸= 0 +1 + q′ +t(y) = +exp +�� ey+T −1(t) +eT −1(t) +A(u) +u du +� +1 + γe−γypt(y) +, +while if γ = 0 +1 + q′ +t(y) = exp +�� ey+T −1(t) +eT −1(t) +A(u) +u +du +� +. +Therefore, if y ∈ (0, −α ln |A(eT −1(t))|) and t > x2 for a sufficiently large value x2 < x∗, +using the bounds in formulas (5.1)–(5.3) and choosing a suitably small ε we deduce +1 + q′ +t(y) ≤ +exp +�� ey+T −1(t) +eT −1(t) +A(u) +u du +� +1 − 1(γ ̸= 0)|γ|e−γy|pt(y)| +≤ exp +� +y|A(eT −1(t))| +� +× +� +� +� +1 +1−ω1|A(eT −1(t))|e2εy , +γ ≥ 0 +1 +1−ω2|A(eT −1(t))|e(γ−ε)y , +γ < 0 +≤ +� +� +� +exp +� +ω3|A(eT −1(t))|e2εy� +, +γ ≥ 0 +exp +� +ω4|A(eT −1(t))|e(γ−ε)y� +, +γ < 0 +for positive constants ωi, i = 1, . . . , 4. Similarly, +1 + q′ +t(y) ≥ +exp +�� ey+T −1(t) +eT −1(t) +A(u) +u du +� +1 + 1(γ ̸= 0)|γ|e−γy|pt(y)| +≥ exp +� +−y|A(eT −1(t))| +� +× +� +� +� +1 +1+ω5|A(eT −1(t))|e2εy , +γ ≥ 0 +1 +1+ω6|A(eT −1(t))|e(γ−ε)y , +γ < 0 +≥ +� +� +� +exp +� +−ω7|A(eT −1(t))|e2εy� +, +γ ≥ 0 +exp +� +−ω8|A(eT −1(t))|e(γ−ε)y� +, +γ < 0 +for positive constants ωi, i = 5, . . . , 8. The result now follows. +Lemma 5.3. If γ > 0 and ρ < 0, there exists a regularly varying function R with +negative index ϱ such that, defining the function +η(t) := (1 + γt)f(t) +1 − F(t) +− 1, +as v → ∞, η(U(v)) = O(R(v)). +Proof. Let v0 > 0 satisfy U(v0) ̸= 0 and U ′(v0) ̸= 0. Then, for v > v0 it holds that +η(U(v)) = 1 + γU(v) +vU′(v) +− 1 += 1 + γU(v0) +vU′(v) ++ γ +� v +v0 +U ′(r) +vU′(v)dr − 1. +12 + +Moreover, by definition of A, we have the identity +γ +� v +v0 +U ′(r) +vU′(v)dr − 1 = +� 1 +v0/v +U ′(zv) +U ′(v) dz − 1 += +� 1 +v0/v +γzγ−1 +� +exp +� +− +� 1 +z +A(vu) +u +du +� +− 1 +� +dz − +�v0 +v +�γ +. +Therefore, denoting by R2(v) the first term on the right-hand side and setting +R1(v) = 1 + γU(v0) +vU′(v) +− +�v0 +v +�γ +, +we have η(U(v)) = R1(v) + R2(v). +On one hand, the function R1(v) is regularly +varying of order −γ. On the other hand, for any β ∈ (0, 1), the function R2(v) can be +decomposed as follows +R2(v) = +� v−(1−β) +v0/v ++ +� 1 +v−(1−β) γzγ−1 +� +exp +� +− +� 1 +z +A(vu) +u +du +� +− 1 +� +dz +=: R2,1(v) + R2,2(v). +Assuming that A is ultimately positive and selecting v0 suitably large, we have +|R2,1(v)| ≤ +� v−(1−β) +v0/v +γzγ−1 +� +1 − exp +� +−A(vz) +z +�� +dz += O(v−γ(1−β)) +and +|R2,2(v)| ≤ +� 1 +v−(1−β) γzγ−1 � +1 − zA(vβ)� +dz += O(v−γ(1−β) ∨ A(vβ)). +Consequently, there exists a regularly varying function R of index ϱ = γ(β − 1) ∨ ρβ +complying with the property in the statement as v → ∞. +Similarly, if A is ultimately negative, choosing β such that β < 2γ and v0 suitably +large, we have +|R2,1(v)| ≤ +� v−(1−β) +v0/v +γzγ−1 � +uA(v0) − 1 +� +dz += O(v−(γ−β/2)(1−β)) +and +|R2,2(v)| ≤ +� 1 +v−(1−β) γzγ−1 � +zA(vβ) − 1 +� +dz += O(v−(γ−β/2)(1−β) ∨ |A(vβ)|) +as v → ∞. Hence, there exists a regularly varying function R of index ϱ = (β − 1)(γ − +β/2)∨ρβ complying with the property in the statement. The proof is now complete. +Lemma 5.4. If γ > 0 and ρ < 0, there exists x3 ∈ (0, ∞) and δ > 0 such that, for all +x ≥ x3, +f(x) = hγ(x) +� +1 + O({1 − Hγ(x)}δ) +� +. +13 + +Proof. Let R∗(t) := R(1/(1 − F(t))), where R is as in Lemma 5.3. Then R∗(t) is regu- +larly varying of index ϱ/γ (Resnick, 2007, Proposition 0.8(iv)). In turn, by Karamata’s +theorem (e.g, Resnick, 2007, Proposition 0.6(a)) we have that for a large t∗ +� ∞ +t∗ +|η(t)| +1 + γtdt < ∞ +and thus, by Proposition 2.1.4 in Falk et al. (2010), we conclude that +τ := lim +t→∞ +1 − F(t) +1 − Hγ(t) ∈ (0, ∞). +(5.4) +As a consequence, for any δ ∈ (0, −ϱ), as t → ∞ +R∗(t) ∼ R +� +1 +τ(1 − Hγ(t)) +� += O({1 − Hγ(t)}δ). +The conclusion now follows by Proposition 2.1.5 in Falk et al. (2010). +Lemma 5.5. If γ < 0 and ρ < 0, there exists a a regularly varying function ˜R with +negative index ˜ϱ = (−1) ∨ (−ρ/γ) such that, defining the function +˜η(y) := (1 − γy)f(x∗ − 1/y) +[1 − F(x∗ − 1/y)]y2 − 1, +as y → ∞, ˜η(y) = O( ˜R(y)). +Proof. By definition, +˜η (y) = +f(x∗ − 1/y) +[1 − F(x∗ − 1/y)]y2 − γ +� +f(x∗ − 1/y) +y(1 − F(x∗ − 1/y)) + 1 +γ +� +=: ˜η1 (y) + ˜η2 (y) . +On one hand, we have that, as y → ∞ +˜η1 (y) = O(1/y). +On the other hand, for v > 1 we have the identity +˜η2 +� +1 +x∗ − U(v) +� += +� ∞ +1 +γzγ−1 +� +1 − exp +�� z +1 +A(uv) +u +du +�� +dz. +Hence, if A is ultimately positive, +˜η2 +� +1 +x∗ − U(v) +� +≤ −γ +� ∞ +1 +zγ−1(zA(v) − 1)dz += O(A(v)) +while, if A is ultimately negative, +����˜η2 +� +1 +x∗ − U(v) +����� ≤ γA(v) +� ∞ +1 +zγ−1 ln zdz += O(|A(v)|). +As a result of the two above inequalities, as v → ∞ +˜η2(t) = O +�����A +� +1 +1 − F(x∗ − 1/y) +����� +� +, +Therefore, by regular variation of 1/(1 − F(x∗ − 1/y)) with index −1/γ, ˜η2(y) is even- +tually dominated by a regularly varing function of index −ρ/γ. The final result now +follows. +14 + +Lemma 5.6. If γ < 0 and ρ < 0, there exist ˜δ > 0 such that, as y → ∞, +f(x∗ − 1/y) +y2 += (1 − γy)1/γ−1 � +1 + O({1 − H−γ(y)} +˜δ) +� +Proof. The function ˜f(y) := f(x∗ − 1/y)y−2 is the density of the distribution function +˜F(y) := F(x∗−1/y), which is in the domain of attraction of G˜γ, with ˜γ = −γ. Moreover, +˜η(y) = (1 + ˜γy) ˜f(y) +1 − ˜F(y) +− 1. +By Lemma 5.5 and regular variation of 1 − H˜γ with index −1/˜γ, we have +˜η(y) = O({1 − H˜γ(y)} +˜δ) +for any ˜δ > 0 such that −˜δ/˜γ > ˜ϱ. Therefore, by Proposition 2.1.5 in Falk et al. (2010), +as y → ∞ it holds that +˜f(y) = h˜γ(y)[1 + O({1 − H˜γ(y)} +˜δ)], +which is the result. +Lemma 5.7. Let ν′ +x(γ) = (∂/∂γ)νx(γ). +(a) If ξ : R �→ (0, ∞), then it holds that +�� ∞ +0 +[ν′x(ξ(x))]2 dx ≤ 1 +2 +�� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) +�x ln(1 + xξ(x)) +ξ(x) +�2 +dx ++ 1 +2 +�� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx. +(b) If instead ξ : R �→ (γ∗, 0), then +�� −1/γ∗ +0 +[ν′x(ξ(x))]2 dx ≤ 1 +2 +�� −1/γ∗ +0 +(1 + xξ(x))−3− +1 +ξ(x) x4dx ++ 1 +2 +�� −1/γ∗ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx. +Proof. Let ϕx(γ) := (∂/∂γ) ln(1 − Hγ(x)). Then, for any x > 0, if ξ(·) > 0, or x ∈ +(0, −1/γ∗), if ξ(·) ∈ (γ∗, 0) we have +ν′ +x(ξ(x)) = 1 +2(1 + xξ(x))− 1 +2 − +1 +2ξ(x) ϕx(ξ(x)) − 1 +2(1 + xξ(x))− 3 +2 − +1 +2ξ(x) x. +If ξ(·) > 0, by Minkowski inequality +�� ∞ +0 +[ν′x(ξ(x))]2 dx ≤ 1 +2 +�� ∞ +0 +(1 + xξ(x))−1− +1 +ξ(x) [ϕx(ξ(x))]2 dx ++ 1 +2 +�� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx. +15 + +The result at point (a) now follows from the above inequality and the fact that, by +equations (B.5)-(B.6) in B¨ucher and Segers (2017), +0 ≤ ϕx(ξ(x)) ≤ x ln(1 + xξ(x)) +ξ(x)(1 + xξ(x)). +If ξ(·) ∈ (γ∗, 0), inequality (B.8) in B¨ucher and Segers (2017) implies that for any +x ∈ (0, −1/γ∗) +0 ≤ ϕx(ξ(x)) ≤ +x2 +1 + xξ(x). +This inequality and an argument by Minkowsi inequality, analogous to the previous one, +now lead to the result at point (b). +Lemma 5.8. Set ψ′ +γ,x(σ) = (∂/∂σ)ψγ,x(σ). +(a) If ς : R �→ (1 ± ϵ), with ϵ ∈ (0, 1), and if γ > 0 +�� ∞ +0 +� +ψ′γ,x(ς(x)) +�2 dx ≤ +�1 +γ + +� +1 +2γ + 1 +� �1 + ϵ +1 − ϵ +�5/2 +. +(b) If γ ∈ (−1/2, 0) and ς(x) ∈ (σ∗, 1), with σ∗ ∈ (0, 1), there is a constant ζ > 0 such +that +�� − σ∗ +γ +0 +� +ψ′γ,x(ς(x)) +�2 dx ≤ +1 +σ∗ +√−γζ +� 1 +γ2 + 1 +� +. +If instead ς(x) > 1, +�� − 1 +γ +0 +� +ψ′γ,x(ς(x)) +�2 dx ≤ +1 +√−γζ +� 1 +γ2 + 1 +� +. +Proof. Note that for x such that 1 + γx/σ > 0 +ψ′ +γ,x(σ) = (1 + γx/σ)− 1 +2γ − 3 +2 +σ5/2 +x +γ + (1 + γx/σ)− 1 +2γ − 3 +2 +σ3/2 +. +Consequently, if ς : R �→ (1 ± ϵ) and γ > 0, by Minkowski inequality +�� ∞ +0 +� +ψ′γ,x(ς(x)) +�2 dx +≤ +�� ∞ +0 +(1 + γx/ς(x))− 1 +γ −3 +ς5(x) +�x +γ +�2 +dx + +�� ∞ +0 +(1 + γx/ς(x))− 1 +γ −3 +ς3(x) +dx +≤ (1 + ϵ) +3 +2 +(1 − ϵ) +5 +2 +1 +γ + (1 + ϵ) +1 +2 +(1 − ϵ) +3 +2 +� +1 +2γ + 1 +� 1 +2 +and the result at point (a) follows. +16 + +If instead, ς(·) ∈ (σ∗, 1), for some σ∗ ∈ (0, 1), and γ ∈ (−1/2, 0), there is a constant +ζ > 0 such that +�� − σ∗ +γ +0 +� +ψ′γ,x(ς(x)) +�2 dx +≤ +�� − σ∗ +γ +0 +(1 + γx/ς(x))− 1 +γ −3 +ς5(x) +�x +γ +�2 +dx + +�� − σ∗ +γ +0 +(1 + γx/ς(x))− 1 +γ −3 +ς3(x) +dx +≤ +�� − σ∗ +γ +0 +(1 + γx/σ∗)−1+ζ +σ3∗ +1 +γ4 dx + +�� − σ∗ +γ +0 +(1 + γx/σ∗)−1+ζ +σ3∗ +dx += +� +1 +ζσ2∗(−γ)5 + +� +1 +ζσ2∗(−γ). +The first half of the statement at point (b) is now established. The second half of the +statement can be proved analogously. +5.3 +Proof of Theorem 3.2 +For every xt > 0, it holds that +H 2(lt; hγ) = +� xt +0 ++ +� ∞ +xt +�� +ft(x) − +� +hγ(x/s(t))/s(t) +�2 +dx +≤ +� φt(xt) +0 +e−y +� +1 − +� +eqt(y)(1 + q′ +t(y)) +�2 +dy ++ +�� +1 − Ft(xt) + +� +1 − Hγ(xt/s(t)) +�2 +=: I1(t) + I2(t). +Let xt be such that the following equality holds +φt(xt) = −α ln |A(eT −1(t))|, +for a positive constant α to be specified later. Then, by Lemmas 5.1-5.2, for a suitably +small ε > 0 there exist constants κ3, κ4 > 0 such that for all sufficiently large t +I1(t) ≤ +� +� +� +� −α ln |A(eT −1(t))| +0 +κ3|A(eT −1(t))|2e(4ε−1)ydy, +γ ≥ 0 +� −α ln |A(eT −1(t))| +0 +κ4|A(eT −1(t))|2e(γ−ε−1)ydy, +γ < 0 +≤ +� +� +� +κ3|A(eT −1(t))|2 � +1 − |A(eT −1(t))|α1 +� +, +γ ≥ 0 +κ4|A(eT −1(t))|2 � +1 − |A(eT −1(t))|α3 +� +, +γ < 0 +, +where α1 := α(1 − 4ε) and α3 := α(1 − 2(ε − γ)) are positive. Moreover, on one hand +we have the identity +1 − Ft(xt) = |A(eT −1(t))|α. +On the other hand, for some constants κ5, κ6 > 0 we have the inequality +1 − Hγ(xt/s(t)) = |A(eT −1(t))|α exp +� +−qt +� +−α ln |A(eT −1(t))| +�� +≤ +� +� +� +|A(eT −1(t))|α exp +� +κ5|A(eT −1(t))|1−2εα� +, +γ ≥ 0 +|A(eT −1(t))|α exp +� +κ6|A(eT −1(t))|1−(ε−γ)α� +, +γ < 0 +. +17 + +Consequently, +I2(t) ≤ +� +� +� +|A(eT −1(t))|α � +1 + exp +� +κ5 +2 |A(eT −1(t))|1−2εα�� +, +γ ≥ 0 +|A(eT −1(t))|α � +1 + exp +� +κ6 +2 |A(eT −1(t))|1−(ε−γ)α�� +, +γ < 0 +. +Now, if γ ≥ 0, we can choose α > 2 and ε small enough, so that +|A(eT −1(t))|α < |A(eT −1(t))|2 +and α2 := 1 − 2εα > 0. If instead γ ∈ (−1/2, 0), we can choose α slightly larger than +2 and ε small enough, so that the inequality in the above display is still satisfied and +α4 := 1−α(ε−γ) > 0. The conclusion then follows noting that T −1(t) = − ln(1−F(t)) +and, in turn, +|A(eT −1(t))| = |A(v)|. +5.4 +Proof of Lemma 3.4 +We analyse the cases where γ > 0 and γ < 0 separately. +Case 1: γ > 0. In this case, ˜s(t) = s(t) = (1 − F(t))/f(t) and ˜lt = lt. By Lemma +5.4, there are positive constants κ, δ and ϵ such that, for all large t and all x > 0 +lt(x) +hγ(x) ≤ hγ(s(t)x + t) +hγ(x) +s(t) +1 − F(t) +� +1 + κ {1 − Hγ(s(t)x + t)}δ� +≤ +� +1 + γx +(1 + γt)/s(t) + γx +�1+1/γ +1 + ϵ +(s(t))1/γ(1 − F(t)). +Moreover, by Lemma 5.3 it holds that as t → ∞ +1 + γt +s(t) += 1 + η(t) = 1 + o(1) +and, in turn, (s(t))1/γ ∼ (1+γt)1/γ. These two facts, combined with the tail equivalence +relation in formula (5.4), imply that for all sufficiently large t and all x > 0 +lt(x) +hγ(x) ≤ +� +1 + γx +1 − ϵ + γx +�1+1/γ +1 + ϵ +(1 − ϵ)τ +≤ +� +1 +1 − ϵ +�1+1/γ +1 + ϵ +(1 − ϵ)τ . +The result now follows. +Case 2: γ < 0. In this case, for any x ∈ (0, (x∗ − t)/˜s(t)) +˜lt(x) = f +� +x∗ − 1 +y +� 1 +y2 +y2˜s(t) +1 − F(t) +where +y ≡ y(x, t) := +1 +˜s(t) +�x∗ − t +˜s(t) +− x +�−1 +Note that y is bounded from below by 1/(x∗ − t), which converges to ∞ as t → x∗. +Thus, by Lemma 5.6 there are positive constants ˜δ, ϵ and ˜κ such that +˜lt(x) ≤ (1 − γy)1/γ−1[1 + ˜κ{1 − H−γ(y)} +˜δ] y2˜s(t) +1 − F(t) +≤ +� +1 + γ ˜s(t) +x∗ − t +� +−1 +γ +� +x +�− 1 +γ −1 � +(x∗ − t) +� +1 − ˜s(t)x +x∗ − t +� +− γ +� 1 +γ −1 ˜s(t)(1 + ϵ) +1 − F(t) (x∗ − t)− 1 +γ −1. +18 + +By hypothesis, it holds that +x∗ − t +˜s(t) +≤ −1 +γ , +thus +� +1 + γ ˜s(t) +x∗ − t +� +−1 +γ +� +x +�− 1 +γ −1 +≤ (1 + γx)−1/γ−1. +Moreover, it holds that +1 − ˜s(t)x +x∗ − t > 0, +thus +� +(x∗ − t) +� +1 − ˜s(t)x +x∗ − t +� +− γ +� 1 +γ −1 +≤ (−γ) +1 +γ −1. +Finally, for all large t, +˜s(t) +x∗ − t ≤ −(1 + ϵ)γ. +Combining all the above inequalities we can now conclude that, for all large t and for +any x ∈ (0, (x∗ − t)/˜s(t)), +˜lt(x) +hγ(x) ≤ (1 + ϵ)2(−γ) +1 +γ (x∗ − t)− 1 +γ +1 − F(t) . +Now, setting t = U(v), we have that v → ∞ if and only if t → x∗ and, by Theorem +2.3.6 in de Haan and Ferreira (2006), there is a constant ϖ > 0 such that for all large t +(x∗ − t)− 1 +γ +1 − F(t) +≤ v[(1 + ϵ)ϖvγ]− 1 +γ = [(1 + ϵ)ϖ]− 1 +γ +The result now follows. +5.5 +Proof of Lemma 3.5 +Note that for any γ′ > −1/2 and σ > 0 +H (hγ; hγ′(σ · )σ) ≤ +�� +R +[νx(γ) − νx(γ′)]2 dx + +�� +R +[ψγ,x(σ) − ψγ,x(1)]2 dx +In what follows, we bound the two terms on the right-hand side for γ′ ∈ (γ ± ϵ) and +σ ∈ (1 ± ϵ), for a suitably small ϵ > 0. We study the the cases where γ > 0, γ < 0 and +γ = 0 separately. +Case 1: γ > 0. An application of the mean-value theorem and Lemma 5.7(a) yields +that, for a function ξ(x) ∈ (γ ∧ γ′, γ ∨ γ′), +�� +R +[νx(γ) − νx(γ′)]2 dx = |γ − γ′| +�� ∞ +0 +[ν′x(ξ(x))]2 dx +≤ |γ − γ′| +2 +�� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) +�x ln(1 + xξ(x)) +ξ(x) +�2 +dx ++ |γ − γ′| +2 +�� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx. +19 + +On one hand, it holds that +� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) +�x ln(1 + xξ(x)) +ξ(x) +�2 +dx +≤ 4 +� ∞ +0 +(1 + x(γ − ϵ))−1− +1 +γ+ϵ +�ln(1 + x(γ − ϵ)) +(γ − ϵ)2 +�2 +dx +≤ 8(γ + ϵ)3 +(γ − ϵ)5 . +On the other hand, it holds that +� ∞ +0 +(1 + ξ(x))−3− +1 +ξ(x) x2dx +≤ +� ∞ +0 +(1 + x(γ − ϵ))−1− +1 +γ+ϵ +1 +(γ − ϵ)2 dx +≤ (γ + ϵ) +(γ − ϵ)3 . +While, an application of the mean-value theorem and Lemma 5.8(a) yields that, for a +function ς(x) ∈ (1 ∧ σ, 1 ∨ σ), +� +R +� +ψγ′,x(σ) − ψγ,x(1) +�2 dx = +� ∞ +0 +� +ψ′ +γ,x(ς(x)) +�2 dx +≤ +� 1 +γ2 + +� +1 +2γ + 1 +�2 �1 + ϵ +1 − ϵ +�5 +. +The result now follows. +Case 2: γ < 0. Assume that γ < γ′, then an application of the mean-value theorem +and Lemma 5.7(b) yields that, for a function ξ(x) ∈ (γ, γ′), +� +R +� +νx(γ) − νx(γ′) +�2 dx = |γ − γ′|2 +� −1/γ +0 +� +ν′ +x(ξ(x)) +�2 dx + 1 − Hγ′(−1/γ) +≤ |γ − γ′|2 +4 +� +� +�� −1/γ +0 +(1 + xξ(x))−3− +1 +ξ(x) x4dx ++ +�� −1/γ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx +� +� +2 ++ 1 − Hγ′(−1/γ). +First, for a constant β satisfying 0 < β < 1/(ϵ − γ) − 2, we have that +� −1/γ +0 +(1 + xξ(x))−3− +1 +ξ(x) x4dx ≤ 1 +γ4 +� −1/γ +0 +(1 + γx)−1+βdx +≤ +1 +(−γ)5 +1 +β . +Similarly, +� −1/γ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx ≤ 1 +γ2 +� −1/γ +0 +(1 + γx)−1+βdx +≤ +1 +(−γ)3 +1 +β . +20 + +Finally, if ϵ is small enough, 1 − Hγ′(−1/γ) ≤ (1 − γ′/γ)2. Thus, we can conclude that +� +R +� +νx(γ) − νx(γ′) +�2 dx ≤ |γ − γ′|2 1 + 1/2β +(−γ)5 . +A similar reasoning when γ > γ′ yields that +� +R +� +νx(γ) − νx(γ′) +�2 dx ≤ |γ − γ′|2 1 + 2/β +(−γ′)5 +≤ |γ − γ′|2 1 + 2/β +(−γ − ϵ)5 . +Next, assuming that σ < 1, an application of the mean-value theorem and the first +half of Lemma 5.8(b) yields that for a function ς(x) ∈ (1 − ϵ, 1) and a constant ζ > 0 +� +R +[ψγ,x(σ) − ψγ,x(1)]2 dx = (1 − σ)2 +� −σ/γ +0 +� +ψ′ +γ,x(ς(x)) +�2 dx + (1 − σ)−1/γ +≤ (1 − σ)2 +� +1 +ζ(1 − ϵ)2(−γ) +� 1 +γ2 + 1 +�2 ++ 1 +� +. +While, if σ > 1, for a function ς(x) ∈ (1, 1 + ϵ) +� +R +[ψγ,x(σ) − ψγ,x(1)]2 dx = (1 − σ)2 +� −1/γ +0 +� +ψ′ +γ,x(ς(x)) +�2 dx + (1 − 1/σ)−1/γ +≤ (1 − σ)2 +� +1 +ζ(−γ) +� 1 +γ2 + 1 +�2 ++ 1 +� +. +The result now follows. +Case 3: γ = 0. Assume that γ′ > 0, then an application of the mean-value theorem +and Lemma 5.7(a) yields that, for a function ξ(x) ∈ (0, γ′), +� +R +� +νx(γ) − νx(γ′) +�2 dx = |γ − γ′|2 +� ∞ +0 +� +ν′ +x(ξ(x)) +�2 dx +≤ |γ − γ′|2 +4 +� +� +�� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) +�x ln(1 + xξ(x)) +ξ(x) +�2 +dx ++ +�� −∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx +� +� +2 +. +On one hand, we have +� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) +�x ln(1 + xξ(x)) +ξ(x) +�2 +dx ≤ +� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) x4dx +≤ +� ∞ +0 +(1 + xγ′)−3− 1 +γ′ x4dx + +� ∞ +0 +e−xx4dx +≤ 36 + Γ(5). +On the other hand, we have +� ∞ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx ≤ +� ∞ +0 +(1 + xγ′)−3− 1 +γ′ x2dx + +� ∞ +0 +e−xx2dx +≤ 3 + Γ(3). +21 + +Assume next that γ′ < 0, then an application of the mean-value theorem and Lemma +5.7(b) yields that, for a function ξ(x) ∈ (−ϵ, 0), +� +R +� +νx(γ) − νx(γ′) +�2 dx = |γ − γ′|2 +� −1/γ′ +0 +� +ν′ +x(ξ(x)) +�2 dx + e1/γ′ +≤ |γ − γ′|2 +4 +� +� +�� −1/γ′ +0 +(1 + xξ(x))−3− +1 +ξ(x) x4dx ++ +�� −1/γ′ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx +� +� +2 ++ e1/γ′. +On one hand, for ϵ sufficiently small we have +� −1/γ′ +0 +(1 + xξ(x))−3− +1 +ξ(x) x4dx ≤ +� −1/γ′ +0 +(1 + xγ′)−3− 1 +γ′ x4dx + +� −1/γ′ +0 +e−xx4dx +≤ 13 +2 Γ(5) +and +� −1/γ′ +0 +(1 + xξ(x))−3− +1 +ξ(x) x2dx ≤ +� −1/γ′ +0 +(1 + xγ′)−3− 1 +γ′ x4dx + +� −1/γ′ +0 +e−xx2dx +≤ 3 +2Γ(3). +On the other hand, for ϵ sufficiently small we have e1/γ′ ≤ |γ′ − γ|2. +Finally, some algebraic manipulations yield +� +R +[ψγ,x(σ) − ψγ,x(1)]2 dx = +� ∞ +0 +�� +e−x/σ 1 +σ − +√ +e−x +�2 +dx +≤ (1 − σ)2 +(1 − ϵ)2 +� +1 + 1 +2 +�1 + ϵ +1 − ϵ +�3/2�2 +. +The proof is now complete. +5.6 +Proof of Corollary 3.6 +By Lemma 8.2 in Ghosal et al. (2000) +K (˜lt; hγ) ≤ 2 +� +� +sup +0 0, we have that for any measurable set sequence Ek +Pk(Ek) = Pk(Ek|X(n−k) ∈ Bk)(1 + o(1)) + o(1) +and +Qk(Ek) = Pk(Ek|X(n−k) ∈ Bk)(1 + o(1)) + o(1) +as n → ∞. Therefore, it suffices to prove that +Pk( · |X(n−k) ∈ Bk) ▷ Qk( · |X(n−k) ∈ Bk). +To do it, we denote by πk and χk the (Lebesgue) densities pertaining to the two condi- +tional probability measures in the formula above and prove that +lim sup +n→∞ K (χk; πk) < ∞. +(5.5) +23 + +Clearly, it holds that for almost every (y, t) ∈ Rk+2 +χk(y, t) = fYk/˜s(X(n−k))(y|X(n−k) = t) +fX(n−k)(t)1(t ∈ Bk) +P(X(n−k) ∈ Bk) +, +where fYk/˜s(X(n−k))(y|X(n−k) = t) and fX(n−k)(t) are the conditional density of Yk/˜s(X(n−k)) +given X(n−k) = t and the marginal density of X(n−k), respectively. Moreover, +πk(y, t) = hZk(y) +fX(n−k)(t)1(t ∈ Bk) +P(X(n−k) ∈ Bk) +, +where hZk(y) is the density of Zk. As a consequence, +K (χk; πk) = +� +Bk +K (fYk/˜s(X(n−k))( · |X(n−k) = t); hZk) +fX(n−k)(t) +P(X(n−k) ∈ Bk)dt. +By Lemma B.11 in Ghosal and van der Vaart (2017) and Lemma 3.4.1 in de Haan and +Ferreira (2006) +K (fYk/˜s(X(n−k))( · |X(n−k) = t); hZk) ≤ kK (˜lt; hγ). +Moreover, by Corollary 3.6, there is a constant Λ > 0 such that for all large n +sup +t∈Bk +K (˜lt; hγ) ≤ Λ +���A +� +(1 − ϵ)n +k +���� +2 +≤ Λ(1 − ϵ)ρ(1 + ϵ) +���A +�n +k +���� +2 +. +Combining the above inequalities we obtain that +K (χk; πk) ≤ Λ(1 − ϵ)ρ(1 + ϵ)k +���A +�n +k +���� +2 +→ Λ(1 − ϵ)ρ(1 + ϵ)λ2 +as n → ∞, where the convergence result in the second line follows from assumption (b). +The result in formula (5.5) is now established and the proof is complete. +Acknowledgements +Simone Padoan is supported by the Bocconi Institute for Data Science and Analytics +(BIDSA), Italy. +References +Balkema, A. A. and L. de Haan (1974). Residual life time at great age. The Annals of +probability 2, 792–804. +Bobbia, B., C. 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Journal of Multivariate Analysis 100, 794–815. +25 + diff --git a/-NA0T4oBgHgl3EQfPP8X/content/tmp_files/load_file.txt b/-NA0T4oBgHgl3EQfPP8X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..59b843987f975b07cc86a8135476e6cc57f328e4 --- /dev/null +++ b/-NA0T4oBgHgl3EQfPP8X/content/tmp_files/load_file.txt @@ -0,0 +1,796 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf,len=795 +page_content='Strong Convergence of Peaks Over a Threshold S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Padoan Department of Decision Sciences, Bocconi University, Italy and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Rizzelli Department of Statistical Sciences, Catholic University, Italy January 6, 2023 Abstract Extreme Value Theory plays an important role to provide approximation re- sults for the extremes of a sequence of independent random variable when their distribution is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' An important one is given by the Generalised Pareto dis- tribution Hγ(x) as an approximation of the distribution Ft(s(t)x) of the excesses over a threshold t, where s(t) is a suitable norming function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In this paper we study the rate of convergence of Ft(s(t)·) to Hγ in variational and Hellinger dis- tances and translate it into that regarding the Kullback-Leibler divergence between the respective densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' We discuss the utility of these results in the statistical field by showing that the derivation of consistency and rate of convergence of estimators of the tail index or tail probabilities can be obtained thorough an alternative and relatively simplified approach, if compared to usual asymptotic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Keywords: Contraction Rate, Consistency, Exceedances, Extreme Quantile, Gener- alised Pareto, Tail Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: Primary 60G70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' secondary 62F12, 62G20 1 Introduction Extreme Value Theory (EVT) develops probabilistic models and methods for describ- ing the random behaviour of extreme observations that rarely occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' These theoretical foundations are very important for studying practical problems in environmental, cli- mate, insurance and financial fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', Embrechts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Dey and Yan, 2016), to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In the univariate setting, the most popular approaches for statistical analysis are the so-called Block Maxima (BM) and Peaks Over Threshold (POT) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' B¨ucher and Zhou, 2021, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , Xn be independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=') random variables according to a common distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The first approach concerns the modelling of k sample maxima derived over blocks of a certain size m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Mm,i = max(X(i−1)m+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , Xim), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In this case, under some regularity conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' de Haan and Ferreira, 2006, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 1), the weak limit theory establishes that F m(amx+bm) converges pointwise to Gγ(x) as m → ∞, for every continuity point x of Gγ, where Gγ is the Generalised Extreme Value (GEV) distribution, am > 0 and bm are suitable norming constants for each m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' and γ ∈ R is the so-called tail index, which describes the tail heaviness of F (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' de Haan and Ferreira, 2006, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The second method concerns the modelling of k random variables out of the n available that exceed a high threshold t, or, equivalently, of k threshold excesses Yj, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='02171v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='PR] 5 Jan 2023 j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , k, which are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' copies of Y = X −t|X > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In this context, the Generalised Pareto (GP) distribution, say Hγ, appears as weak limit law of appropriately normalised high threshold exceedances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' for all x > 0, Ft(s(t)x) converges pointwise to Hγ(x) as t → x∗, for all the continuity points x of Hγ(x), where Ft(x) = P(Y ≤ x) and s(t) > 0 is a suitable scaling function for any t ≤ x∗, with x∗ = inf(x : F(x) < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' This result motivates the POT approach, which was introduced decades ago by the seminal paper Balkema and de Haan (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Since then, few other convergence results emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' For instance, the uniform convergence of Ft(s(t) · ) to Hγ and the coresponding convergence rate have been derived by Pickands III (1975) and Raoult and Worms (2003), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Similar results but in Wasserstein distance have been recently established by Bobbia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' As for the GEV distribution, more results are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In particular, there are sufficient conditions to ensure, in addition to weak convergence, that F m(am · +bm) converges to Gγ for example uniformly and in variational distance and the density of F m(am · +bm) converges pointwise, locally uniformly and uniformly to that of Gγ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Falk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', 2010, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Resnick, 2007, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The main contribution of this article is to provide new convergence results that can be useful in practical problems for the POT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Motivated by the utility in the statistical field to asses the asymptotic accuracy of estimation procedures, we study stronger forms of convergence than the pointwise one, as limt→x∗ D(Ft(s(t) · ), Hγ) = 0, where D( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' · ) is either the variational distance, the Hellinger distance or the Kullback- Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In particular, we provide upper bounds for the rate of convergence to zero of D(Ft(s(t) · );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Hγ) in the case that D( · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' · ) is the variational and Hellinger dis- tance, and further translate them into bounds on Kullback-Leibler divergence between the densities of Ft(s(t)·) and Hγ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Estimators of the tail index γ (and other related quantities) are typically defined as functionals of the random variables (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , Yk), as for instance the popular Hill (Hill, 1975), Moment (Dekkers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', 1989), Pickands (Pickands III, 1975), Maximum Likelihood (ML, Jenkinson, 1969), Generalised Probability Weighted Moment (GPWM, Hosking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', 1985) estimators, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In real applications, the distribution F is typically unknown and so is F(s(t) · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Although, for large t, Hγ provides a model approximation for Ft(s(t) · ), when one wants to derive asymptotic properties as the consistency and especially the rate of convergence of the tail index estimators (or other related quantities), still the fact that (after rescaling) the random variables (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , Yk) are actually distributed according to Ft(s(t) · ) needs to be taken into account, which makes asymptotic derivations quite burdensome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' These are even more complicated if t is determined on the basis of the (k + 1)-th largest order statistic of the original sample X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , Xn, which is the most common situation in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In this case, the threshold is in fact random and, up to rescaling, Ft(s(t) · ) only gives a conditional model for the variables Yj given a fixed value t of the chosen statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Asymptotic properties for POT methods have been studied in the last fifty years, see for example Hall and Welsh (1984), Drees (1998), Dekkers and de Haan (1993) and the reference therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Leveraging on our strong convergence results we can show that, for random sequences (such as sequences of estimators) convergence results in probability that hold under the limit model Hγ, are also valid for a rescaled sample of excesses over a large order statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Precisely, we show that the distribution of the latter, up to rescaling and reordering, is contiguous to that of an ordered i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' sample from Hγ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', van der Vaart, 2000, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' As a by product of this result, one can derive the consistency and rate of convergence of a tail index estimator (or an estimator of a related quantity) by defining it as a functional of the random sequence (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' , Zk) which is distributed according the 2 limit model Hγ, and, if density of Ft(s(t)·) satisfies some regularity conditions, then the same asymptotic results hold even when such estimator is defined through the sequence of excesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' This approach simplifies a lot the computations as asymptotic properties are easily derivable under the limit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The article is organised as follows, Section 2 of the paper provides a brief summary of the probabilistic context on which our results are based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Section 3 provides our new results on strong convergence to a Pareto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Section 4 explains in what applications concerning statistical estimation our results are useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Section 5 provides the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 2 Background Let X be a random variable with a distribution function F that is in the domain of attraction of the GEV distribution Gγ, shortly denoted as F ∈ D(Gγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' This means that there are norming constants am > 0 and bm ∈ R for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' such that lim m→∞ F m(amx + bm) = exp � − (1 + γx)−1/γ� =: Gγ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1) for all x ∈ R such that 1 + γx > 0, where γ ∈ R, and this is true if only if there is a scaling function s(t) > 0 with t < x∗ such that lim t→x∗ Ft(s(t)x) = 1 − (1 + γx)−1/γ =: Hγ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2) e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', de Haan and Ferreira (2006, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The densities of Hγ and Gγ are hγ(x) = (1 + γx)−(1/γ+1) and gγ(x) = Gγ(x)hγ(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Let U(v) := F ←(1 − 1/v), for v ≥ 1, where F ← is the left-continuous inverse function of F and G←(exp(−1/x)) = (xγ − 1)/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Then, we recall that the first-order condition in formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1) is equivalent to the limit result lim v→∞ U(vx) − U(v) a(v) = xγ − 1 γ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3) for all x > 0, where a(v) > 0 is a suitable scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In particular, we have that s(t) = a(1/(1 − F(t))), see de Haan and Ferreira (2006, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 1) for possible selections of the function a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' A stronger convergence form than that in formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2) is the uniform one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' sup x∈[0, x∗−t s(t) ) |Ft(s(t)x) − Hγ(x)| → 0, t → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' To establish the speed at which Ft(s(t)x) converges uniformly to Hγ(x), Raoult and Worms (2003) relied on a specific formulation of the well-known second-order condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In its general form, the second order condition requires the existence of a posi- tive function a and a positive or negative function A, named rate function, such that limv→∞ |A(v)| = 0 and lim v→∞ U(vx)−U(v) a(v) − xγ−1 γ A(v) = D(x), x > 0, 3 where D is a non-null function which is not a multiple of (xγ − 1)/γ, see de Haan and Ferreira (2006, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The rate function A is necessarily regularly varying at infinity with index ρ ≤ 0, named second-order parameter (de Haan and Ferreira, 2006, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In the sequel, we use the same specific form of second order condition of Raoult and Worms (2003) to obtain decay rates for stronger metrics than uniform distance between distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 3 Strong results for POT In this section, we discuss strong forms of convergence for the distribution of rescaled exceedances over a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' First, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1, we discuss convergence to a GP distribution in variational and Hellinger distance, drawing a connection with known results for density convergence of normalized maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2 we quantify the speed of convergence in variational and Hellinger distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Finally, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3, we show how these can be used to also bound Kullback-Leibler divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Throughout, for a twice differentiable function W(x) on R, we denote with W ′(x) = (∂/∂x)W(x) and W ′′(x) = (∂2/∂x2)W(x) the first and second order derivatives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1 Strong convergence under classical assumptions Let the distribution function F be twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In the sequel, we denote f = F ′, gm = (F m(am · +bm))′ and ft = F ′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Under the following classical von Mises-type conditions lim x→∞ xf(x) 1 − F(x) = 1 γ , γ > 0, lim x→x∗ (x∗ − x)f(x) 1 − F(x) = −1 γ , γ < 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1) lim x→x∗ f(x) � x∗ x (1 − F(v)dv) (1 − F(x))2 = 0, γ = 0, we know that the first-order condition in formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3) is satisfied and it holds that lim v→∞ va(v)f(a(v)x + U(v)) = (1 + γx)−1/γ−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2) locally uniformly for (1 + γx) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Since the equality gm(x) = F m−1(amx + bm)hm(x) holds true, with bm = U(m), am = a(m) and hm(x) = mamf(amx + bm), and since F m−1(amx+bm) converges to Gγ(x) locally uniformly as m → ∞, the convergence result in formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2) thus implies that gm(x) converges to gγ(x) locally uniformly (Resnick, 2007, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' On the other hand, the density pertaining to Ft(s(t)x) is lt(x) := ft(s(t)x)s(t) = s(t)f(s(t)x + t) 1 − F(t) and, setting v = 1/(1 − F(t)), we have a(v) = s(t) and v → ∞ as t → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Therefore, a further implication of the convergence result in formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2) is that lt(x) converges to hγ(x) locally uniformly for x > 0, if γ ≥ 0, or x ∈ (0, −1/γ), if γ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In turn, by Scheffe’s lemma we have lim t→x∗ V (Pt, P) = 0, where V (Pt, P) = sup B∈B |Pt(B) − P(B)| 4 is the total variation distance between the probability measures Pt(B) := P �X − t s(t) ∈ B ����X > t � and P(B) := P(Z ∈ B), and where Z is a random variable with distribution Hγ and B is a set in the Borel σ-field of R, denoted by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Let H 2(lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) := � �� lt(x) − � hγ(x) �2 dx be the square of the Hellinger distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' It is well know that the Hellinger and total variation distances are related as H 2(lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) ≤ 2V (Pt, P) ≤ 2H (lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Ghosal and van der Vaart (2017, Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Therefore, the conditions in formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1) ultimately entail that also the Hellinger distance between the density of rescaled peaks over a threshold lt and the GP density hγ converges to zero as t → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In the next subsection we introduce a stronger assumption, allowing us to also quantify the speed of such convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2 Convergence rates As in Raoult and Worms (2003) we rely on the following assumption, in order to derive the convergence rate for the variational and Hellinger distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Assume that F is twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Moreover, assume that there exists ρ ≤ 0 such that A(v) := vU′′(v) U ′(v) + 1 − γ defines a function of constant sign near infinity, whose absolute value |A(v)| is regularly varying as v → ∞ with index of variation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' When Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1 holds then the classical von-Mises conditions in formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1) are also satisfied for the cases where γ is positive, negative or equal to zero, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Furthermore, Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1 implies that an appropriate scaling function for the exceedances of a high threshold t < x∗, which complies with the equivalent first-order condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2), is defined as s(t) = (1 − F(t))/f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' With such a choice of the scaling function s, we establish the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Assume Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='1 is satisfied with γ > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Then, there exist constants ci > 0 with i = 1, 2, αj > 0 with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=', 4, K > 0 and t0 < x∗ such that H 2(lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) K|A(v)|2 ≤ S(v) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='4) for all t ≥ t0, where v = 1/(1 − F(t)) and S(v) := � 1 − |A(v)|α1 + 4 exp (c1|A(v)|α2) , if γ ≥ 0 1 − |A(v)|α3 + 4 exp (c2|A(v)|α4) , if γ < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 5 Given the relationship between the total variation and Hellinger distances in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3), the following result is a direct consequence of Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2, for all t ≥ t0 V (Pt, P) ≤ |A(v)| � KS(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2 implies that the Hellinger and variational distances of the probability density and measure of rescaled exceedances from their GP distribution counterparts are bounded from above by C|A(v)|, for a positive constant C, as the threshold t approaches the end-point x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Since for a fixed x ∈ ∩t≥t0(0, x∗−t s(t) ) it holds that |Ft(s(t)x) − Hγ(x)| ≤ V (Pt, P) and since Raoult and Worms (2003, Theorem 2(i)) implies that |Ft(s(t)x)−Hγ(x)|/|A(v)| converges to a positive constant, there also exists c > 0 such that, for all large t, c|A(v)| is a lower bound for variational and Hellinger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Therefore, since c|A(v)| ≤ V (Pt, P) ≤ H (lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) ≤ C|A(v)|, the decay rate of variational and Hellinger distances is precisely |A(v)| as t → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Differently from the result on uniform convergence in Raoult and Worms (2003), our results on convergence rates in the stronger total variation and Hellinger topologies are given for γ > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Although the bound in formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='4) remains mathematically valid also for tail indices below −1/2, the restriction γ > −1/2 is imposed to guarantee that constants α3, α4 in the definition of S(v) are positive, so that S(v) is positive and bounded as t approaches x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Note that such a behaviour of S is essential to deduce from the bound in formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='4) that the rate of convergence is |A(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='3 Kullback-Leibler divergences A further implication of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2 concerns the speed of convergence to zero of the Kullback-Leibler divergence K (˜lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) := � ln � ˜lt(x)/hγ(x) � ˜lt(x)dx, and the divergences of higher order p ≥ 2 Dp(˜lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) := � ���ln � ˜lt(x)/hγ(x) ���� p ˜lt(x)dx, where ˜lt = (Ft(˜s(t) · ))′ and ˜s(t) is a scaling function possibly different from s(t), which ensures that the support of the conditional distribution Ft(˜s(t)x) is contained in that of the GP distribution Hγ when γ < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' x ∈ R : x < (x∗ − t)/˜s(t) < −1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' We recall indeed that, when γ is negative, the end-point (x∗ − t)/s(t) of lt converges to −1/γ as t approaches x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Nevertheless, for t < x∗ it can be that (x∗ − t)/s(t) > −1/γ, entailing that K (lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) = Dp(lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' hγ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' The introduction of a more flexible scaling function ˜s is thus meant to rule out this uninteresting situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' In order to exploit Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2 to give bounds on Kullback-Leibler and higher order divergences, we first introduce by the next two lemmas a uniform bound on density ratios and a Lipschitz continuity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NA0T4oBgHgl3EQfPP8X/content/2301.02171v1.pdf'} +page_content='2, if ρ < 0 and γ ̸= 0, and if ˜s(t)/s(t) → 1 as t → x∗, then there exist a t1 < x∗ and a constant M ∈ (0, ∞) such that sup t≥t1 sup 010 layers), the values are on the order of 30 Wm-1K-1 for +the supported in-plane, 60 Wm-1K-1 for the suspended in-plane and 3 Wm-1K-1 for the cross-plane +conductivities. For the substrate interface conductance, we used 1 MWm-2K-1. + + +Figure S10 a. Finite element model for the SThM probe on a MoS2 suspended sample. b. Zoomed-in +view of the model where the temperature gradient is visible on the sample surface. +Using those material parameters, we estimated a ratio between the probe temperature and the tip +apex temperature of 4.9. The model also accounts for the tip-sample thermal resistance. This method + +(a) +(b)and model were experimentally confirmed elsewhere4,5,17. Taking these into consideration, we obtain +a sample temperature rise Δ𝑇𝑠𝑎𝑚𝑝𝑙𝑒 of 7.4 ± 0.7 K. This gives a Seebeck variation of 72±10 µVK-1. + +References +(1) +Ng, H. K.; Chi, D.; Hippalgaonkar, K. Effect of Dimensionality on Thermoelectric Powerfactor of +Molybdenum Disulfide. J Appl Phys 2017, 121 (20), 204303. +https://doi.org/10.1063/1.4984138. +(2) +Wu, J.; Liu, Y.; Liu, Y.; Liu, Y.; Cai, Y.; Zhao, Y.; Ng, H. K.; Watanabe, K.; Taniguchi, T.; Zhang, G.; +Qiu, C. W.; Chi, D.; Neto, A. H. C.; Thong, J. T. L.; Loh, K. P.; Hippalgaonkar, K. Large +Enhancement of Thermoelectric Performance in MoS2/h-BN Heterostructure Due to Vacancy- +Induced Band Hybridization. Proc Natl Acad Sci U S A 2020, 117 (25), 13929–13936. +https://doi.org/10.1073/pnas.2007495117. +(3) +Cui, X.; Lee, G. H.; Kim, Y. D.; Arefe, G.; Huang, P. Y.; Lee, C. H.; Chenet, D. A.; Zhang, X.; Wang, +L.; Ye, F.; Pizzocchero, F.; Jessen, B. S.; Watanabe, K.; Taniguchi, T.; Muller, D. A.; Low, T.; Kim, +P.; Hone, J. Multi-Terminal Transport Measurements of MoS2 Using a van Der Waals +Heterostructure Device Platform. Nature Nanotechnology 2015 10:6 2015, 10 (6), 534–540. +https://doi.org/10.1038/nnano.2015.70. +(4) +Tovee, P.; Pumarol, M.; Zeze, D.; Kjoller, K.; Kolosov, O. Nanoscale Spatial Resolution Probes +for Scanning Thermal Microscopy of Solid State Materials. 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Reduced Thermal Conductivity of +Supported and Encased Monolayer and Bilayer MoS2. 2d Mater 2020, 8 (1), 011001. +https://doi.org/10.1088/2053-1583/ABA4ED. +(12) +Zhang, X.; Sun, D.; Li, Y.; Lee, G. H.; Cui, X.; Chenet, D.; You, Y.; Heinz, T. F.; Hone, J. C. +Measurement of Lateral and Interfacial Thermal Conductivity of Single- and Bilayer MoS2 and +MoSe2 Using Refined Optothermal Raman Technique. ACS Appl Mater Interfaces 2015, 7 (46), +25923–25929. https://doi.org/10.1021/ACSAMI.5B08580/ASSET/IMAGES/LARGE/AM-2015- +085805_0003.JPEG. +(13) +Jo, I.; Pettes, M. T.; Ou, E.; Wu, W.; Shi, L. Basal-Plane Thermal Conductivity of Few-Layer +Molybdenum Disulfide. Appl Phys Lett 2014, 104 (20), 201902. +https://doi.org/10.1063/1.4876965. +(14) +Yuan, P.; Wang, R.; Wang, T.; Wang, X.; Chemistry, Y. X.-P.; 2018, undefined. Nonmonotonic +Thickness-Dependence of in-Plane Thermal Conductivity of Few-Layered MoS 2: 2.4 to 37.8 +Nm. pubs.rsc.org. +(15) +Bae, J. J.; Jeong, H. Y.; Han, G. H.; Kim, J.; Kim, H.; Kim, M. S.; Moon, B. H.; Lim, S. C.; Lee, Y. H. +Thickness-Dependent in-Plane Thermal Conductivity of Suspended MoS2 Grown by Chemical +Vapor Deposition. Nanoscale 2017, 9 (7), 2541–2547. https://doi.org/10.1039/C6NR09484H. +(16) +Meng, X.; Pandey, T.; Jeong, J.; Fu, S.; Yang, J.; Chen, K.; Singh, A.; He, F.; Xu, X.; Zhou, J.; +Hsieh, W. P.; Singh, A. K.; Lin, J. F.; Wang, Y. Thermal Conductivity Enhancement in MoS2 +under Extreme Strain. Phys Rev Lett 2019, 122 (15), 155901. +https://doi.org/10.1103/PHYSREVLETT.122.155901/FIGURES/3/MEDIUM. +(17) +Gehring, P.; Harzheim, A.; Spièce, J.; Sheng, Y.; Rogers, G.; Evangeli, C.; Mishra, A.; Robinson, +B. J.; Porfyrakis, K.; Warner, J. H.; Kolosov, O. v.; Briggs, G. A. D.; Mol, J. A. Field-Effect Control +of Graphene-Fullerene Thermoelectric Nanodevices. Nano Lett 2017, 17 (11), 7055–7061. +https://doi.org/10.1021/ACS.NANOLETT.7B03736/ASSET/IMAGES/NL-2017- +03736Q_M017.GIF. + + diff --git a/09AzT4oBgHgl3EQfDPrL/content/tmp_files/load_file.txt b/09AzT4oBgHgl3EQfDPrL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d643a95c750d88780aee20216cc7c020da78428a --- /dev/null +++ b/09AzT4oBgHgl3EQfDPrL/content/tmp_files/load_file.txt @@ -0,0 +1,2242 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf,len=2241 +page_content='Single-material MoS2 thermoelectric junction enabled by substrate engineering Mohammadali Razeghi1, Jean Spiece3, Oğuzhan Oğuz1, Doruk Pehlivanoğlu2, Yubin Huang3, Ali Sheraz2, Phillip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Dobson4, Jonathan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Weaver4, Pascal Gehring3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Serkan Kasırga1,2* 1 Bilkent University UNAM – Institute of Materials Science and Nanotehcnology, Bilkent 06800 Ankara, Turkey 2 Department of Physics, Bilkent University, Bilkent 06800 Ankara, Turkey 3 IMCN/NAPS, Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium 4 James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='K Corresponding Author: kasirga@unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='tr Abstract To realize a thermoelectric power generator, typically a junction between two materials with different Seebeck coefficient needs to be fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Such difference in Seebeck coefficients can be induced by doping, which renders difficult when working with two-dimensional (2d) materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Here, we employ substrate effects to form a thermoelectric junction in ultra-thin few-layer MoS2 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We investigated the junctions with a combination of scanning photocurrent microscopy and scanning thermal microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This allows us to reveal that thermoelectric junctions form across the substrate- engineered parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We attribute this to a gating effect induced by interfacial charges in combination with alterations in the electron-phonon scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This work demonstrates that substrate engineering is a promising strategy to develop future compact thin-film thermoelectric power generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Main Text In ultra-thin materials with large surface-to-bulk ratio, interactions with the substrate can have strong impact on the materials properties 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' It is therefore important to understand this so-called substrate- effect, especially in order to optimize the reliability of future devices based on two-dimensional (2d) semiconducting materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As an example, the choice of substrate for mono- and few-layer MoS2 has been shown to strongly affect its Raman modes and photoluminescence (PL)7, electronic8, and thermal transport9 properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In this work, we employ the substrate effect to enable completely new functionalities in a 2d semiconductor device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To this end, we engineer the substrate that atomically thin MoS2 is deposited on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Using a combination of scanning photocurrent microscopy (SPCM) along with scanning thermal microscopy (SThM) we demonstrate that substrate engineering is a powerful way to build a thermoelectric junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Schematic of a substrate-engineered device: a MoS2 flake is suspended over a circular hole drilled in the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Metal contacts are used for scanning photocurrent microscopy (SPCM), scanning thermal gate microscopy (SThGM) and I-V measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The inset shows a magnification of the area indicated by the dashed yellow square, where Seebeck coefficients of supported and suspended parts are labelled with 𝑆1 and 𝑆2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical microscope image of a multi- layered device over circular holes with indium contacts, marked with grey overlays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM reflection map and the corresponding open-circuit photocurrent map acquired from the yellow dashed rectangle in b with 532 nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' {𝐼𝑚𝑖𝑛, 𝐼𝑚𝑎𝑥} = {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5} nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Photocurrent map from the red dashed rectangle region in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Black circle is the position of the hole determined from the reflection image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Right panel shows the photocurrent, 𝐼𝑃𝐶 vs bias taken from point 1 (red dots) and point 2 (blue dots) over the suspended part of the crystal marked on the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Lower graph is the derived photoconductance, 𝐺𝑃𝐶 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In the following we predict that a thermoelectric junction with a Seebeck coefficient difference of tens of µV/K can be fabricated when connecting regions of suspended MoS2 to supported regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We assume that the Seebeck coefficient 𝑆 in thermal equilibrium is composed of contributions from the energy-dependent diffusion (𝑆𝑁), scattering (𝑆τ) and the phonon-drag (𝑆𝑝𝑑), so that 𝑆 = 𝑆𝑁 + 𝑆𝜏 + 𝑆𝑝𝑑 9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Here, 𝑆𝑁 and 𝑆τ terms can be written from the Mott relation assuming that MoS2 is in the highly conductive state and electrons are the majority carriers: 𝑆τ = − 𝜋2𝑘𝐵 2𝑇 3𝑒 𝜕ln𝜏 𝜕𝐸 | 𝐸=𝐸𝐹 and 𝑆𝑁 = ± 𝑘𝐵 𝑒 [ 𝐸𝐹−𝐸𝐶 𝑘𝐵𝑇 − (𝑟+2)𝐹𝑟+1(𝜂) (𝑟+1)𝐹𝑟(𝜂) ] where 𝑇 is the temperature, 𝑘𝐵 is the Boltzmann constant, 𝑒 is the electron’s charge, 𝜏 is the relaxation time, 𝐸𝐹 is the Fermi energy, 𝐸𝐶 is the conduction band edge energy, 𝑟 is scattering parameter and 𝐸 is the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝐹𝑚(𝜂) is the m-th order Fermi integral11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In the 2d limit, 𝜏 is energy independent, thus 𝑆𝜏 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝑆𝑝𝑑 term can be estimated from the theory of phonon-drag in semiconductors in the first order as 𝑆𝑝𝑑 = − 𝛽𝑣𝑝𝑙𝑝 𝜇𝑇 where, 𝑣𝑝 and 𝑙𝑝 are the group velocity and the mean free path of a phonon, 𝛽 is a parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and 𝜇 is the electron mobility, respectively10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Importantly, 𝑙𝑝 and 𝜇 are heavily affected by the presence of a substrate12 which implies that the 𝑆𝑝𝑑 term gets strongly modified when the MoS2 flake is suspended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' pended ported Reflection Map 0mV S1 noelectric Photocurrent Map 0mVIndeed, we find that for suspended MoS2 at room temperature 𝑆𝑝𝑑 ≈ −100 µV/K and for MoS2 on SiO2 at room temperature 𝑆𝑝𝑑 ≈ −230 µV/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Similarly, 𝑆𝑁 is heavily influenced by the presence or absence of the substrate as electron density depends on the interfacial Coulomb impurities and short- ranged defects11–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We estimate that that for MoS2, 𝑆𝑁 ranges from -400 µV/K to -200 µV/K for carrier concentrations ranging from 1012 cm-2 (suspended few layer MoS2) to 3 x 1013 cm-2 (SiO2 supported few layer MoS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='18–20 As a result, a substrate engineered thermoelectric junction with a Seebeck coefficient difference of Δ𝑆 ≈ 70 µV/K can be formed along the MoS2 flake (see Figure 1a and Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To test this hypothesis, we fabricated substrate-engineered MoS2 devices by mechanical exfoliation and dry transfer21 of atomically thin MoS2 flakes on substrates (sapphire or oxidized silicon) with pre- patterned trenches/holes formed by focused ion beam (FIB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We contacted the flakes with Indium needles22–24 which are suitable for achieving Ohmic contacts to MoS225,26 (gold-contacted device measurements are shown in Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' A typical device is shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We then used scanning photocurrent microscopy, to locally heat up the junction with a focused laser beam and to measure the photothermoelectric current that is generated (see Methods for experimental details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 1c shows the greyscale reflection intensity map and the corresponding photocurrent distribution over the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For the few-layer suspended MoS2 devices we observe a bipolar photoresponse at the junctions between the supported and the suspended part of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The spatial distribution of the signal agrees well with the finite element analysis simulations, given in the supporting information, and suggests the formation of a thermoelectric junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' When applying a voltage bias 𝑉 to the junction, the photocurrent, 𝐼𝑃𝐶 changes linearly with bias, while the photoconductance, 𝐺𝑃𝐶 = 𝐼𝑃𝐶 𝑉 −𝐼𝑃𝐶 0 𝑉 (𝐼𝑃𝐶 𝑉 , 𝐼𝑃𝐶 0 : photocurrent under 𝑉 and 0 mV bias, respectively) stays constant (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Such bias-independent photoconductance is typically an indication for an photothermoelectric nature of the observed signal22,24,27–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Although we propose that the photocurrent in substrate-engineered MoS2 devices is dominated by the photothermal effect (PTE)30,31, other possible mechanisms have been reported that may lead to a photovoltaic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' These include (1) strain related effects such as strain modulation of materials properties and flexo- photovoltaic effect13, and (2) substrate proximity related effects that forms a built-in electric field32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Next, we present experimental evidence for a thermoelectric origin of the observed photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To this end, we employed scanning thermal gate microscopy (SThGM), where a hot AFM tip heats up the junction locally while the resulting voltage build-up on the devices is recorded (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Since no laser-illumination of the sample is required in this method, it can be used to ultimately exclude photovoltaic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 2 compares SPCM and SThGM maps of the same holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We observed the same bipolar signals in the suspended regions with both experimental methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Thanks to its sub-100 nm lateral resolution, SThGM further allows us to observe local variations of the thermovoltage in supported MoS2 that can be attributed to charge puddles induced by local doping via the substrate33– 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We confirmed that the SThGM signal disappears when no power is dissipated in the probe heater, which rules out parasitic effects induced by the laser used for AFM feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Furthermore, SThGM allows us to estimate the magnitude of the local Seebeck coefficient variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Using the probe- calibration data we obtain a value of Δ𝑆 = 72 ± 10 µV/K (See supporting information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Despite the uncertainties regarding the real sample temperature, the obtained Δ𝑆 value is very close to the theoretically predicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM reflection map and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' photocurrent map of the device shown in the inset of panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The yellow rectangle indicates the region that was investigated by SThGM in c (AFM height map) and d (SThGM thermovoltage map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM map of the same region excerpted from the map given in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Color scale is the same as in panel b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bars in c, d and e: 3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To understand why suspending MoS2 alters its Seebeck coefficient, we first would like to discuss the possibility of strain induced changes in the materials properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' MoS2, like graphene, is nominally compressed when deposited on a substrate36–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Upon suspending the crystals, the free-standing part either adheres to the sidewalls of the hole and dimples or, bulges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As a result, strain might be present in the free-standing part of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Strain can affect both the bandgap and the Seebeck coefficient of MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The indirect optical gap is modulated by -110 meV/%-strain for a trilayer MoS236,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Ab initio studies show a ~10% decrease in the Seebeck coefficient of monolayer MoS2 per 1% tensile strain 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To estimate the biaxial strain, we performed atomic force microscopy (AFM) height trace mapping on the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Most samples, regardless of the geometry of the hole exhibit slight bulging of a few nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For the MoS2 flakes suspended on the circular holes in the device shown in Figure 3a, the bulge height is 𝛿𝑡 ≈ 25 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Similar 𝛿𝑡 values were measured for other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The biaxial strain can then be calculated using an uniformly loaded circular membrane model, and is as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='0025% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Such a small strain on MoS2 is not sufficient to induce a significant change in bandgap or Seebeck coefficient 43–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Next, we consider the substrate induced changes on the material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The presence or the absence of the substrate can cause enhanced or diminished optical absorption due to the screening effects, Fermi level pinning46 and charges donated by the substrate7,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' More significantly, the doping effect due to the trapped charges at the interface with the substrate can locally gate the MoS2 and modify the number of charge carriers48 and thus its Seebeck coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To investigate the electrostatic impact of the substrate on the MoS2 membrane, we investigated the surface potential difference (SPD) on devices using Kelvin Probe Force Microscopy (KPFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPD can provide an insight on the band bending of the MoS2 due to the substrate effects49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 3b-d shows the AFM height trace map and the uncalibrated SPD map of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPD across the supported and suspended part of the flake is on the order of 50 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This shift in the SPD value hints that there is a slight change in the Fermi level of the suspended part with respect to the supported part of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The same type of charge carriers is dominant on both sides of the junction formed by the suspended and supported parts of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The band structure formed by such a junction in zero bias cannot be used in separation of photoinduced carriers50, however, it can lead to the formation of a thermoelectric junction11,51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This is in line with the SThGM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' AFM height trace map of a device suspended over circular holes show a bulge of 𝛿𝑡 ≈ 25 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The line trace is overlayed on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' AFM height trace map of the sample shows the bulged and dimpled parts of the flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' KPFM map of the sample shows the variation in the surface potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Line traces taken along the numbered lines in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Direction of the arrows in c indicates the direction of the line plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In the remainder of the paper, we aim at controlling the electrostatics that are responsible for the formation of a thermoelectric junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Charge transport in MoS2 is dominated by electrons due to unintentional doping52,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Modulating the density and the type of free charge carriers can be done by applying a gate voltage 𝑉𝑔 to the junction54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This significantly modifies the magnitude and the sign of the Seebeck coefficient as demonstrated in previous studies16,30,31,55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The Mott relation56 can be used to model the Seebeck coefficient as a function of 𝑉𝑔: 𝑆 = 𝜋2𝑘𝐵 2𝑇 3𝑒 1 𝑅 𝑑𝑅 𝑑𝑉𝑔 𝑑𝑉𝑔 𝑑𝐸 | 𝐸=𝐸𝐹 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' (1) Here, 𝑇 is the temperature, 𝑘𝐵 is the Boltzmann constant, 𝑒 is the electron’s charge, 𝑅 is the device resistance, 𝐸𝐹 is the Fermi energy and 𝐸 is the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Since hole transport is limited due to substrate induced Fermi level pinning on SiO2 supported MoS2 field-effect devices,46 to observe the sign inversion of the Seebeck coefficient (see the Supporting Information for measurements on device fabricated on SiO2 and Al2O3 coated SiO2) we followed an alternative approach to emulate suspension: we fabricated heterostructure devices where the crystal is partially supported by hexagonal boron nitride (h-BN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' h-BN is commonly used to encapsulate two- dimensional materials thanks to its hydrophobic and atomically smooth surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This leads to less unintentional doping due to the interfacial charge trapping and reduced electron scattering7,57,58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' A ~10 ML MoS2 is placed over a 10 nm thick h-BN crystal to form a double-junction device (see supporting information for a single-junction device formed by a MoS2 flake which is partially placed over a h-BN flake) and indium contacts are placed over the MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The device is on 1 µm thick oxide coated Si substrate where Si is used as the back-gate electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 4a shows the optical micrograph of the device and its schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The presence of h-BN modifies the SPD by 80 mV – a value very similar to the values we find for suspended devices (see SI) – which is consistent with the relative n-doping by the h-BN substrate32,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We therefore attribute this difference to the Fermi level shift due to the difference in interfacial charge doping by the different substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' ot 0 1234μm Figure 4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical micrograph of a Si back-gated MoS2 device partially placed over h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Its cross- sectional schematic is shown in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM reflection map and the photocurrent map of the device shown in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝐼𝑚𝑎𝑥 = 3 nA and 𝐼𝑚𝑖𝑛 = −3 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar: 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Current-Voltage graph versus 𝑉𝐺 from -40 to 40 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Inset shows the resistance versus 𝑉𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝐼𝑃𝐶 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝑉𝐺 recorded at the points marked in the SPCM map in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure 4b shows the SPCM map under zero gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We observe a bipolar photocurrent signal from the junctions between h-BN and SiO2 supported MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Raman mapping (see the Supporting Information) reveals slight intensity decrease and a small shift of the A1𝑔 peak over the h-BN supported part of the MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This is consistent with the stiffening of the Raman mode due to the higher degree of charged impurities in SiO2 as compared to h-BN7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' By applying a gate voltage to the device, its resistance can be tuned significantly as free charges are depleted (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Under large positive gate voltages, the I-V characteristic becomes asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To investigate the dependence of the photocurrent on carrier type and concentration, the laser is held at specific positions on the device as marked in Figure 4d, and the gate is swept from positive to negative voltages with respect to the ground terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For positive gate voltages, the magnitude of the photoresponse from both junctions, between h-BN and SiO2 supported MoS2, (points 2 and 3) decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' When a negative gate voltage is applied, the magnitude of the photoresponse at both junctions increases by almost a factor of two at 𝑉𝐺 = −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Once this maximum is reached, the amplitude of the photocurrent at both points decreases and has the same value as the photocurrent generated over the MoS2 (point 4) at 𝑉𝐺 = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' These observations can be qualitatively explained as follows: at a gate voltage of 𝑉𝐺 = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 𝑉, the majority charge carrier type in the h-BN supported part changes from electrons to holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As a consequence, the Seebeck coefficients of MoS2 resting on h-BN and SiO2, respectively, become similar, which leads to ∆𝑆 ≈ 0, and curves 2,3 and 4 in Figure 4d cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The photocurrent signal recorded near the indium contacts (points 1 and 5) decreases non-monotonically with decreasing 𝑉𝐺 and reaches SiO2 Si Imax Iminzero at 𝑉𝐺 = −40 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' At this voltage the Seebeck coefficient of MoS2 on SiO2 reaches that of Indium (SIn = + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='7 µV/K)59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In conclusion we demonstrated that substrate engineering can be used to generate a thermoelectric junction in atomically thin MoS2 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Similar strategies can be employed in other low dimensional materials that exhibit large and tunable Seebeck coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This might in particular be promising at low temperature where effects like band-hybridization and Kondo scattering can produce a very strong photothermoelectric effect9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Author Contributions T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' designed and conceived the experiments, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' prepared the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' fabricated devices, performed the experiment and analyzed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' prepared the substrates, performed simulations, and helped with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' performed the AFM and KPFM measurements and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' performed some of the earlier measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' performed the SThGM measurements and analyzed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='D and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='W contributed discussions on the implementation of VITA-DM-GLA-1 SThM probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' All authors discussed the results and reviewed the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Competing Interests The Authors declare no Competing Financial or Non-Financial Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Methods SPCM setup is a commercially available setup from LST Scientific Instruments Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' which offers a compact scanning head with easily interchangeable lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Two SR-830 Lock-in amplifiers are employed, one for the reflection map and the other for the photocurrent/voltage measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In the main text we reported the photocurrent (a measurement of the photovoltage is given in Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The incident laser beam is chopped at a certain frequency and focused onto the sample through a 40x objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The electrical response is collected through gold probes pressed on the electrical contacts of the devices and the signal is amplified by a lock-in amplifier set to the chopping frequency of the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Although various wavelengths (406, 532, 633 nm) are employed for the measurements, unless otherwise stated we used 532 nm in the experiments reported in the main text (see Figure S3 for SPCM measurements with different wavelengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' All the excitation energies are above the indirect bandgap of the few layer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scanning Thermal Microscopy measurements were performed with a Dimension Icon (Bruker) AFM under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The probe used in the experiments is VITA-DM-GLA-1 made of a palladium heater on a silicon nitride cantilever and tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The radius is typically in the order of 25-40 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The heater is part of a modified Wheatstone bridge and is driven by a combined 91 kHz AC and DC bias, as reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The signal is detected via a SR830 lock-in amplifier and fed in the AFM controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This signal monitors the probe temperature and thus allows to locally map the thermal conductance of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In this work, the power supplied to the probe gives rise to a 45K excess temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' While the probe is scanning the sample, we measure the voltage drop across the device using a low noise preamplifier (SR 560).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This voltage is created by the local heating induced by the hot SThM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' It is then fed also to the AFM controller and recorded simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In this study, the thermovoltage measurements were performed without modulating the heater power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We note that it is also possible to generate similar maps by varying the heater temperature and detecting thermovoltage via lock-in detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Data Availability Source data available from the corresponding authors upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' References (1) Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='843415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Supporting Information: Single-material MoS2 thermoelectric junction enabled by substrate engineering Mohammadali Razeghi1, Jean Spiece3, Oğuzhan Oğuz1, Doruk Pehlivanoğlu2, Yubin Huang3, Ali Sheraz2, Phillip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Dobson4, Jonathan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Weaver4, Pascal Gehring3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Serkan Kasırga1,2* 1 Bilkent University UNAM – Institute of Materials Science and Nanotehcnology, Bilkent 06800 Ankara, Turkey 2 Department of Physics, Bilkent University, Bilkent 06800 Ankara, Turkey 3 IMCN/NAPS, Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium 4 James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='K Corresponding Author: kasirga@unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='tr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Theoretical prediction of the substrate-effect induced Seebeck coefficient difference in MoS2 As discussed in the main text, we assume that the Seebeck coefficient S in thermal equilibrium is composed of contributions from the energy-dependent diffusion (𝑆𝑁), scattering (𝑆τ) and the phonon- drag (𝑆𝑝𝑑), so that 𝑆 = 𝑆𝑁 + 𝑆𝜏 + 𝑆𝑝𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Here, 𝑆𝑁 and 𝑆τ terms can be written from the Mott relation assuming that MoS2 is in the highly conductive state and electrons are the majority carriers: 𝑆τ = − 𝜋2𝑘𝐵 2𝑇 3𝑒 𝜕ln𝜏 𝜕𝐸 | 𝐸=𝐸𝐹 and 𝑆𝑁 = ± 𝑘𝐵 𝑒 [ 𝐸𝐹−𝐸𝐶 𝑘𝐵𝑇 − (𝑟+2)𝐹𝑟+1(𝜂) (𝑟+1)𝐹𝑟(𝜂) ] As mentioned in the main text, 𝑆τ is zero as 𝜏 is energy independent in the 2d limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝑆𝑁 term is composed of constants related to material properties, scattering parameter 𝑟 and the Fermi integral of the 𝑟-th order: 𝐹𝑟 = ∫ [ 𝑥𝑚 𝑒𝑥−𝜂 + 1]𝑑𝑥 ∞ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The scattering parameters of 2d materials are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='1,2 Here, as discussed in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 1, 𝑟 = 0 adequately accounts for the acoustic phonon scattering and small deviations of experimental data from the calculated values is due to the other scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As a result, at the room temperature 𝑆𝑁 for suspended MoS2 (1012 cm-2) is about -400 µV/K and for SiO2 supported MoS2 (1013 cm-2) is about -200 µV/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scattering parameters 𝒓 of 2d materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scattering mechanism 𝒓 Charged Impurity Scattering 3/2 Acoustic Phonon Scattering 0 Intervalley Scattering 0 Strongly Screened Coulomb Scattering 1/2 𝑆𝑝𝑑 term can be estimated from the theory of phonon-drag in semiconductors in the first order as 𝑆𝑝𝑑 = − 𝛽𝑣𝑝𝑙𝑝 𝜇𝑇 where, 𝑣𝑝 and 𝑙𝑝 are the group velocity and the mean free path of a phonon, 𝛽 is a parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and 𝜇 is the electron mobility, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As the dominant charge carriers are electrons, 𝑆𝑝𝑑 term has a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We use the parameters given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Based on the values given in the table we obtain 𝑆𝑝𝑑 𝑆𝑖𝑂2 = −230 µV/K and 𝑆𝑝𝑑 𝑆𝑢𝑠 = −100 µV/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The total 𝑆 = 𝑆𝑁 + 𝑆𝑝𝑑 for suspended and SiO2 supported parts can be calculated by adding both contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝑆𝑆𝑢𝑠 = −500 µV/K and 𝑆𝑆𝑖𝑂2 = −430 µV/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Of course, we consider this to be a rough estimate as we ignore charged impurity scattering and strongly screeded Coulomb scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Also there are certain errors associated with the measurement of the parameters used for the calculation of the Seebeck coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' However, overall, this calculation shows that the substrate induced effect must be present under right experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Parameters used for 𝑺𝒑𝒅 calculation Parameter On SiO2 (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 3) Suspended (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 3) 𝑣𝑝 7 105 cm/s 7 105 cm/s 𝑙𝑝 5 nm 20 nm 𝜇 5 cm2/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='s 50 cm2/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM map on a gold electrode substrate engineered MoS2 device Throughout the study we used indium contacted devices thanks to their rapid fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To compare our indium device results, we fabricated gold contacted devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S1 shows the optical microscope images and corresponding SPCM reflection and photocurrent maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' There is no qualitative difference between the indium contacted devices and gold contacted device in the substrate- engineered photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Despite IV measurement is collected from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='25 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='25 V its rectifying behaviour can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Power dependence of the photocurrent from the substrate engineered junction is also comparable to the one reported in indium contacted devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical microscope micrograph of a gold contacted substrate engineered MoS2 device is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar is 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM reflection map and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' photocurrent map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' IV curve shows signs of rectifying nature of the contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Power dependence of the photocurrent at one of the side of the junction is plotted in a log-log graph and the exponent is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scanning photovoltage microscopy, AFM and KPFM measurements on a parallel trench device Figure S2 shows an MoS2 device fabricated on trenches drilled on sapphire with different depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We performed SPVM, AFM and KPFM Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' First, AFM measurements show that the crystal is stuck to the bottom of the 100 nm deep trench (Figure S2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For the rest of the trenches the flake bulges about 10 nm above the surface (Figure S2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' AFM height trace map also reveals a peculiar wrinkle formation over the suspended part of the flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In this measurement we operated the scanning microscope at photovoltage mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S2d shows the reflection map and the corresponding photovoltage map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The bipolar response is evident with slightly lower positive signal in some of the trenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This asymmetry can be explained by lower heating of one side of the samples due to the scan direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' One important observation that agrees well with the photothermoelectric photoresponse is that the 100 nm trench shows very small photovoltage as compared to other trenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S2e shows the KPFM and AFM profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The suspended part of the crystal has 60 meV lower surface potential difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This is consistent with other KPFM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The lower panel shows the variation in the height over the wrinkles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The workfunction is calculated with a calibrated tip and it follows the wrinkles of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' However, the difference in the SPD is not due to the variations in the height profile of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The change in the workfunciton is a good indication of the changes in the electronic landscape of the device upon suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Small variations along the wrinkles are also expected due to formation of varying stress regions along the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical microscope image of the device with trench depths labelled next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' AFM height trace map and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' line trace taken along the height trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The bulge of the crystal over the trenches is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Reflection and photovoltage maps obtained by operating the scanning microscope in photovoltage mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar is 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Left panel shows the workfunction and height taken over Profile1 103nm Onmthe red lines marked on the maps given in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The variation of the workfunction along the trench is very small and correlated with the wrinkles of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM maps taken at different laser wavelengths and incidence polarization We used three different wavelengths, 406, 532 and 633 nm, in our experiments all of them which are at an energy larger than the band gap of MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S3 shows the SPCM results collected with different laser wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Also, polarization dependence of the photocurrent measured at each end of the trench as well as a point over the contact is given in Figure S3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' There is no polarization dependence of the photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This shows that The effect is not due to built in polarization fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical microscope image of a two-terminal substrate-engineered MoS2 device with different trench widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar is 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM reflection map of the region marked with yellow rectangle in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c, d and e show photocurrent map taken at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' At each run laser power is set to ~40 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The measured signal in all three measurements are very close and the overall photocurrent features are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Incident polarization of the 633 nm laser is rotated and 𝐼𝑃𝐶 is measured at three different points marked by colored arrows on d, black dots- near contact, red dots- at the positive side and blue dots- at the negative side of the trench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' There is no polarization dependence of the measured photocurrent at the three points where photocurrent is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Finite element simulation of a substrate modified thermoelectric junction To understand how a substrate modified thermoelectric junction would behave depending on how the contacts are configured, we performed finite element analysis simulations using COMSOL Multiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' An irregularly shaped crystal is modelled over a substrate with a hole and voltage at a floating terminal is measured with respect to different laser positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The observed pattern agrees with our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S4 shows the thermoelectric emf generated and temperature distribution maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 406 nm 532 nm 633 nm Figure S4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' An arbitrary crystal is modelled over a SiO2 substrate with a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The outline drawn over the voltage distribution map shows the outline of the crystal and the outline of the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The red line indicates the ground terminal and the blue line indicates the floating voltage terminal where the photothermoelectric emf is measured from like in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For comparison, another terminal is simulated as the floating terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Temperature distribution vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' laser position is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As clear, the maximum temperature rise is achieved at the center of the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' KPFM on h-BN supported and suspended MoS2 Figure S5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical micrograph of an MoS2 crystal partially suspended over a trench and partially supported on h-BN (outlined by blacked dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' White dashed square shows the AFM region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' AFM height trace map and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' corresponding SPD map is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPD line traces from the colored lines in c are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The difference between the SPD in h-BN supported, suspended and SiO2 supported parts are evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bars are 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Gate dependent measurements We performed gate dependent SPCM measurements both on suspended and h-BN supported MoS2 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' In both cases, we used 1 µm SiO2 coated Si wafers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Si is used as the back gate in both device configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We reported the h-BN supported junctions in the main text as devices over holes showed significant change upon application of negative gate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S5 shows the degradation of the suspended device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' After application of a few volts the device irreversibly shows a contrast change hBN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='10 hBNSupported 58mV Suspended :35mV Sio,Supported 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='0 lenght [um]starting from the edges of the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We fabricated a long trench with open ends to see if the trapped air within the hole is causing the observed contrast change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' However, same contrast change is observed after applying negative gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We observe that the contrast change starts from near the hole and expands from there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' At the moment we are not fully aware of the reasons leading this contrast change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We consider that the release of the adsorbed molecules on the surface of the substrate under large negative gate voltages lead to such degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S6 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Optical microscope micrograph of indium contacted MoS2 on SiO2/Si with stair-like holes before and after application of gate voltages down to 𝑉𝐺 = −20 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Lower panel shows a clear contrast change around the holes extending to the indium contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SPCM maps taken at 𝑉𝐺 = 0 𝑉 with 532 nm of 86 µW on sample: (i) before gating, (ii) after 𝑉𝐺 = −15 𝑉 scan and (iii) after the scan in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 𝐼𝑚𝑎𝑥 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 nA and 𝐼𝑚𝑖𝑛 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='5 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scan starts from top left corner to the bottom left corner with progressing to the right in raster scan pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bars are 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' To prevent the sample degradation problem under large negative gate voltages, we coated the substrate surface with 5 nm thick Al2O3 using atomic layer deposition (ALD) method after milling the holes with FIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Then, the device is fabricated over the ALD coated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The device didn’t show any sign of degradation and produced pronounced photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Measurements from the device is given in Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Although the device exhibits the expected gate dependent response, as discussed in the main text, there is no carrier inversion induced reduction in the photovoltage due to the Fermi level pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' ii ili BeforeV After VG Figure S7 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Schematic of the device along with the optical image is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The sample is coated with 30 nm thick Al2O3 to passivate the SiO2 surface and to minimize the pinholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scale bar is 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Photovoltage map collected in DC mode without the Lock-in amplifier and chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' i is the reflection map, and photovoltage maps at ii is the 𝑉𝐺 =-60 V, iii 𝑉𝐺 = 0 V, iv 𝑉𝐺 =60 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Here, 𝑉𝑚𝑎𝑥 = 20 mV and 𝑉𝑚𝑖𝑛 = -20 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Photovoltage line trace taken along the dashed arrow given in b-ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Large signal corresponds to the more negative gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Photovoltage data collected from points indicated on b-ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This sample showed no Seebeck coefficient inversion due to possible Fermi level pinning induced by the substrate as discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' H-BN supported devices performed better and showed no sign of such a contrast change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S7 shows the reflection and the photocurrent maps reported in the main text and the photocurrent from point 2 and 3 subtracted from point 4, marked on the photocurrent map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Both junctions of the h-BN show almost identical response under gate voltage (point 3 data is multiplied by -1 for viewing convenience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' ii MoS2 B in In A/2O3 SiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Si ili IV Point A Point B Point C Figure S8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Same figure from the main text is copied here for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Raman intensity map and the 𝐴1𝑔 peak shift map is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Gate dependent signal from point 4 is subtracted from the gate dependent data from point 2 (red curve) and point 3 (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Blue curve is multiplied by -1 for viewing convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Scanning Thermal Microscope Calibration and Seebeck variation estimation The Scanning Thermal Microscope (SThM) measurements were performed on a commercial Bruker Icon instrument with a VITA-GLA-DM-1 probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The probe, consisting of the silicon nitride lever with a Pd heater/thermometer has been calibrated on a hot plate to relate the temperature to its electrical resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The calibration curves are shown on figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S9 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' SThM probe calibration of the electrical resistance with the supplied power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Temperature as a function of electrical resistance As described elsewhere4,5, the probe is part of a modified Wheatstone bridge which is balanced at low voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' During the measurements, we applied a combined AC (91 kHz) and DC bias on the bridge which heats the probe and creates a bridge offset that directly measures the probe heater temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For most experiments, we applied 1mW on the probe creating a Δ𝑇 of 50 ± 2 K, when the probe was far away from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' When the SThM tip is brought into contact with the devices, it locally heats the materials below its apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' While the probe scans the surface, the device open circuit voltage is recorded and amplified via a SR830 voltage preamplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This voltage is referred to as the thermovoltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We excluded any E Imax 3(a) (b) 368 100 80 367 (Ohms) 60 366 40 365 R = 363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='68 + 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='73 P 20 ,= -1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='39 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='48 R applied probe probe 364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='35 364 368 372 376 380 384 P applied (mW) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Rprobe (Ohms)shortcut between the probe and the device as no leakage current could be measured between the probe and both contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The thermovoltage can be written analytically as6,7, 𝑉𝑡ℎ(𝑥) = − ∫ 𝑆(𝑥) 𝜕𝑇 𝜕𝑥 (𝑥)𝑑𝑥 𝐵 𝐴 where 𝑆(𝑥) is the position dependent Seebeck coefficient and 𝜕𝑇 𝜕𝑥 (𝑥) is the position dependent temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Both are integrated over the whole device length from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As shown elsewhere6,7, it is possible to deconvolute the Seebeck coefficient from the temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This however requires a precise estimation of the temperature gradient and thus the sample temperature rise under the tip, Δ𝑇𝑠𝑎𝑚𝑝𝑙𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' As we know the probe temperature far away from the sample (50 ± 2 K) and we monitor its temperature via the Wheatstone bridge, we know that the probe temperature in contact with the sample is 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='8 ± 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The probe cooling occurs because of several heat transfer mechanisms4,5 (solid- solid conduction, air conduction, water meniscus, …).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For those probes, the Pd heater is however distributed over the whole triangular shaped silicon nitride tip4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This implies that the tip temperature and probe temperature are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We turned to finite element modelling (COMSOL Multiphysics) to estimate the tip temperature over the MoS2 suspended and supported sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S10 shows the overall simulated probe and sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' We used reported values for the in-plane and out-of-plane MoS2 thermal conductivity as well as for the MoS2-glass interface conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Reported values vary greatly in literature8–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' However, to the best of our knowledge, for a thick sample (>10 layers), the values are on the order of 30 Wm-1K-1 for the supported in-plane, 60 Wm-1K-1 for the suspended in-plane and 3 Wm-1K-1 for the cross-plane conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' For the substrate interface conductance, we used 1 MWm-2K-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Figure S10 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Finite element model for the SThM probe on a MoS2 suspended sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Zoomed-in view of the model where the temperature gradient is visible on the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Using those material parameters, we estimated a ratio between the probe temperature and the tip apex temperature of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' The model also accounts for the tip-sample thermal resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This method (a) (b)and model were experimentally confirmed elsewhere4,5,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Taking these into consideration, we obtain a sample temperature rise Δ𝑇𝑠𝑎𝑚𝑝𝑙𝑒 of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='7 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' This gives a Seebeck variation of 72±10 µVK-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' References (1) Ng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Chi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Hippalgaonkar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Effect of Dimensionality on Thermoelectric Powerfactor of Molybdenum Disulfide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' J Appl Phys 2017, 121 (20), 204303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='4984138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' (2) Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Ng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Qiu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Chi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Neto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Thong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Loh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Hippalgaonkar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Large Enhancement of Thermoelectric Performance in MoS2/h-BN Heterostructure Due to Vacancy- Induced Band Hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' Proc Natl Acad Sci U S A 2020, 117 (25), 13929–13936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content='2007495117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfDPrL/content/2301.00974v1.pdf'} +page_content=' (3) Cui, X.' 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We study two families of lattices whose number of elements are given by +the numbers in even (respectively odd) positions in the Fibonacci sequence. The even +Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially +ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even +one. We give a combinatorial proof of the lattice property, relying on a description of +words for the Garside element in terms of Schröder trees, and on a recursive description +of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins +in the lattice. As a byproduct we also obtain that the number of words for the Garside +element is given by a little Schröder number. +Contents +1. +Introduction +1 +2. +Definition and structure of the poset +3 +2.1. +Definition of the poset +3 +2.2. +Lattice property +4 +3. +Schröder trees and words for the Garside element +7 +3.1. +labelling of Schröder trees +7 +3.2. +Words for the Garside element in terms of Schröder trees +10 +4. +Enumerative results +17 +4.1. +Number of simple elements +17 +4.2. +Number of left-divisors of the lcm of the atoms and odd Fibonacci lattice +17 +4.3. +Number of words for the divisors of the Garside element +18 +References +20 +1. Introduction +Several algebraic structures naturally yield examples of lattices: as elementary examples, +one can cite the lattice of subsets of a given set ordered by inclusion, or the lattice of +subgroups of a given group. +One can then study which properties are satisfied by the +obtained lattices, or conversely, starting from a known lattice, wondering for instance if +it can be realized in a given algebraic framework, or if a property of the lattice implies +properties of the attached algebraic structure(s) and vice-versa. +The aim of this paper is to give a combinatorial description of a finite lattice that +appeared in the framework of Garside theory. We will not recall results and principles +of Garside theory as they will not be used in this paper, but the interested reader can +look at [5, 4] for more on the topic. +This is a branch of combinatorial group theory +which aims at establishing properties of families of infinite groups such as the solvability +of the word problem, the conjugacy problem, the structure of the center, etc. Roughly +speaking, a Garside group is a group of fraction of a monoid (called a Garside monoid) with + +2 +THOMAS GOBET AND BAPTISTE ROGNERUD +particularly nice divisibility properties, which ensures that the above-mentioned problems +can be solved. Such a monoid M has no nontrivial invertible element, and comes equipped +with a distinguished element ∆ (called a Garside element) whose left- and right-divisors are +finite, coincide, generate the monoid, and form a lattice under left- and right-divisibility. +The left- or right-divisors of ∆ are called the simples. +The fundamental example of a Garside group is the n-strand Artin braid group [7]. +It admits several non-equivalent Garside structures (i.e., nonisomorphic Garside monoids +whose group of fractions are isomorphic to the n-strand braid group), and the lattice of +simples in the first discovered such Garside structure is isomorphic to the weak Bruhat +order on the symmetric group. Several widely studied lattices can be realized as lattices +of simples of a Garside monoid: this includes the lattices of left and right weak Bruhat +order on any finite Coxeter group [3, 6], the lattice of (generalized) noncrossing partitions +attached to a finite Coxeter group [1, 2], etc. (see also [12] for many other examples). This +suggests the following question: +Question. Which lattices can appear as lattices of simples of Garside monoids ? +The aim of this paper is to study a family Pn of lattices arising as simples of a family Mn, +n ≥ 2 of Garside monoids introduced by the first author [8]. For n = 2, the corresponding +Garside group is isomorphic to the 3-strand braid group B3, while in general it is isomorphic +to the (n, n + 1)-torus knot group, which for n > 3 is a (strict) extension of the (n + 1)- +strand braid group Bn+1. The lattice property of Pn follows from the fact proven in op. +cit. that Mn is a Garside monoid, but it gives very little information about the structure +and properties of the lattice. For instance, one does not have a formula enumerating the +number of simples, and only an algorithm to calculate meet and joins in the lattice. +In Section 2 we give a new proof of the lattice property of Pn (Theorem 2.8) by exhibiting +the recursive structure of the poset. Every lattice Pn turns out to contain the lattices Pi, +i < n as sublattices. Note that an ingredient of the proof of Theorem 2.8 is proven later on +in the paper, as it relies on a combinatorial description for the set of words for the Garside +element in terms of Schröder trees. +More precisely, in Section 3 we establish a simple +bijection between the set of words for ∆n and the set of Schröder trees on n+1 leaves, in such +a way that applying a defining relation of Mn to a word amounts to applying what we call a +"local move" on the corresponding Schröder tree (Theorem 3.12 and Corollary 3.13). These +local moves are given by specific edge contraction and are related to the notion of refinement +considered in [10]. This allows us to establish in Proposition 3.16 an isomorphism of posets +between subposets of Pn and Pi, i < n, required in the proof of Theorem 2.8. +Finally, the obtained recursive description of Pn together with the description of words +for ∆n in terms of Schröder trees allows us to derive a few enumerative results. This is +done in Section 4. The first one is that the number of elements of Pn is given by F2n, where +Fi is the i-th Fibonacci number (Lemma 4.1). We thus call Pn the even Fibonacci lattice. +The atoms of Mn turn out to have the same left- and right-lcm, which is strictly less than +∆n. We also show that the sublattice of Pn defined as the order ideal of this lcm has F2n−1 +elements (Lemma 4.3), and thus call it the odd Fibonacci lattice. Other enumerative results +include the determination of the number of words for the Garside elements (Corollary 3.14), +and the number of words for the whole set of simples (Theorem 4.7). +Recall that the Garside monoid Mn under study in this paper has group of fractions +isomorphic to the (n, n + 1)-torus knot group. This Garside structure was generalized to +all torus knot groups in [9]. It would be interesting to have a description of the lattices of +simples of this bigger family of Garside monoids. + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +3 +1 +ρ1 +ρ2 +ρ2ρ1 +ρ3 +ρ1ρ3 +ρ3ρ1 +ρ1ρ3ρ1 +ρ3ρ2 +ρ2ρ1ρ3 +ρ3ρ2ρ1 +ρ2 +3 +ρ3ρ1ρ3 +ρ2 +3ρ1 +(ρ1ρ3)2 +(ρ3ρ1)2 +ρ3 +3 +ρ3ρ2ρ1ρ3 +ρ2 +3ρ1ρ3 +(ρ3ρ1)2ρ3 +ρ4 +3 +Figure 1. The even Fibonacci lattice for n = 3 and (in blue) the odd Fibonacci +lattice inside it. +2. Definition and structure of the poset +2.1. Definition of the poset. The beginning of this section is devoted to explaining how +the poset under study is defined. We recall the definition of the monoid from which it is +built, as well as a few properties of this monoid (all of which are proven in [8]). +Let M be a monoid and a, b ∈ M. We say that a is a left divisor of b (or that b is a +right multiple of a) if there is c ∈ M such that ac = b. We similarly define right divisors +and left multiples. +Let M0 be the trivial monoid and for n ≥ 1, let Mn be the monoid defined by the +presentation +(2.1) +� +ρ1, ρ2, . . . , ρn +���� ρ1ρnρi = ρi+1ρn for all 1 ≤ i ≤ n − 1 +� +. +We denote by S the set of generators {ρ1, ρ2, . . . , ρn}, and by R the defining relations of +Mn. This monoid was introduced by the first author in [8, Definition 4.1]. Note that this +monoid is equipped with a length function λ : Mn −→ Z≥0 given by the multiplicative +extension of λ(ρi) = i for all i = 1, . . . , n, which is possible since the defining relations +do not change the length of a word. As a corollary, the only invertible element in Mn is + +4 +THOMAS GOBET AND BAPTISTE ROGNERUD +the identity, and the left- and right-divisibility relations are partial orders on Mn. We will +write a ≤L b or simply a ≤ b if a left-divides b, and a ≤R b if a right-divides b. +This monoid was shown to be a so-called Garside monoid (see [8, Theorem 4.18]), with +corresponding Garside group (which has the same presentation as Mn) isomorphic to the +(n, n + 1)-torus knot group, that is, the fundamental group of the complement of the +torus knot Tn,n+1 in S3. +Garside monoids have several important properties. +Among +them, the left- and right-divisibility relations equip Mn with two lattice structures, and +Mn comes equipped with a distinguished element ∆n, called a Garside element, which has +the following two properties +(1) The set of left divisors of ∆n coincides with its set of right divisors, and forms a +finite set. +(2) The set of left (or right) divisors of ∆n generates Mn. +This Garside element is given by ∆n = ρn+1 +n +. In particular, as any Garside monoid is a +lattice for both left- and right-divisibility, the set Div(∆n) of left (or right) divisors of ∆n +is a finite lattice if equipped by the order relation given by the restriction of left- (or right-) +divisibility on Mn. The set Div(∆n) is the set of simple elements or simples of Mn. In +general (Div(∆n), ≤L) and (Div(∆n), ≤R) will not be isomorphic as posets. But we always +have +(Div(∆n), ≤L) ∼= (Div(∆n), ≤R)op +(see for instance [8, Lemma 2.19]; such a property holds in any Garside monoid). +We will give a new proof that (Div(∆n), ≤) (and hence (Div(∆n), ≤R) is a lattice, in +a way which will exhibit a recursive structure of the poset. To this end, we will require +(sometimes without mentioning it) a few basic results on the monoid Mn which are either +explained above or proven in [8]: +(1) The left- and right-divisibility relations on Mn are partial orders. +(2) The monoid Mn is both left- and right-cancellative, i.e., for a, b, c ∈ Mn, we have +that ab = ac ⇒ b = c, and ba = ca ⇒ b = c (see [8, Propositions 4.9 and 4.12]), +(3) The set of left- and right-divisors of ∆n coincide. In fact, the element ∆n is central +in Mn, hence as Mn is cancellative, for a, b ∈ Mn such that ab = ∆n, we have +ab = ba (see [8, Proposition 4.15]) +2.2. Lattice property. The aim of this subsection is to prove a few properties of simple +elements of Mn, and to derive a new algebraic proof that Div(∆n) is a lattice. +Proposition 2.1. Let x1x2 · · · xk be a word for ∆n, with xi ∈ S for all i = 1, . . . , k. There +are i1 = 1 < i2 < · · · < iℓ ≤ k such that +• For all j = 1, . . . , ℓ, the word yj := xijxij+1 · · · xij+1−1 (with the convention that +iℓ+1 = k + 1) is a word for a power of ρn, +• The decomposition y1|y2| · · · |yℓ of the word x1x2 · · · xk is maximal in the sense that +no word among the yj can be decomposed as a product of two nonempty words which +are words for powers of ρn. +Morever, a decomposition with the above properties is unique. +Proof. The existence of the decomposition is clear using the fact that Mn is cancellative: +given the word x1x2 · · · xk, consider the smallest i ∈ {1, 2, . . . , k} such that x1x2 · · · xk is +a word for a power of ρn. Such an i has to exist, as x1x2 · · · xk is a word for a power of +ρn. Then set i2 := i + 1. By cancellativity in Mn, since x1 · · · xi and x1 · · · xk are both +words for a power of ρn, the word xi+1 · · · xk must also be a word for a power of ρn. Hence +one can go on, arguing the same with the word xi+1 · · · xk. Again by cancellativity, this +decomposition must be maximal. + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +5 +Now assume that the decomposition is not unique, that is, assume that y1|y2| · · · |yℓ +and z1|z2| · · · |zℓ′ are two decompositions of the word x1x2 · · · xk satisfying the properties +of the statement. As both y1 and z1 are words for a power of ρn, if y1 ̸= z1, then one +word must be strict prefix of the other, say z1 is a strict prefix of y1. But this contradicts +the maximality of the decomposition y1|y2| · · · |yℓ: indeed if y1 = x1x2 · · · xi2−1 and z1 = +x1x2 · · · xp with p < i2 − 1, we can decompose y1 nontrivially as x1x2 · · · xp|xp+1 · · · xi2−1, +and by cancellativity both x1 · · · xp and xp+1 · · · xi2−1 are words for powers of ρn. +□ +Example 2.2. Consider the word ρ3ρ1ρ7ρ1ρ7ρ5ρ4ρ7ρ7ρ1ρ7ρ6 in M7. We claim that this is a +word for the Garside element ρ8 +7 of M7. Indeed, using the defining relation ρ1ρ7ρi = ρi+1ρ7 +with i = 5 and 6, we get that +ρ3(ρ1ρ7ρ1ρ7ρ5)ρ4ρ7ρ7(ρ1ρ7ρ6) = ρ3ρ3 +7ρ4ρ4 +7, +and we observe also applying defining relations that +ρ3ρ3 +7ρ4ρ4 +7 = ρ1ρ7ρ2ρ2 +7ρ4 = ρ1ρ7ρ1ρ7ρ1ρ7ρ4 = ρ1ρ7ρ1ρ7ρ5ρ7 = ρ1ρ7ρ6ρ2 +7 = ρ4 +7. +The decomposition according to Proposition 2.1 is given by +ρ3ρ1ρ7ρ1ρ7ρ5ρ4 +� +�� +� +:=y1 +| ρ7 +���� +:=y2 +| ρ7 +���� +:=y3 +| ρ1ρ7ρ6 +� �� � +:=y4 +. +It is indeed clear by considering λ(u) for u prefixes of y1 or y4 that whenever u is a proper +prefix, we do not have λ(u) equal to a multiple of 7, which is a necessary condition for a +word to represent a power of ρ7. +Lemma 2.3. Let 1 ≤ k ≤ n. Then +S ∩ {x ∈ Div(∆n) | x ≤ ρkρk +n} = {ρ1, ρ2, . . . , ρk}. +Proof. We argue by induction on k. The result is clear for k = 1, as no defining relation of +Mn can be applied to the word ρ1ρn. Now let k > 1. Observe that +ρkρk +n = (ρ1ρn)k = (ρ1ρn)(ρ1ρn)k−1. +In particular we have ρ1 ≤ ρkρk +n and by induction, we get ρ1ρnρi ≤ ρkρk +n for all 1 ≤ i ≤ k−1. +As ρ1ρnρi = ρi+1ρn we get that {ρ1, ρ2, . . . , ρk} ⊆ S ∩ {x ∈ Div(∆n) | x ≤ ρkρk +n}. +It remains to show that no other ρi can be a left-divisor of ρkρk +n. Hence assume that +i > k and ρi ≤ ρkρk +n. Hence there is a word x1x2 · · · xp for ρkρk +n, where xi ∈ S for all i, +such that x1 = ρi. As the words x1x2 · · · xp and ρkρk +n represent the same element, they +can be related by a finite sequence of words w0 = x1x2 · · · xp, w1, . . . , wq = ρkρk +n, where +each wi is a word with letters in S and wi+1 is obtained from wi by applying a single +relation somewhere in the word. As the first letter of w0 differs from the first letter of +wq, there must exist some 0 ≤ ℓ < q such that wℓ begins by ρi but wℓ+1 does not. It +follows that the relation allowing one to pass from wℓ to wℓ+1 has to be applied at the +beginning of the word wℓ. But the only possible relation with one side beginning by ρi +is ρiρn = ρ1ρnρi−1. It follows that ρ1ρnρi−1 ≤ ρkρk +n = (ρ1ρn)k. By cancellativity, we get +that ρi−1 ≤ (ρ1ρn)k−1 = ρk−1ρk−1 +n +. By induction this forces one to have i − 1 ≤ k − 1, +contradicting our assumption that i > k. +□ +Similarly, we have +Lemma 2.4. Let 1 ≤ k ≤ n. Then +S ∩ {x ∈ Div(∆n) | x ≤R ρk +n} = {ρn, ρn−1, . . . , ρn−k+1}. + +6 +THOMAS GOBET AND BAPTISTE ROGNERUD +Proof. As for Lemma 2.3, we argue by induction on k. The result is clear for k = 1. Hence +assume that k > 1. As (ρ1ρn)n−jρj = ρn−j+1 +n +, we get that ρj ≤R ρk +n for all j such that +n − j + 1 ≤ k, that is, for all j ≥ n − k + 1. It remains to show that no other ρj can +right-divide ρk +n. Hence assume that ρj ≤R ρk +n, where j < n − k + 1. Arguing as in the +proof of Lemma 2.3, we see that ρ1ρnρj = ρj+1ρn must be a right-divisor of ρk +n, hence by +cancellativity that ρj+1 ≤R ρk−1 +n +. By induction this forces j + 1 ≥ n − k + 2, contradicting +our assumtion that j < n − k + 1. +□ +For x ∈ Div(∆n), let d(x) := max{k ≥ 0 | ρk +n ≤ x}. Let 0 ≤ i ≤ n + 1 and let +Di +n := {x ∈ Div(∆n) | d(x) = i}. +Note that +Div(∆n) = +� +0≤i≤n+1 +Di +n. +We have Dn +n = {ρn +n}, Dn+1 +n += {∆n}. +Lemma 2.5. Let x ∈ Div(∆n) and i = d(x). Let x′ ∈ Mn such that x = ρi +nx′. Note that +x′ ∈ D0 +n. Let x1x2 · · · xk be a word for x, where xi ∈ S for all i = 1, . . . , k. Then there is +1 ≤ ℓ ≤ k such that x1x2 · · · xℓ is a word for ρi +n (and hence xℓ+1 · · · xk is a word for x′ by +cancellativity). In other words, any word for x has a prefix which is a word for ρi +n. +Proof. It suffices to show that if z1z2 · · · zp is an expression for x such that z1z2 · · · zq is +an expression for ρi +n (q ≤ p, then one cannot apply a defining relation of Mn on the word +z1z2 · · · zp simultaneously involving letters of the word z1z2 · · · zq and letters of the word +zq+1 · · · zp. Let us consider the three possible cases where this could occur: one could have +ρ1ρn|ρj, ρ1|ρnρj, or ρj+1|ρn (1 ≤ j < n), where the | separates the letters zq and zq+1. +The last two cases cannot happen, since one would have zq+1 = ρn, hence zq+1 · · · zp would +be a word for x′ beginning by ρn, contradicting the fact that x′ ∈ D0 +n. It remains to show +that the case ρ1ρn|ρj cannot happen. Hence assume that zq−1 = ρ1, zq = ρn, zq+1 = ρj. +By cancellativity, as z1z2 · · · zq is a word for ρi +n, it implies that ρ1 ≤R ρi−1 +n +. By lemma 2.4, +this implies that n − (i − 1) + 1 = 1, hence that i = n + 1. +Since x ∈ Div(∆n) and +x = ρn+1 +n +x′ = ∆nx′, we get x′ = 1, contradicting the fact that zq+1 = ρj. +□ +Lemma 2.6. Let i, j ∈ {0, 1, . . . , n + 1}, with i ̸= j. Let x ∈ Di +n, y ∈ Dj +n. Assume that +x ≤ y. Then i < j and x < ρj +n ≤ y. +Proof. It is clear that i < j, since ρi +n ≤ y as ρi +n ≤ x, hence j < i would contradict y ∈ Dj +n. +In particular x < y. Let x′, y′ such that x = ρi +nx′ and y = ρj +ny′. Note that x′, y′ both lie +in D0 +n. Since x ≤ y and Mn is cancellative, we get that x′ < ρj−i +n y′. It implies that there +exists a word x1x2 · · · xk for ρj−i +n +y′ (xi ∈ S) and 1 ≤ ℓ < k such that x1x2 · · · xℓ is a word +for x′. Now by lemma 2.5, there is 0 ≤ ℓ′ ≤ k such that x1x2 · · · xℓ′ is a word for ρj−i +n . If +ℓ′ ≤ ℓ, then ρj−i +n +≤ x′, contradicting the fact that x′ ∈ D0 +n. Hence ℓ′ > ℓ, and x′ < ρj−i +n . +Multiplying by ρi +n on the left we get x < ρj +n. +□ +Lemma 2.7. Let z1, z2 ∈ Di +n. Let 1 ≤ k1 < k2 ≤ n and assume that there are two cover +relations z1 ≤· ρk1 +n , z2 ≤· ρk2 +n in (Div(∆n), ≤). Then z1 < z2. +Proof. As z1 ≤ ρk1 +n , z2 ≤ ρk2 +n +are cover relations, there are 1 ≤ j1, j2 ≤ n such that +z1ρj1 = ρk1 +n , z2ρj2 = ρk2 +n . By lemma 2.4, for ℓ ∈ {1, 2} we have jℓ ∈ {n − kℓ + 1, . . . , n} and +ρkℓ +n = ρjℓ+kℓ−1−n +n +(ρ1ρn)n−jℓρjℓ. +In particular, we have +zℓ = ρjℓ+kℓ−1−n +n +(ρ1ρn)n−jℓ + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +7 +and as (ρ1ρn)n−jℓ = ρn−jℓρn−jℓ +n +, by lemma 2.3 we see that ρn cannot be a left divisor of +(ρ1ρn)n−jℓ, and hence that d(zℓ) = jℓ + kℓ − 1 − n. But d(zℓ) = i for ℓ ∈ {1, 2}, and since +k1 < k2 we deduce that j1 > j2. Since zℓ = ρi +n(ρ1ρn)n−jℓ, we get that z1 < z2, which +concludes the proof. +□ +Theorem 2.8. The poset (Div(∆n), ≤) is a lattice. Given i, j ∈ {0, 1, . . . , n+1} with i ≤ j +and x ∈ Di +n, y ∈ Dj +n, we have +x ∧ y = x ∧i +�� +i +{z ∈ Di +n | z ≤ y} +� +, +where ∨i and ∧i denote the meet and join on the restriction of the left-divisibility order on +Di +n, which itself forms a lattice. Note that if i = j we simply get x ∧ y = x ∧i y. +Proof. The proof is by induction on n. We have Div(∆0) = {•}, and Div(∆1) = {1, ρ1, ρ2 +1}, +which is a lattice. Hence assume that n ≥ 2. By Proposition 3.16 below, the restriction of +the left-divisibility to Di +n yields an isomorphism of poset with Div(∆n−i) if i ̸= 0, n+1, while +the restriction to D0 +n yields an isomorphism of poset with Div(∆n−1), and the restriction to +Dn+1 +n +an isomorphism of posets with Div(∆0) = {•}. In particular, by induction, all these +posets are lattices. As the poset (Div(∆n), ≤) is finite and admits a maximal element, it +suffices to show that x ∧ y as defined by the formula above is indeed the join of x and y. +It is clear that x∧y ≤ x. Let us show that x∧y ≤ y. If i = j this is clear, hence assume +that i < j. By lemma 2.6 we see that +� +i +{z ∈ Di +n | z ≤ y} = +� +i +{z ∈ Di +n | z ≤ ρj +n}. +It suffices to check that � +i{z ∈ Di +n | z ≤ ρj +n} ≤ ρj +n. Note that +� +i +{z ∈ Di +n | z ≤ ρj +n} = +� +i +{z ∈ Di +n | z ≤ ρj +n and (z ≤· x ≤ ρj +n ⇒ x /∈ Di +n)}. +Now by lemma 2.6, if z ∈ Di +n and x is any element such that z ≤· x ≤ ρj +n and x /∈ Di +n, then +x = ρk +n for some k (necessarily smaller than or equal to j). It implies that +� +i +{z ∈ Di +n | z ≤ ρj +n} = +� +i +{z ∈ Di +n | z ≤ ρj +n and z ≤· ρk +n for some k ≤ j}. +By lemma 2.7, we have that +� +i +{z ∈ Di +n | z ≤ ρj +n and z ≤· ρk +n for some k ≤ j} +has to be an element of the set {z ∈ Di +n | z ≤ ρj +n and z ≤· ρk +n for some k ≤ j}, hence that +it is in particular a left-divisor of ρj +n (and hence of y). +Now assume that u ≤ x, y. We can assume that u ∈ Di +n, otherwise by lemma 2.6 we +have u < ρi +n ≤ x ∧ y. As u ≤ y, we have that u ≤ � +i{z ∈ Di +n | z ≤ y}. And hence, that +u ≤ x ∧i +�� +i{z ∈ Di +n | z ≤ y} +� += x ∧ y. +□ +3. Schröder trees and words for the Garside element +3.1. labelling of Schröder trees. A rooted plane tree is a tree embedded in the plane +with one distinguished vertex called the root. The vertices of degree 1 are called the leaves +of the tree and the other vertices are called inner vertices. One can consider rooted trees +as directed graphs by orienting the edges from the root toward the leaves. If there is an +oriented edge from a vertex v to a vertex w, we say that v is the parent of w and w is a +child of v. As can be seen in Figure 2, we draw the trees with their root on the top and the + +8 +THOMAS GOBET AND BAPTISTE ROGNERUD +leaves on the bottom. The planar embedding induces a total ordering (from left to right) +on the children of each vertex, hence we can speak about the leftmost child of a vertex. +Alternatively one has a useful recursive definition of a rooted plane tree: it is either the +empty tree with no inner vertex and a single leaf or a tuple T = (r, Tr) where r is the root +vertex and Tr is an ordered list of rooted plane trees. If T is a tree with the first definition, +the vertex r is its root and the list Tr is the list of subtrees, ordered from left to right, +obtained by removing the root r and all the edges adjacent to r in T. +Definition 3.1. +(1) A Schröder tree is a rooted plane tree in which each inner vertex has at least two +children. +(2) A binary tree is a rooted plane tree in which each inner vertex has exactly two +children. +(3) The size of a tree is its number of leaves. +(4) The height of a tree is the number of vertices in a maximal chain of descendants. +(5) The Schröder tree on n leaves in which every child of the root is a leaf is called +the Schröder bush. We denote it by δn. +(6) The Schröder tree given by the binary tree in which every right child (resp. every +left child) is a leaf is called a left comb (resp. a right comb). +· · · +Figure 2. From left to right: the unique Schröder tree with 1 leaf, the unique +Schröder tree with two leaves, the three Schröder trees with 3 leaves. +Then the +Schröder bush and on its right a left comb. +The Schröder trees are counted by the so-called little Schröder numbers. The sequence +starts with 1, 1, 3, 11, 45, 197, 903, 4279, 20793, ... and is referred as A001003 in [11]. +We will label (and read the labels of) the vertices and the leaves of our trees using the +so-called post-order traversal. This is a recursive algorithm that visits each vertex and leaf +of the tree exactly once. Concretely, if T = +� +r, (T1, . . . , Tk) +� +is a rooted planar tree, then +we recursively apply the algorithm to T1, T2 until Tk and finally we visit the root r. When +the algorithm meets an empty tree it visits its leaf and then, the recursion stops and it +goes up one level in the recursive process. The first vertex visited by the algorithm is the +leftmost leaf of T, then the algorithm moves to its parent v (but does not visit v) and visits +the second subtree of v starting with the leftmost leaf and so on. We refer to Figure 3 for +an illustration where the first vertex visited by the algorithm is labeled by 1, the second +by 2 and so on. The last vertex visited by the algorithm is always the root of T. Let m, n +be two integers such that m ≥ n − 1. We then label a Schröder tree T with n ≥ 2 leaves +by labelling its vertices one after the other with respect to the total order defined by the +post-order traversal, using the following rules: +(1) Let v be the leftmost child of a vertex w. Then w is the root of a Schröder tree +� +w, (T1, · · · , Tk) +� +and v is the root of T1. The label λ(v) of v is equal to the number +of leaves of the forest consisting of all the trees T2, · · · , Tk. + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +9 +20 +11 +3 +1 +2 +6 +4 +5 +10 +7 +8 +9 +12 +19 +15 +13 14 +18 +16 17 +Figure 3. Post-order traversal of a Schröder tree of size 12. +(2) If v is not the leftmost child of a vertex of T, we consider LD(v) the set of its +leftmost descendants consisting of the leftmost child of v and its leftmost child and +so one. Then the label of v is m − � +w∈LD(v) λ(w). Note that using the post-order +traversal, the label of the leftmost descendants of a vertex v are already determined +when we visit v. +The result is a labelled Schröder tree that we denote by Lm(T). +This procedure is +illustrated in Figure 4. +Definition 3.2. Let Lm(T) be a labelled Schröder tree with n leaves labelled by m ≥ n−1. +The sum of the labels of the vertices of T is called its weight (with respect to m). +Lemma 3.3. Let T be a Schröder tree with n leaves and m ≥ n − 1. Then the integers +labelling Lm(T) are strictly nonnegative with the exception of the root which may be labelled +by 0. +Proof. If a vertex is a leftmost child, then its label is a number of leaves, hence it is positive. +If v is not a leftmost child, then it is labelled by m − � +w∈LD(v) λ(w). Each λ(w) is equal +to a certain number of leaves of T and the set of leaves associated to distinct vertices of +LD(v) do not intersect. Moreover, exactly one element of LD(v) is a leaf and this leaf is +not counted in � +w∈LD(v) λ(w). We therefore have +(3.1) + + +� +w∈LD(v) +λ(w) + + + 1 ≤ n, +hence m − � +w∈LD(v) λ(w) ≥ 0. Moreover if � +w∈LD(v) λ(w) = m, then by (3.1) we must +have m = n − 1. It follows that v has n descendants since the leftmost leaf which is a +descendant of v is not counted, hence v is the root of T. +□ +This labelling is almost determined by the recursive structure of the tree, as shown by +the following result. +Lemma 3.4. Let T = +� +r, (T1, . . . , Tk) +� +be a Schröder tree and v be a vertex of Ti for +i ∈ {1, . . . , k}. Then, +(1) If v is not the root of T1, then its label in Lm(T) is equal to its label in Lm(Ti). +(2) If v is the root of T1, then its label in Lm(T1) is equal to the sum of the labels of v +and of the root of T in Lm(T). +Proof. Let v be a vertex of Ti. If v is a leftmost child in T which is not the root of T1, +then its label is a number of leaves of a certain forest which is contained in Ti. Hence this +number is the same in the big tree T or in the extracted tree Ti. If v is not a leftmost +child, then its label is determined by the labels of its leftmost descendants, hence it is the +same in the tree T as in the extracted tree Ti since we have just shown that the labels of +leftmost descendants which are not the root of T1 agree. The root of T1 has a different +behaviour since in T it is a leftmost child and this is not the case in T1. Hence if v is the +root of T1, denoting by λ1 the label of v in T1, we have λ1(v) = m − � +w∈LD(v) λ1(w). The + +10 +THOMAS GOBET AND BAPTISTE ROGNERUD +labels of the descendants of v are the same in T and in T1, that is, we have λ1(w) = λ(w) +for all w ∈ LD(v). In T, the label of the root r is given by +λ(r) = m − λ(v) − +� +w∈LD(v) +λ(w) = m − λ(v) − +� +w∈LD(v) +λ1(w). +Hence we have λ(r) + λ(v) = λ1(v). +□ +3.2. Words for the Garside element in terms of Schröder trees. Reading the la- +belled tree Lm(T) using the post-order traversal and associating the generator ρi to the +letter i with the convention that ρ0 = e, gives a map Φm from the set of Schröder trees +labelled by m to the set S⋆ of words for the elements of the monoid Mm. We refer to +Figure 4 for an illustration. +0 +5 +5 +1 +11 +10 +1 +11 +9 +2 +11 11 +11 +8 +2 +1 +11 +10 +1 +11 +Figure +4. Example +of +the +labelling +of +a +Schröder +tree +of +size +12 +with +m += +11. +The +corresponding +element +in +the +monoid +M11 +is +ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5ρ11ρ1ρ11ρ2ρ1ρ11ρ10ρ8. +Definition 3.5. Let T be a non-empty Schröder tree. If T has a subtree T1 satisfying the +three following properties: +(1) The root r1 of T1 is not the root of T, hence it has a parent r0 which has at least +two children, +(2) The root r1 has exactly two children, +(3) The right subtree of T1 is the empty tree with only one leave. +Then, we can construct another tree �T by contracting the edge r0 − r1, in other words by +removing the root r1 of T1 and attaching the two subtrees of T1 to r0. See Figure 5 for an +illustration. We call such a transformation, or the inverse transformation, a local move. +Note that, since r0 has at least two children in the configuration described above (see also +the left picture in Figure 5), we get that r0 has at least three children in the configuration +obtained after applying the local move. In particular, to apply a local move in the other +direction, we need to have a Schröder tree �T with a subtree T1 satisfying : +(1) The parent r0 of T1 (which is allowed to be the root of T) has at least three children, +(2) The tree T1 is not the last child of r0, and is directly followed by an empty tree +with only one leaf. +r0 +Sk +r1 +r2 +A1 +Sk+2 +←→ +r0 +Sk +r2 +A1 +Sk+2 +Figure 5. Local move. + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +11 +Formally, if the subtree of T with root r0 is S = +� +r0, (S1, · · · , Sk, T1, Sk+2, · · · , Sr) +� +and +the subtree T1 is +� +r1, (A1, A2) +� +, then we obtain the tree �T by replacing S by +� +r0, (S1, · · · , Sk, A1, A2, Sk+2, · · · , Sr) +� +. +Lemma 3.6. Let T and S be two Schröder trees with n leaves. Then one can pass from +the tree T to the tree S by applying a sequence of local moves. +Proof. It is enough to show that T can be transformed into the Schröder bush δn–recall +that this is the Schröder tree in which every child of the root is a leaf–by a sequence of +local moves. The Schröder tree S can then be transformed as well into δn, and hence T +can be transformed into S. We argue by induction on the number of leaves. For n = 1 and +n = 2 there is nothing to prove. If T has a subtree S which is not of the form δk, then we +can transform S into δk for some k by applying the induction hypothesis to S. Hence we +can assume that T = +� +r, (δn1, · · · , δnk) +� +with � ni = n. If T is not equal to δn, then it has +at least a non-empty subtree S. If S has only two leaves, then we can apply a local move to +remove its root and to attach the two leaves to the root of T. If it has more than 3 leaves, +by induction there is a sequence of local moves from S to a left comb. Then, by repeatedly +applying a local move at the root of the left comb, we remove all the inner vertices of the +left comb and attach all its leaves to the root of T. Applying this to all subtrees S of T +wich are not empty, we end up getting δn. +□ +Lemma 3.7. Let T be a Schröder tree with n leaves and m ≥ n − 1. Then Φm(T) is a +word for ρn−1(ρm)n−1ρm−n+1 in Mm. In particular if m = n − 1, then it is a word for the +Garside element of Mn−1. +Proof. If T = δn is the Schröder tree with only one root and n leaves, then Φm(T) = +ρn−1(ρm)n−1ρm−n+1. If T is another Schröder tree, then by Lemma 3.6 there is a sequence +of local moves from T to δn. To finish the proof it is enough to show that applying a local +move to a Schröder tree T amounts to applying a relation of the monoid Mm to Φm(T). +This is easily obtained by staring at Figure 5. +Indeed, if T is the tree at the left of Figure 5, then the label of r2 is 1, the label of the +leaf on its right is m and the label of r1 is a certain integer ℓ. Since r1 is not the root of +T, we have 1 ≤ ℓ. Moreover, since r1 is not a leaf of T, we have ℓ < m. Hence in Φm(T) +we have the factor ρ1ρmρℓ with 1 ≤ ℓ ≤ m − 1. +If �T denotes the right tree of Figure 5, then the label of r2 is ℓ + 1. Indeed r2 is a +leftmost child in �T if and only if r1 is a leftmost child in T. In this case its label is the +number of leaves of the forest in its right and in �T there is precisely one more leaf in this +forest than in T. In the other case, the label of r2 in �T is m − � +w∈LD(r2) λ(w). The label +of r1 is ℓ = m − 1 − � +w∈LD(r2) λ(w). So the label of r2 is ℓ + 1. The leaf on the right of r2 +in �T is labelled by m, hence Φm( �T) is obtained by replacing ρ1ρmρℓ in Φm(T) by ρℓ+1ρm, +and vice-versa. +□ +Proposition 3.8. For m = n−1, the map Φm from the set of Schröder trees with n leaves +to the set of words for ρn +n−1 in Mn−1 is surjective. +Proof. We have to show that to each word y for ρn +n−1 ∈ Mn−1, we can attach a Schröder +tree T with n leaves, in such a way that Φm(T) = y. The word y and the word ρn +n−1 +can be transformed into each other by applying a sequence of defining relations of Mm. +We already know that the word ρn +n−1 is in the image of Φm since it is the image of the +Schröder bush. To conclude the proof, we therefore need to show the following claim: given +a Schröder tree S, if the corresponding labelling has a substring of the form 1mℓ (resp. +(ℓ + 1)m) with 1 ≤ ℓ ≤ m − 1, then we are necessarily in the configuration of the left + +12 +THOMAS GOBET AND BAPTISTE ROGNERUD +picture in Figure 5 (resp. the right picture), and hence we can apply a local move. Indeed, +as one can pass from the word ρn +n−1 to the word y by a sequence of defining relations +let y0 = ρn +n−1, y1, . . . , yk = y be expressions of ρn +n−1 such that yi is obtained from yi−1 +by applying a single relation in Mm. +Applying the relation on y0 = Φm(T) to get y1 +corresponds to applying a local move on T to get a Schröder tree T1 and as seen in the +proof of lemma 3.7, we get Φm(T1) = y1. +To show the claim, assume that S is a Schröder tree with labelling having a substring +of the form 1mℓ with 1 ≤ ℓ ≤ m − 1. Note that m can only be the label of a leaf. Let v be +the parent of that leaf. It is a root of a family of trees, say (v, T1, . . . , Tk) and our leaf with +label m corresponds to one of the trees Ti (which has to be empty). It is clear that such a +tree cannot be T1: indeed, as T1 is the leftmost child of v, in that case m = n − 1 would +be the number of leafs in the forest T2, . . . , Tk, which is at most n − 1. As m = n − 1, +the only possibility would be that v is the root of S, hence m would be the first label and +therefore could not be preceded by a label 1. Hence m labels one of the trees T2, . . . , Tk, +say Ti. It follows that the label 1 preceding m is the label of the root of Ti−1. If i = 2 +then k = 2 as the label 1 is then the label of the leftmost child of v, meaning that there is +only one leaf in the forest T2, . . . , Tk. In that case, it only remains to show that v cannot +be the root of S to match the configuration in the left picture of Figure 5. But this is +clear for if v was the root of S, the last label would be m corresponding to T2, hence +no ℓ could appear. Hence v is not the root of S, and its label is ℓ. Now if i ̸= 2, then +i − 1 ̸= 1. The root v′ of Ti−1 is labelled by 1 and as v′ is not the leftmost child of v, we +have 1 = λ(v′) = m−� +w∈LD(v′) λ(w), yielding � +w∈LD(v′) λ(w) = m−1. This means that +there are m leaves in Ti−1, and as there is one leaf in Ti and m = n−1, the only possibility +is that i − 1 = 1 and k = 2, contradicting i ̸= 2. +Now, assume that S is a Schröder tree with labelling having a substring of the form +(ℓ + 1)m with 1 ≤ ℓ ≤ m − 1. Again, m can only label a leaf. Let v be the parent of that +leaf as above, which is a root of a family T1, . . . , Tk of trees with Ti corresponding to our +leaf for some i. We need to show that i ̸= 1 and k ≥ 3. In the previous case we have seen +that if i = 1, then m = n − 1 is the number of leaves in T2, . . . , Tk, forcing v to be the root +of S and m to be the first label in S. Hence i ≥ 2. If k = 2 (hence i = 2), then the root of +T1 is labelled by 1 = ℓ + 1, contradicting 1 ≤ ℓ. Hence k ≥ 2. +□ +Lemma 3.9. +(1) Let T be a Schröder tree with n leaves labelled by m ≥ n − 1. Then, +the weight of T is nm. +(2) Let w be a vertex of T which is not a leaf and v its leftmost child, that is w is the +root of a Schröder tree +� +w, (T1, · · · , Tk) +� +and v is the root of T1. Then the weight of +the forest F = (T2, · · · , Tk) attached to w is λ(v)m, and the labelling of of a vertex +in a tree Ti for i ≥ 2 is the same as its labelling inside T. +Proof. The first result is proved by induction on the number of leaves. If the tree has one +leaf the result holds by definition of our labelling. Let T = (r, T1, · · · , Tk) be a Schröder +tree, where Ti has ni leaves. By induction, the tree Ti has weight mni for i ≥ 1. Us- +ing Lemma 3.4, the sum of the labels of the vertices of the tree Ti (in T) is equal to mni +for i ≥ 2 and the sum of the labels of the vertices of T1 and of the root of T is equal to +mn1. Hence, the tree T has weight �k +i=1 mni = mn. For the second point, the number of +leaves of the forest F is equal to λ(v). Hence by the first point, the forest F has weight +λ(v)m. +□ +Proposition 3.10. Let m ≥ n − 1. Then the map Φm from the set of Schröder trees with +n leaves to the set of words for the element ρn−1(ρm)n−1ρm−n+1 ∈ Mm is injective. + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +13 +Proof. Let T = (r, T1, · · · , Tk) be a Schröder tree with n leaves labelled by m. This proof is +purely combinatorial and it only involves the word W in N obtained by reading the labels +of the tree in post-order. The first step of the proof is to remark that one can recover the +decomposition ‘root and list of subtrees’ of a Schröder tree just by looking at W. We will +illustrate the algorithm in Example 3.11 below. Precisely we want to split the word W +into a certain number of factors W = W1 · · · Wk such that each subword Wi is equal to the +word obtained by reading the labels of Lm(Ti) in post-order. +The first letter w1 of W is the label of the leftmost leaf r1 of T and by induction we will +find the letters w2, w3, · · · , wi corresponding to the ancestors r2, r3, · · · , ri of r1. Since the +labels of these vertices count a number of leaves of T, when �i +j=1 wj = n − 1, then all the +leaves of T have been counted so ri is the root of T1 and we stop the induction. +If we have found the letter wk corresponding to rk ̸= r, then wk is the number of leaves +of the right forest attached to the parent rk+1 of rk. By Lemma 3.9, the weight of F is +m · wk, hence the word obtained by reading the vertices of F is wk+1 · · · wi where i is the +smallest integer such that �i +j=k+1 wj = mwk. All these letters correspond to the vertices +of F, hence the next letter is the label of the vertex read after F in the post-order traversal, +which is the vertex rk+1. +Since the word W only contains strictly non-negative integers (except possibly the label +of the root of T), at each step of the induction the value w1 +· · ·+wi strictly increases and +the induction stops. If wn1 is the letter corresponding to the leftmost child of the root of T, +then the word w1 · · · wn1 is the word obtained by reading all the vertices of the subtree T1. +By Lemma 3.4, this is almost the word obtained by reading Lm(T1) we just need to ‘correct’ +the label of the root of T1 by adding the label of the root of T which is the last letter wl of +W. To conclude the word consisting of the labels of T1 is WT1 = w1 · · · wn1−1(wn1 + wl). +Let � +W be the word obtained by removing the letters w1, · · · , wn1 and wl. We use the +same procedure to extract the subwords corresponding to the other subtrees of T. Due +to the asymmetry of Lemma 3.4, there is a slight difference. We have found all the labels +w1, w2, · · · , wt of the vertices r1, r2, · · · , rt of the left branch of Ti when �t +j=1 wj = m and +there is no need to ‘correct’ the word as above. +We are now ready to prove that Φm is injective. +If the words of two trees T = +(r, (T1, · · · , Tk)) and S = (s, (S1, · · · , Sl)) obtained by reading the labels of their vertices +in post-order are equal, then by the discussion above we have k = l and for i ∈ {1, · · · , k}, +the words obtained by reading the vertices of the subtrees Lm(Ti) and Lm(Si) are equal. +By induction on the number of leaves, we have Si = Ti for i = 1, · · · , k and we get that +T = S. +□ +Example 3.11. We illustrate the decomposition involved in the proof of Proposition 3.10 +with the example of Figure 4. We consider the leftmost subtree T1 of T with n = 7 leaves +and which is labelled by m = 11. We have Φ11(T1) = ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5. The +first letter 1 tels us that the forest on the right of the leftmost leaf r1 has 1 vertex. Its +weight is m = 11. Hence ρ11 labels the only vertex of the forest and the next letter 5 +corresponds to the parent r2 of r1. Since 1 + 5 = 6 we know that it is the leftmost child +of the root. Hence the word ρ1ρ11ρ5 is obtained by reading the vertices of the leftmost +subtree S of T1. We apply the ‘correction’ and we get ρ1ρ11ρ10 = Φ11(S). The rest of the +word ρ1ρ11ρ10ρ2ρ11ρ11ρ9 corresponds to the other subtrees of T1 and it splits as ρ1ρ11ρ10 +and ρ2ρ11ρ11ρ9. +Combining Proposition 3.8 and Proposition 3.10 we get our main result of the section: +Theorem 3.12. For m = n − 1, the map Φm from the set of Schröder trees with n leaves +to the set of words for ρn +n−1 in Mn−1 is bijective. + +14 +THOMAS GOBET AND BAPTISTE ROGNERUD +Corollary 3.13. The following two graphs are isomorphic under Φn−1: +(1) The graph of words for ρn +n−1 in Mn−1, where vertices are given by expressions of +ρn +n−1 and there is an edge between two expressions whenever they differ by applica- +tion of a single relation, +(2) The graph of Schröder trees with n leaves, where vertices are given by Schröder trees +and there is an edge between two trees whenever they differ by application of a local +move. +Proof. The previous theorem gives the bijection between the sets of vertices. The proof of +Lemma 3.7 shows that whenever one can apply a local move, one can apply a relation on +the corresponding words. The proof of Proposition 3.8 shows that whenever one can apply +a relation on words, a local move can be applied on the corresponding trees. +□ +We illustrate the situation for M3 in Figure 6 below. +Corollary 3.14. The number of words for the Garside element of Mn is a little Schröder +number A001003 [11]. +Lemma 3.15. Let T = (r, S1, · · · , Sk) be a Schröder tree with n leaves labelled by m = n−1. +Then, the word obtained by reading all the labels of a subtree Sj is a word for ρljn where lj +is the number of leaves of Sj. +Proof. Let us assume that the tree Sj has s + 1 leaves. By Lemma 3.7, the labels of the +subtree Sj is a word for ρsρs +n−1ρn−1−s. If s = 0, then we have a word for ρn−1. Otherwise, +we can apply the relations [8, Lemma 4.5] with i = s and j = n−1. Alternatively, using the +Schröder trees, it is easy to see that these relations comes from the following modifications +of the trees. +The word ρsρs +n−1ρn−1−s correspond to the case where the tree Sj is the +Schröder bush with s + 1 leaves. Using our local moves, we can modify it to the left comb. +The corresponding word is now (ρ1ρs)sρn−1−s. Now we can inductively apply the local +move to contract the edge between the root of T and the root left comb. The result is s+1 +empty trees attached to the root of T and the corresponding word is ρs+1 +n−1. +□ +Proposition 3.16. Let n ≥ 1. We have the following isomorphisms of posets: +(1) D0 +n ∼= Div(∆n−1), Dn+1 +n +∼= Div(∆0) = {•}, +(2) For all 1 ≤ i ≤ n, Di +n ∼= Div(∆n−i), +where every set is ordered by the restriction of the left-divisibility order in the monoid Mk +for suitable k. +Proof. We begin by proving the second statement. An element x of Di +n can be written in the +form ρi +nx′, where x′ is uniquely determined by cancellativity, and such that ρn is not a left- +divisor of x′. In particular, there is y a divisor of ∆n such that ρi +nx′y = ρn+1 +n +, and y ̸= 1. We +associate a tree (or rather a family of trees) to x as follows. Write x′ as a product a1a2 · · · aj +of elements of S. Complete the word ρi +na1a2 · · · aj to a word ρi +na1a2 · · · ajb1b2 · · · bℓ for ∆n, +i.e., choose a word b1b2 · · · bℓ for y. +There are several possibilités for the bi’s, but the +condition that x ∈ Di +n ensures that, writing the corresponding Schröder tree in the form +(r, T1, T2, . . . , Ti, S1, S2, · · · Sd), where the i first trees are empty trees with a single leaf, +then a1a2 · · · aj has all its labels inside S1. Indeed, the labelling a1a2 · · · aj begins at the +beginning (in the post-order convention) of the tree S1 since the trees T1, T2, . . . , Ti yield +the label ρi +n, and if another tree among S2, . . . , Sd was partly labelled by the ai’s, then a +power of ρn would left-divide x′, since the word obtained from S1 is a power of ρn (lemma +3.15). It is then possible to reduce all the trees S1, S2, . . . , Sd to a single tree S still having +the labelling a1, a2, . . . , aj at the beginning, by first reducing S2, . . . , Sd to a set of empty + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +15 +ρ2ρ1ρ3ρ2ρ1ρ3 +ρ3ρ1ρ3ρ2ρ3 +ρ3ρ1ρ3ρ1ρ3ρ1 +ρ3ρ2ρ3ρ3ρ1 +ρ2ρ3ρ3ρ1ρ3 +ρ3ρ3ρ3ρ3 +ρ3ρ3ρ1ρ3ρ2 +ρ3ρ2ρ1ρ3ρ2ρ1 +ρ1ρ3ρ1ρ3ρ1ρ3 +ρ1ρ3ρ2ρ3ρ3 +ρ1ρ3ρ2ρ1ρ3ρ2 +1 +2 +2 +1 +3 +3 +3 +2 +1 +3 +3 +3 +1 +1 +1 +3 +3 +3 +1 +2 +3 +3 +1 +2 +3 +3 +3 +3 +3 +3 +3 +3 +3 +2 +1 +3 +3 +1 +2 +2 +1 +3 +1 +1 +1 +3 +3 +3 +2 +1 +3 +3 +3 +2 +1 +3 +2 +1 +3 +Figure 6. Illustration of Corollary 3.13 for n = 4: the graph of reduced words for +∆3 and the isomorphic graph of Schröder trees on 4 = 3 + 1 leaves. +trees �T2, . . . , �Td′ with single leafs, and then merging S1 and �T2 using a local move, then +merging the resulting tree with �T3, and so on (see Figure 7 for an illustration). In this way +we associate to x a Schröder tree of the form (r, T1, T2, . . . , Ti, S), where the Tk’s are empty +trees with a single leaf, and the labelling corresponding to the chosen word a1a2 · · · aj is an +initial section of the tree S (in fact, in algebraic terms, what we did is modify the word for +y to get a suitable one yielding a unique tree after the empty trees). Note that by initial +section we mean a prefix of the word obtained from the labelling of S read in post-order, +where we exclude the label of the root, i.e., if the root has a label, then the prefix is strict. +We denote by Sn,i the set of such Schröder trees, that is, those Schröder trees on n leaves +with i + 1 child of the root, and such that the i first child are leafs. Note that the tree +that we attached to x depends on a choice of word for x, but applying a defining relation + +16 +THOMAS GOBET AND BAPTISTE ROGNERUD +in the word x corresponds to applying a local move in the tree S, and this cannot make S +split into several trees since the root of S is frozen (its label corresponds to the last letter +of y ̸= 1). Hence we can apply all local moves with all labels in the (strict) initial section +corresponding to a word for x, and we keep a Schröder tree on n−i+1 leaves. In this way, +forgetting the i first empty trees, what we attached to x is an equivalence class of a (strict) +initial section of a Schröder tree on n − i + 1 leaves under local moves, that is, a divisor +of ∆n−i. This mapping is injective since one can recover a word for x from the obtained +Schröder tree on n − i + 1 leaves easily by mapping S to (r, T1, . . . , Ti, S), labelling such a +tree, and reading the word obtained by reading the i first empty trees and then the initial +section. +It remains to show that it is surjective. Hence consider an initial section of a Schröder +tree S on n − i leaves. We must show that, in the tree (r, T1, T2, . . . , Ti, S), the initial +section of S is a word a1a2 · · · aj which labels an element x′ of D0 +n. Assume that ρn is a +left-divisor of x′. Then, using local moves only involving those labels in the initial section +of S corresponding to a word for x′, one can transform (r, T1, T2, . . . , Ti, S) into a tree of the +form (r, T1, T2, . . . , Ti, Ti+1, S′ +1, . . . , S′ +e), i.e., S can be split into several trees, the first one +(corresponding to ρn) being an empty tree. This is a contradiction: to split S into several +trees, one would need to apply a local move involving the root of S, which is frozen since +the initial section does not cover the root. Hence x′ ∈ D0 +n, and our mapping is surjective. +This completes the proof of the second point, as it is clear that our mappings preserve +left-divisibility. +For the first point, we have Dn+1 +n += {ρn+1 +n +}, hence there is nothing to prove. To show +that D0 +n ∼= Div(∆n−1), one proceeds in a similar way as in the proof of point 1. Let x ∈ D0 +n +and let y such that xy = ∆n. Choose words for x and y, and consider the corresponding +Schröder tree T = (r, T1, . . . , Tk). Since x ∈ D0 +n, the initial section of T corresponding to +the word for x must be a proper initial section of T1. Using local moves on T2, . . . , Tk (which +amounts to changing the word for y), we can find a Schröder tree that is equivalent to T +under local moves, and that is of the form (r, �T1, �T2), where �T1 still has the chosen word +for x as a proper initial section, and �T2 is the empty tree with only one leaf. In particular +�T1 is a Schröder tree on n leaf. Applying defining relations to words for x amounts to +applying local moves inside the first tree, and arguing as in the first point this establishes +the isomorphism of posets between D0 +n and Div(∆n−1). +□ +r +Ti +S1 +�T1 +�T2 +−→ +r +Ti +r1 +S1 +�T1 +�T2 +−→ +r +Ti +r2 +r1 +S1 +�T1 +�T2 += +r +Ti +S +Figure 7. Illustration for the proof of Proposition 3.16. + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +17 +4. Enumerative results +We have already seen (Theorem 3.12) that the words for ρn+1 +n +are in bijection with +Schröder trees on n+1 leaves. In this section, we give some additional enumerative results +for several families of particular elements of Mn. +4.1. Number of simple elements. +Corollary 4.1. Let n ≥ 2, and let An := |Div(∆n)|. Then +An = 2A0 + 2An−1 + +n−2 +� +i=1 +Ai. +(4.1) +It follows that An = F2n, where F0, F1, F2, . . . denotes the Fibonacci sequence 1, 2, 3, 5, 8, ... +inductively defined by F0 = 1, F1 = 2, and Fi = Fi−1 + Fi−2 for all i ≥ 2. The sequence of +the Ans is referred as A001906 in [11]. +Proof. The equality (4.1) follows immediately from the disjoint union Div(∆n) = � +0≤i≤n+1 Di +n +and Proposition 3.16. We have A0 = F0, A1 = 3 = F2, and it is elementary to check that +the inductive formula given by 4.1 is also satisfied by the sequence F2n. This shows that +An = F2n for all n ≥ 0. +□ +Definition 4.2. We call the lattice (Div(∆n), ≤) the even Fibonacci lattice. +4.2. Number of left-divisors of the lcm of the atoms and odd Fibonacci lat- +tice. The set DivL(ρn +n) of left-divisors of ρn +n also forms a lattice under the restriction +of left-divisibility, since it is an order ideal in the lattice (Div(∆n), ≤). In terms of the +Garside monoid Mn, the element ρn +n is both the left- and right-lcm of the generators +S = {ρ1, ρ2, . . . , ρn} (see [8, Corollary 4.17]). For n ≥ 1 we set Bn := |DivL(ρn +n)|. +Lemma 4.3. We have Bn = F2n−1 for all n ≥ 1. The sequence of the Bns is referred as +A001519 in [11]. +Proof. Let x ∈ Div(∆n). We claim that x ∈ DivL(ρn +n) if and only if ρnx ∈ Div(∆n). Indeed, +if x ≤ ρn +n, there is y ∈ Mn such that xy = ρn +n. We then have ρnxy = ρn+1 +n += ∆n, hence +ρnx is a left-divisor of ∆n. Conversely, assume that ρnx ∈ Div(∆n). It follows that there +is y ∈ Div(∆n) such that ρnxy = ∆n = ρn+1 +n +. By cancellativity we get that xy = ρn +n, hence +x ∈ DivL(ρn +n). +It follows that DivL(ρn +n) is in bijection with the set +{ρnx | x ∈ Div(∆n)} ∩ Div(∆n). +But this set is nothing but � +1≤i≤n+1 Di +n. It follows that +Bn = |Div(∆n)| − |D0 +n| = |Div(∆n)| − |Div(∆n−1)|, +where the last equality follows from point (1) of Proposition 3.16. By Corollary 4.1 we +thus get that +Bn = An − An−1 = F2n − F2n−2 = F2n−1, +which concludes the proof. +□ +Definition 4.4. We call the lattice (DivL(ρn +n), ≤) the odd Fibonacci lattice. +Both lattices for M3 are depicted in Figure 1. +Remark 4.5. Note that the set of right-divisors of ρn +n also has cardinality Bn: in fact, the +two posets (DivL(ρn +n), ≤L) and (DivR(ρn +n), ≤R) are anti-isomorphic via x �→ x, where x is +the element of Mn such that xx = ρn +n (this element is unique by right-cancellativity). + +18 +THOMAS GOBET AND BAPTISTE ROGNERUD +4.3. Number of words for the divisors of the Garside element. +Lemma 4.6. Let T1 and T2 be two Schröder trees with n leaves labelled by m ≥ n − 1, and +denote by m1 and m2 the corresponding words obtained by reading the labels in post-order. +If the words m1 and m2 have a common prefix x1x2 · · · xl, then xi labels a leftmost child in +T1 if and only if it labels a leftmost child in T2. +Proof. We prove the result by induction on the number of leaves. If x1 · · · xl is obtained +by reading all the vertices of T1 = (r, S1, · · · , Sk), then m1 = x1 · · · xl = m2 and by +Proposition 3.10, we have T1 = T2, hence there is nothing to prove. Otherwise, let Sj be +the first subtree of T1 which is not covered by the word x1 · · · xl, similarly let Uk the first +subtree of T2 = (r, U1, · · · Uv) which is not covered by x1 · · · xl. Looking at the proof of +Proposition 3.10, we see that the first subtrees S1, · · · , Sj−1 are completely determined by +the word x1 · · · xl, hence we have j = k and Si = Ui for all i < k. Let xs be the letter of +x1 · · · xl labelling the first vertex of Sj. Let m′ +1 be the subword of m1 and m′ +2 the subword +of m2 starting at the xs. As explained in the proof of Proposition 3.10, we can determine +the subword mj +1 of m′ +1 which correspond to Sj. The trees Uj and Sj do not need to have the +same number of leaves. If one of the trees, say Uj, has less leaves, then one can apply local +moves in the trees Uj, Uj+1, · · · Uv as in the proof of Proposition 3.16 in order to obtain +a tree ˜Uj with the same number of leaves as Sj. This will modify the word m′ +2, but not +the prefix xs · · · xl, and xi labels a leftmost child in Uj if and only if it labels a leftmost +child in ˜Uj (see Figure 7 for an illustration). After doing the modification, we consider the +subword mj +2 corresponding to the tree ˜Uj and apply the induction hypothesis to mj +1 and +mj +2. +□ +Theorem 4.7. The set of words for the left-divisors of ρn+1 +n +is in bijection with the set of +Schröder trees with n + 2 leaves. +Proof. Let us denote by sk the number of Schröder trees with k + 1 leaves, and dk the +number of words for the divisors of ρk+1 +k +. +Recall that Div(∆n) = � +0≤i≤n+1 Di +n, and let di +n be the number of words for the elements +of Di +n. If i = n + 1, then ρn+1 +n +is the only element of Di +n and by Theorem 3.12, there are +sn words for this element, hence we have dn+1 +n += sn. +Let 0 ≤ i ≤ n and w = x1 · · · xl be a word for an element of Di +n. The word w is a strict +prefix of a Schröder tree T = (r, S1, · · · , Sk). By Lemma 2.5, w = w1w2 where w1 is a word +for ρi +n and ρn is not a left divisor of w2 (when i = 0 the word w1 is empty). Let Sj be the +last subtree of T which has a vertex labelled by a letter of w1. We can apply a succession +of defining relations to w1 in order to obtain ρi +n. These relations correspond to local move +in the trees S1, · · · , Sj which collapse all the trees S1, · · · Sj to empty trees. In order to +reduce Sj to a list of empty trees we must use its root. Since the root is always the last +label of the tree in post-order, the word w1 covers all the first j trees which have in total +i leaves. Since ρn does not divide w2, we see that w2 is a (possibly empty) strict prefix +of Sj+1. It is also possible to modify the trees Sj+2, · · · , Sk without changing the first j +trees. Indeed, as in the proof of Proposition 3.16 we can reduce the trees Sj+2, · · · , Sk to +empty trees and then merge them (until we can) to Sj+1. +• When i = 0, after modification we obtain a tree ˜T = (r, ˜S, L) where L the empty +tree, ˜S is a tree with n leaves and w = w2 is a strict prefix of ˜S. +• When 1 ≤ i ≤ n, we obtain a tree ˜T = (r, S1, · · · , Sj, �Sj+1) and w2 is a strict prefix +of the tree �Sj+1 with n + 1 − i leaves. +In both cases, the tree �Sj+1 is obtained by possibly introducing new vertices to Sj+1, +and as Figure 7 shows, these new vertices occur after the vertices of Sj+1, in post-order, + +ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID +19 +hence w2 is still a strict prefix of �Sj+1. Hence, we see that in the decomposition w = w1w2 +of Lemma 2.5, the word w1 is obtained by reading all the vertices of a Schröder tree with +i leaves and w2 is a strict prefix of a Schröder tree, denoted by ˜S, with li + 1 leaves where +li = n − i leaves if i ̸= 0 and li = n − 1 if i = 0. +Let w = w1w2 be a word of an element of Di +n with w2 having t letters. Let ˜S be a +Schröder tree with li + 1 leaves having w2 as a strict prefix. Then, we construct a word +γ(w) by first extracting ˜S, then labelling it accordingly to its number of leaves (i.e., with +m = li) and finally taking its first t letters in post-order. Algebraically, it is easy to see +how the word γ(w) is obtained from w2: if wi is the label of a leftmost child in ˜S, we have +γ(w)i = wi. Otherwise, since the tree ˜Sj has li + 1 leaves, we have γ(w)i = wi − n + (li). +A priori γ(w) depends on the choice of a tree ˜S, but Lemma 4.6 tells us that γ(w) only +depends on w2. The word γ(w) is a prefix of a Schröder tree with li + 1 leaves, hence it +is a word for a divisor of ∆li. We have obtained a map γ from the set of words for the +elements of Di +n to the set of words for the divisors of ∆li. +Conversely, if z is a word of length k for a divisor of ∆li, it is a prefix (strict since the +root is not contributing) of a Schröder tree S with li +1 leaves. We can view S as a subtree +of a Schröder tree with n + 1 leaves by considering: +• T = (r, S, L) when i = 0; +• T = (r, δi, S) when i ≥ 1. +Reading up to the first k letters of the subtree S produces a word w = w1w2 of an +element of Di +n such that γ(w) = z. Hence γ is surjective and we set ǫ(z) = w2. As before +ǫ(z) only depends on z, not on the tree having z as a prefix. +When i = 0, the map γ is injective, indeed if w and z are two words such that γ(w) = +γ(z), then by Lemma 4.6 the labels of the leftmost child in γ(w) and γ(z) are the same, +hence w and z are equal. This proves that d0 +n = dn−1. +When i ≥ 1, then γ is far from being injective, since it forgets the first part of the tree. +The set of words for the elements of Di +n is the disjoint union of two sets E1 and E2 where +E1 is the set of words w = w1w2 where w1 covers exactly one tree S1 and E2 is the set +of words where w1 covers at least two trees. Note that when i = 1, the set E2 is empty +otherwise both sets are non-empty. Indeed E2 contains at least all the words of the form +ρi +nw2 and E1 contains at least the words of the form ρi−1ρi−1 +n +ρn−i−1w2 which correspond +to the Schröder bush δi attached as the leftmost subtree of a Schröder tree. +If z is a word for a divisor of ∆li, we compute the cardinality of the preimage of z by γ +by looking at γ−1(z) ∩ E1 and γ−1(z) ∩ E2. If i = 1, we obviously only consider the first +case. The elements of γ−1(z) ∩ E1 are obtained by concatenation of the word of a single +Schröder tree with i leaves and ǫ(z), and the elements of γ−1(z) ∩ E2 are concatenation +of the words of a forest with i leaves made of at least two Schröder tree and ǫ(z). Such a +forest is nothing but a Schröder tree with i-leaves from which the root has been removed. +So we have +|γ−1(z) ∩ E1| = si−1 = |γ−1(z) ∩ E2|. +Taking the sum on all possible words z, we have d1 +n = s0 · dn−1 and di +n = 2 · si−1 · dn−i +when n ≥ i ≥ 2. +We have obtained: +d0 +n = dn−1; +d1 +n = s0 · dn−1 = dn−1; +and +di +n = 2 · si−1 · dn−i when n ≥ i ≥ 2 and dn+1 +n += sn. + +20 +THOMAS GOBET AND BAPTISTE ROGNERUD +By induction on the number of leaves, we have di = si+1, for every i ≤ n − 1, and +dn = 2sn + 2 +n +� +i=2 +si−1sn−i+1 + sn += 3sn + 2 +n−1 +� +i=1 +sisn−i. +Using generating functions, it is not difficult to check that this implies that dn = sn+1, see +for example [13, Theorem 5]. +□ +References +[1] D. Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. 36 (2003), 647-683. +[2] J. Birman, K.H. Ko, and S.J. Lee, A New Approach to the Word and Conjugacy Problems in the +Braid Groups, Adv. in Math. 139 (1998), 322–353. +[3] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271. +[4] P. Dehornoy, F. Digne, D. Krammer, E. Godelle, and J. Michel. Foundations of Garside theory, Tracts +in Mathematics 22, Europ. Math. Soc. (2015). +[5] P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, +Proc. London Math. Soc. (3) 79 (1999), no. 3, 569-604. +[6] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302. +[7] F.A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. 20 (1969), no. 2, +235–254. +[8] T. Gobet, On some torus knot groups and submonoids of the braid groups, J. Algebra 607 (2022), +Part B, 260-289. +[9] T. Gobet, A new Garside structure on torus knot groups and some complex braid groups, preprint +(2022), https://arxiv.org/abs/2209.02291. +[10] J-L. Loday, Realization of the Stasheff polytope, Arch. Math., 83 (2004), 267-278. +[11] OEIS Foundation Inc, The On-Line Encyclopedia of Integer Sequences, Published electronically at +https://oeis.org +[12] M. Picantin, Petits groupes gaussiens, PhD Thesis, Université de Caen, 2000. +[13] F. Qi, B. Guo Some explicit and recursive formulas of the large and little Schröder numbers, Arab +Journal of Mathematical Sciences Vol: 23, Issue: 2, Page: 141-147 (2017). +Institut Denis Poisson, CNRS UMR 7350, Faculté des Sciences et Techniques, Université +de Tours, Parc de Grandmont, 37200 TOURS, France +Email address: thomas.gobet@lmpt.univ-tours.fr +Institut de Mathématiques de Jussieu, Paris Rive Gauche (IMJ-PRG), Campus des Grands +Moulins, Université de Paris - Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 PARIS +Cedex 13, France +Email address: baptiste.rognerud@imj-prg.fr + diff --git a/39AyT4oBgHgl3EQf1_mj/content/tmp_files/load_file.txt b/39AyT4oBgHgl3EQf1_mj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..117fd86cf22372a9ab2dcccf283fdf4db743dd69 --- /dev/null +++ b/39AyT4oBgHgl3EQf1_mj/content/tmp_files/load_file.txt @@ -0,0 +1,917 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf,len=916 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='00744v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='CO] 2 Jan 2023 ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID THOMAS GOBET AND BAPTISTE ROGNERUD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schröder trees, and on a recursive description of the even Fibonacci lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This yields an explicit formula to calculate meets and joins in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As a byproduct we also obtain that the number of words for the Garside element is given by a little Schröder number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition and structure of the poset 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition of the poset 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lattice property 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Schröder trees and words for the Garside element 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' labelling of Schröder trees 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Words for the Garside element in terms of Schröder trees 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Enumerative results 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Number of simple elements 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Number of left-divisors of the lcm of the atoms and odd Fibonacci lattice 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Number of words for the divisors of the Garside element 18 References 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Introduction Several algebraic structures naturally yield examples of lattices: as elementary examples, one can cite the lattice of subsets of a given set ordered by inclusion, or the lattice of subgroups of a given group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' One can then study which properties are satisfied by the obtained lattices, or conversely, starting from a known lattice, wondering for instance if it can be realized in a given algebraic framework, or if a property of the lattice implies properties of the attached algebraic structure(s) and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The aim of this paper is to give a combinatorial description of a finite lattice that appeared in the framework of Garside theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We will not recall results and principles of Garside theory as they will not be used in this paper, but the interested reader can look at [5, 4] for more on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This is a branch of combinatorial group theory which aims at establishing properties of families of infinite groups such as the solvability of the word problem, the conjugacy problem, the structure of the center, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Roughly speaking, a Garside group is a group of fraction of a monoid (called a Garside monoid) with 2 THOMAS GOBET AND BAPTISTE ROGNERUD particularly nice divisibility properties, which ensures that the above-mentioned problems can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Such a monoid M has no nontrivial invertible element, and comes equipped with a distinguished element ∆ (called a Garside element) whose left- and right-divisors are finite, coincide, generate the monoid, and form a lattice under left- and right-divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The left- or right-divisors of ∆ are called the simples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The fundamental example of a Garside group is the n-strand Artin braid group [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It admits several non-equivalent Garside structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=', nonisomorphic Garside monoids whose group of fractions are isomorphic to the n-strand braid group), and the lattice of simples in the first discovered such Garside structure is isomorphic to the weak Bruhat order on the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Several widely studied lattices can be realized as lattices of simples of a Garside monoid: this includes the lattices of left and right weak Bruhat order on any finite Coxeter group [3, 6], the lattice of (generalized) noncrossing partitions attached to a finite Coxeter group [1, 2], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (see also [12] for many other examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This suggests the following question: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Which lattices can appear as lattices of simples of Garside monoids ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The aim of this paper is to study a family Pn of lattices arising as simples of a family Mn, n ≥ 2 of Garside monoids introduced by the first author [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For n = 2, the corresponding Garside group is isomorphic to the 3-strand braid group B3, while in general it is isomorphic to the (n, n + 1)-torus knot group, which for n > 3 is a (strict) extension of the (n + 1)- strand braid group Bn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The lattice property of Pn follows from the fact proven in op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' that Mn is a Garside monoid, but it gives very little information about the structure and properties of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For instance, one does not have a formula enumerating the number of simples, and only an algorithm to calculate meet and joins in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In Section 2 we give a new proof of the lattice property of Pn (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8) by exhibiting the recursive structure of the poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Every lattice Pn turns out to contain the lattices Pi, i < n as sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that an ingredient of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8 is proven later on in the paper, as it relies on a combinatorial description for the set of words for the Garside element in terms of Schröder trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' More precisely, in Section 3 we establish a simple bijection between the set of words for ∆n and the set of Schröder trees on n+1 leaves, in such a way that applying a defining relation of Mn to a word amounts to applying what we call a "local move" on the corresponding Schröder tree (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='12 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' These local moves are given by specific edge contraction and are related to the notion of refinement considered in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This allows us to establish in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16 an isomorphism of posets between subposets of Pn and Pi, i < n, required in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Finally, the obtained recursive description of Pn together with the description of words for ∆n in terms of Schröder trees allows us to derive a few enumerative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This is done in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The first one is that the number of elements of Pn is given by F2n, where Fi is the i-th Fibonacci number (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We thus call Pn the even Fibonacci lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The atoms of Mn turn out to have the same left- and right-lcm, which is strictly less than ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We also show that the sublattice of Pn defined as the order ideal of this lcm has F2n−1 elements (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3), and thus call it the odd Fibonacci lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Other enumerative results include the determination of the number of words for the Garside elements (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='14), and the number of words for the whole set of simples (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Recall that the Garside monoid Mn under study in this paper has group of fractions isomorphic to the (n, n + 1)-torus knot group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This Garside structure was generalized to all torus knot groups in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It would be interesting to have a description of the lattices of simples of this bigger family of Garside monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 3 1 ρ1 ρ2 ρ2ρ1 ρ3 ρ1ρ3 ρ3ρ1 ρ1ρ3ρ1 ρ3ρ2 ρ2ρ1ρ3 ρ3ρ2ρ1 ρ2 3 ρ3ρ1ρ3 ρ2 3ρ1 (ρ1ρ3)2 (ρ3ρ1)2 ρ3 3 ρ3ρ2ρ1ρ3 ρ2 3ρ1ρ3 (ρ3ρ1)2ρ3 ρ4 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The even Fibonacci lattice for n = 3 and (in blue) the odd Fibonacci lattice inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition and structure of the poset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition of the poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The beginning of this section is devoted to explaining how the poset under study is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We recall the definition of the monoid from which it is built, as well as a few properties of this monoid (all of which are proven in [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let M be a monoid and a, b ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We say that a is a left divisor of b (or that b is a right multiple of a) if there is c ∈ M such that ac = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We similarly define right divisors and left multiples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let M0 be the trivial monoid and for n ≥ 1, let Mn be the monoid defined by the presentation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1) � ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ρn ���� ρ1ρnρi = ρi+1ρn for all 1 ≤ i ≤ n − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We denote by S the set of generators {ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ρn}, and by R the defining relations of Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This monoid was introduced by the first author in [8, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that this monoid is equipped with a length function λ : Mn −→ Z≥0 given by the multiplicative extension of λ(ρi) = i for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , n, which is possible since the defining relations do not change the length of a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As a corollary, the only invertible element in Mn is 4 THOMAS GOBET AND BAPTISTE ROGNERUD the identity, and the left- and right-divisibility relations are partial orders on Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We will write a ≤L b or simply a ≤ b if a left-divides b, and a ≤R b if a right-divides b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This monoid was shown to be a so-called Garside monoid (see [8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='18]), with corresponding Garside group (which has the same presentation as Mn) isomorphic to the (n, n + 1)-torus knot group, that is, the fundamental group of the complement of the torus knot Tn,n+1 in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Garside monoids have several important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Among them, the left- and right-divisibility relations equip Mn with two lattice structures, and Mn comes equipped with a distinguished element ∆n, called a Garside element, which has the following two properties (1) The set of left divisors of ∆n coincides with its set of right divisors, and forms a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (2) The set of left (or right) divisors of ∆n generates Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This Garside element is given by ∆n = ρn+1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular, as any Garside monoid is a lattice for both left- and right-divisibility, the set Div(∆n) of left (or right) divisors of ∆n is a finite lattice if equipped by the order relation given by the restriction of left- (or right-) divisibility on Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The set Div(∆n) is the set of simple elements or simples of Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In general (Div(∆n), ≤L) and (Div(∆n), ≤R) will not be isomorphic as posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' But we always have (Div(∆n), ≤L) ∼= (Div(∆n), ≤R)op (see for instance [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' such a property holds in any Garside monoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We will give a new proof that (Div(∆n), ≤) (and hence (Div(∆n), ≤R) is a lattice, in a way which will exhibit a recursive structure of the poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' To this end, we will require (sometimes without mentioning it) a few basic results on the monoid Mn which are either explained above or proven in [8]: (1) The left- and right-divisibility relations on Mn are partial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (2) The monoid Mn is both left- and right-cancellative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=', for a, b, c ∈ Mn, we have that ab = ac ⇒ b = c, and ba = ca ⇒ b = c (see [8, Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='12]), (3) The set of left- and right-divisors of ∆n coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In fact, the element ∆n is central in Mn, hence as Mn is cancellative, for a, b ∈ Mn such that ab = ∆n, we have ab = ba (see [8, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='15]) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lattice property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The aim of this subsection is to prove a few properties of simple elements of Mn, and to derive a new algebraic proof that Div(∆n) is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x1x2 · · · xk be a word for ∆n, with xi ∈ S for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' There are i1 = 1 < i2 < · · · < iℓ ≤ k such that For all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ℓ, the word yj := xijxij+1 · · · xij+1−1 (with the convention that iℓ+1 = k + 1) is a word for a power of ρn, The decomposition y1|y2| · · · |yℓ of the word x1x2 · · · xk is maximal in the sense that no word among the yj can be decomposed as a product of two nonempty words which are words for powers of ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Morever, a decomposition with the above properties is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The existence of the decomposition is clear using the fact that Mn is cancellative: given the word x1x2 · · · xk, consider the smallest i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , k} such that x1x2 · · · xk is a word for a power of ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Such an i has to exist, as x1x2 · · · xk is a word for a power of ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then set i2 := i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By cancellativity in Mn, since x1 · · · xi and x1 · · · xk are both words for a power of ρn, the word xi+1 · · · xk must also be a word for a power of ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence one can go on, arguing the same with the word xi+1 · · · xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Again by cancellativity, this decomposition must be maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 5 Now assume that the decomposition is not unique, that is, assume that y1|y2| · · · |yℓ and z1|z2| · · · |zℓ′ are two decompositions of the word x1x2 · · · xk satisfying the properties of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As both y1 and z1 are words for a power of ρn, if y1 ̸= z1, then one word must be strict prefix of the other, say z1 is a strict prefix of y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' But this contradicts the maximality of the decomposition y1|y2| · · · |yℓ: indeed if y1 = x1x2 · · · xi2−1 and z1 = x1x2 · · · xp with p < i2 − 1, we can decompose y1 nontrivially as x1x2 · · · xp|xp+1 · · · xi2−1, and by cancellativity both x1 · · · xp and xp+1 · · · xi2−1 are words for powers of ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Consider the word ρ3ρ1ρ7ρ1ρ7ρ5ρ4ρ7ρ7ρ1ρ7ρ6 in M7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We claim that this is a word for the Garside element ρ8 7 of M7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed, using the defining relation ρ1ρ7ρi = ρi+1ρ7 with i = 5 and 6, we get that ρ3(ρ1ρ7ρ1ρ7ρ5)ρ4ρ7ρ7(ρ1ρ7ρ6) = ρ3ρ3 7ρ4ρ4 7, and we observe also applying defining relations that ρ3ρ3 7ρ4ρ4 7 = ρ1ρ7ρ2ρ2 7ρ4 = ρ1ρ7ρ1ρ7ρ1ρ7ρ4 = ρ1ρ7ρ1ρ7ρ5ρ7 = ρ1ρ7ρ6ρ2 7 = ρ4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The decomposition according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1 is given by ρ3ρ1ρ7ρ1ρ7ρ5ρ4 � �� � :=y1 | ρ7 ���� :=y2 | ρ7 ���� :=y3 | ρ1ρ7ρ6 � �� � :=y4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is indeed clear by considering λ(u) for u prefixes of y1 or y4 that whenever u is a proper prefix, we do not have λ(u) equal to a multiple of 7, which is a necessary condition for a word to represent a power of ρ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then S ∩ {x ∈ Div(∆n) | x ≤ ρkρk n} = {ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ρk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We argue by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The result is clear for k = 1, as no defining relation of Mn can be applied to the word ρ1ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now let k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Observe that ρkρk n = (ρ1ρn)k = (ρ1ρn)(ρ1ρn)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular we have ρ1 ≤ ρkρk n and by induction, we get ρ1ρnρi ≤ ρkρk n for all 1 ≤ i ≤ k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As ρ1ρnρi = ρi+1ρn we get that {ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ρk} ⊆ S ∩ {x ∈ Div(∆n) | x ≤ ρkρk n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It remains to show that no other ρi can be a left-divisor of ρkρk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence assume that i > k and ρi ≤ ρkρk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence there is a word x1x2 · · · xp for ρkρk n, where xi ∈ S for all i, such that x1 = ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As the words x1x2 · · · xp and ρkρk n represent the same element, they can be related by a finite sequence of words w0 = x1x2 · · · xp, w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , wq = ρkρk n, where each wi is a word with letters in S and wi+1 is obtained from wi by applying a single relation somewhere in the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As the first letter of w0 differs from the first letter of wq, there must exist some 0 ≤ ℓ < q such that wℓ begins by ρi but wℓ+1 does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that the relation allowing one to pass from wℓ to wℓ+1 has to be applied at the beginning of the word wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' But the only possible relation with one side beginning by ρi is ρiρn = ρ1ρnρi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that ρ1ρnρi−1 ≤ ρkρk n = (ρ1ρn)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By cancellativity, we get that ρi−1 ≤ (ρ1ρn)k−1 = ρk−1ρk−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By induction this forces one to have i − 1 ≤ k − 1, contradicting our assumption that i > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Similarly, we have Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then S ∩ {x ∈ Div(∆n) | x ≤R ρk n} = {ρn, ρn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ρn−k+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 6 THOMAS GOBET AND BAPTISTE ROGNERUD Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As for Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3, we argue by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The result is clear for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence assume that k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As (ρ1ρn)n−jρj = ρn−j+1 n , we get that ρj ≤R ρk n for all j such that n − j + 1 ≤ k, that is, for all j ≥ n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It remains to show that no other ρj can right-divide ρk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence assume that ρj ≤R ρk n, where j < n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Arguing as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3, we see that ρ1ρnρj = ρj+1ρn must be a right-divisor of ρk n, hence by cancellativity that ρj+1 ≤R ρk−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By induction this forces j + 1 ≥ n − k + 2, contradicting our assumtion that j < n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ For x ∈ Div(∆n), let d(x) := max{k ≥ 0 | ρk n ≤ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let 0 ≤ i ≤ n + 1 and let Di n := {x ∈ Div(∆n) | d(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that Div(∆n) = � 0≤i≤n+1 Di n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have Dn n = {ρn n}, Dn+1 n = {∆n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x ∈ Div(∆n) and i = d(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x′ ∈ Mn such that x = ρi nx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that x′ ∈ D0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x1x2 · · · xk be a word for x, where xi ∈ S for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then there is 1 ≤ ℓ ≤ k such that x1x2 · · · xℓ is a word for ρi n (and hence xℓ+1 · · · xk is a word for x′ by cancellativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In other words, any word for x has a prefix which is a word for ρi n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It suffices to show that if z1z2 · · · zp is an expression for x such that z1z2 · · · zq is an expression for ρi n (q ≤ p, then one cannot apply a defining relation of Mn on the word z1z2 · · · zp simultaneously involving letters of the word z1z2 · · · zq and letters of the word zq+1 · · · zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let us consider the three possible cases where this could occur: one could have ρ1ρn|ρj, ρ1|ρnρj, or ρj+1|ρn (1 ≤ j < n), where the | separates the letters zq and zq+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The last two cases cannot happen, since one would have zq+1 = ρn, hence zq+1 · · · zp would be a word for x′ beginning by ρn, contradicting the fact that x′ ∈ D0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It remains to show that the case ρ1ρn|ρj cannot happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence assume that zq−1 = ρ1, zq = ρn, zq+1 = ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By cancellativity, as z1z2 · · · zq is a word for ρi n, it implies that ρ1 ≤R ρi−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4, this implies that n − (i − 1) + 1 = 1, hence that i = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since x ∈ Div(∆n) and x = ρn+1 n x′ = ∆nx′, we get x′ = 1, contradicting the fact that zq+1 = ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let i, j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , n + 1}, with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x ∈ Di n, y ∈ Dj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Assume that x ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then i < j and x < ρj n ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is clear that i < j, since ρi n ≤ y as ρi n ≤ x, hence j < i would contradict y ∈ Dj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular x < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x′, y′ such that x = ρi nx′ and y = ρj ny′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that x′, y′ both lie in D0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since x ≤ y and Mn is cancellative, we get that x′ < ρj−i n y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It implies that there exists a word x1x2 · · · xk for ρj−i n y′ (xi ∈ S) and 1 ≤ ℓ < k such that x1x2 · · · xℓ is a word for x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now by lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5, there is 0 ≤ ℓ′ ≤ k such that x1x2 · · · xℓ′ is a word for ρj−i n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If ℓ′ ≤ ℓ, then ρj−i n ≤ x′, contradicting the fact that x′ ∈ D0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence ℓ′ > ℓ, and x′ < ρj−i n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Multiplying by ρi n on the left we get x < ρj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let z1, z2 ∈ Di n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let 1 ≤ k1 < k2 ≤ n and assume that there are two cover relations z1 ≤· ρk1 n , z2 ≤· ρk2 n in (Div(∆n), ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then z1 < z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As z1 ≤ ρk1 n , z2 ≤ ρk2 n are cover relations, there are 1 ≤ j1, j2 ≤ n such that z1ρj1 = ρk1 n , z2ρj2 = ρk2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4, for ℓ ∈ {1, 2} we have jℓ ∈ {n − kℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , n} and ρkℓ n = ρjℓ+kℓ−1−n n (ρ1ρn)n−jℓρjℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular, we have zℓ = ρjℓ+kℓ−1−n n (ρ1ρn)n−jℓ ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 7 and as (ρ1ρn)n−jℓ = ρn−jℓρn−jℓ n , by lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3 we see that ρn cannot be a left divisor of (ρ1ρn)n−jℓ, and hence that d(zℓ) = jℓ + kℓ − 1 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' But d(zℓ) = i for ℓ ∈ {1, 2}, and since k1 < k2 we deduce that j1 > j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since zℓ = ρi n(ρ1ρn)n−jℓ, we get that z1 < z2, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The poset (Div(∆n), ≤) is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Given i, j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , n+1} with i ≤ j and x ∈ Di n, y ∈ Dj n, we have x ∧ y = x ∧i �� i {z ∈ Di n | z ≤ y} � , where ∨i and ∧i denote the meet and join on the restriction of the left-divisibility order on Di n, which itself forms a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that if i = j we simply get x ∧ y = x ∧i y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The proof is by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have Div(∆0) = {•}, and Div(∆1) = {1, ρ1, ρ2 1}, which is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence assume that n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16 below, the restriction of the left-divisibility to Di n yields an isomorphism of poset with Div(∆n−i) if i ̸= 0, n+1, while the restriction to D0 n yields an isomorphism of poset with Div(∆n−1), and the restriction to Dn+1 n an isomorphism of posets with Div(∆0) = {•}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular, by induction, all these posets are lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As the poset (Div(∆n), ≤) is finite and admits a maximal element, it suffices to show that x ∧ y as defined by the formula above is indeed the join of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is clear that x∧y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let us show that x∧y ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If i = j this is clear, hence assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6 we see that � i {z ∈ Di n | z ≤ y} = � i {z ∈ Di n | z ≤ ρj n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It suffices to check that � i{z ∈ Di n | z ≤ ρj n} ≤ ρj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that � i {z ∈ Di n | z ≤ ρj n} = � i {z ∈ Di n | z ≤ ρj n and (z ≤· x ≤ ρj n ⇒ x /∈ Di n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now by lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6, if z ∈ Di n and x is any element such that z ≤· x ≤ ρj n and x /∈ Di n, then x = ρk n for some k (necessarily smaller than or equal to j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It implies that � i {z ∈ Di n | z ≤ ρj n} = � i {z ∈ Di n | z ≤ ρj n and z ≤· ρk n for some k ≤ j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7, we have that � i {z ∈ Di n | z ≤ ρj n and z ≤· ρk n for some k ≤ j} has to be an element of the set {z ∈ Di n | z ≤ ρj n and z ≤· ρk n for some k ≤ j}, hence that it is in particular a left-divisor of ρj n (and hence of y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now assume that u ≤ x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We can assume that u ∈ Di n, otherwise by lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6 we have u < ρi n ≤ x ∧ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As u ≤ y, we have that u ≤ � i{z ∈ Di n | z ≤ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' And hence, that u ≤ x ∧i �� i{z ∈ Di n | z ≤ y} � = x ∧ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Schröder trees and words for the Garside element 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' labelling of Schröder trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' A rooted plane tree is a tree embedded in the plane with one distinguished vertex called the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The vertices of degree 1 are called the leaves of the tree and the other vertices are called inner vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' One can consider rooted trees as directed graphs by orienting the edges from the root toward the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If there is an oriented edge from a vertex v to a vertex w, we say that v is the parent of w and w is a child of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As can be seen in Figure 2, we draw the trees with their root on the top and the 8 THOMAS GOBET AND BAPTISTE ROGNERUD leaves on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The planar embedding induces a total ordering (from left to right) on the children of each vertex, hence we can speak about the leftmost child of a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Alternatively one has a useful recursive definition of a rooted plane tree: it is either the empty tree with no inner vertex and a single leaf or a tuple T = (r, Tr) where r is the root vertex and Tr is an ordered list of rooted plane trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If T is a tree with the first definition, the vertex r is its root and the list Tr is the list of subtrees, ordered from left to right, obtained by removing the root r and all the edges adjacent to r in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (1) A Schröder tree is a rooted plane tree in which each inner vertex has at least two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (2) A binary tree is a rooted plane tree in which each inner vertex has exactly two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (3) The size of a tree is its number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (4) The height of a tree is the number of vertices in a maximal chain of descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (5) The Schröder tree on n leaves in which every child of the root is a leaf is called the Schröder bush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We denote it by δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (6) The Schröder tree given by the binary tree in which every right child (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' every left child) is a leaf is called a left comb (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' a right comb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' · · Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' From left to right: the unique Schröder tree with 1 leaf, the unique Schröder tree with two leaves, the three Schröder trees with 3 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then the Schröder bush and on its right a left comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The Schröder trees are counted by the so-called little Schröder numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The sequence starts with 1, 1, 3, 11, 45, 197, 903, 4279, 20793, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' and is referred as A001003 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We will label (and read the labels of) the vertices and the leaves of our trees using the so-called post-order traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This is a recursive algorithm that visits each vertex and leaf of the tree exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Concretely, if T = � r, (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk) � is a rooted planar tree, then we recursively apply the algorithm to T1, T2 until Tk and finally we visit the root r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' When the algorithm meets an empty tree it visits its leaf and then, the recursion stops and it goes up one level in the recursive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The first vertex visited by the algorithm is the leftmost leaf of T, then the algorithm moves to its parent v (but does not visit v) and visits the second subtree of v starting with the leftmost leaf and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We refer to Figure 3 for an illustration where the first vertex visited by the algorithm is labeled by 1, the second by 2 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The last vertex visited by the algorithm is always the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let m, n be two integers such that m ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We then label a Schröder tree T with n ≥ 2 leaves by labelling its vertices one after the other with respect to the total order defined by the post-order traversal, using the following rules: (1) Let v be the leftmost child of a vertex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then w is the root of a Schröder tree � w, (T1, · · · , Tk) � and v is the root of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The label λ(v) of v is equal to the number of leaves of the forest consisting of all the trees T2, · · · , Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 9 20 11 3 1 2 6 4 5 10 7 8 9 12 19 15 13 14 18 16 17 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Post-order traversal of a Schröder tree of size 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (2) If v is not the leftmost child of a vertex of T, we consider LD(v) the set of its leftmost descendants consisting of the leftmost child of v and its leftmost child and so one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then the label of v is m − � w∈LD(v) λ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that using the post-order traversal, the label of the leftmost descendants of a vertex v are already determined when we visit v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The result is a labelled Schröder tree that we denote by Lm(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This procedure is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let Lm(T) be a labelled Schröder tree with n leaves labelled by m ≥ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The sum of the labels of the vertices of T is called its weight (with respect to m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T be a Schröder tree with n leaves and m ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then the integers labelling Lm(T) are strictly nonnegative with the exception of the root which may be labelled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If a vertex is a leftmost child, then its label is a number of leaves, hence it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If v is not a leftmost child, then it is labelled by m − � w∈LD(v) λ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Each λ(w) is equal to a certain number of leaves of T and the set of leaves associated to distinct vertices of LD(v) do not intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Moreover, exactly one element of LD(v) is a leaf and this leaf is not counted in � w∈LD(v) λ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We therefore have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1) \uf8eb \uf8ed � w∈LD(v) λ(w) \uf8f6 \uf8f8 + 1 ≤ n, hence m − � w∈LD(v) λ(w) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Moreover if � w∈LD(v) λ(w) = m, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1) we must have m = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that v has n descendants since the leftmost leaf which is a descendant of v is not counted, hence v is the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ This labelling is almost determined by the recursive structure of the tree, as shown by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T = � r, (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk) � be a Schröder tree and v be a vertex of Ti for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, (1) If v is not the root of T1, then its label in Lm(T) is equal to its label in Lm(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (2) If v is the root of T1, then its label in Lm(T1) is equal to the sum of the labels of v and of the root of T in Lm(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let v be a vertex of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If v is a leftmost child in T which is not the root of T1, then its label is a number of leaves of a certain forest which is contained in Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence this number is the same in the big tree T or in the extracted tree Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If v is not a leftmost child, then its label is determined by the labels of its leftmost descendants, hence it is the same in the tree T as in the extracted tree Ti since we have just shown that the labels of leftmost descendants which are not the root of T1 agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The root of T1 has a different behaviour since in T it is a leftmost child and this is not the case in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence if v is the root of T1, denoting by λ1 the label of v in T1, we have λ1(v) = m − � w∈LD(v) λ1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The 10 THOMAS GOBET AND BAPTISTE ROGNERUD labels of the descendants of v are the same in T and in T1, that is, we have λ1(w) = λ(w) for all w ∈ LD(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In T, the label of the root r is given by λ(r) = m − λ(v) − � w∈LD(v) λ(w) = m − λ(v) − � w∈LD(v) λ1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence we have λ(r) + λ(v) = λ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Words for the Garside element in terms of Schröder trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Reading the la- belled tree Lm(T) using the post-order traversal and associating the generator ρi to the letter i with the convention that ρ0 = e, gives a map Φm from the set of Schröder trees labelled by m to the set S⋆ of words for the elements of the monoid Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We refer to Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 0 5 5 1 11 10 1 11 9 2 11 11 11 8 2 1 11 10 1 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Example of the labelling of a Schröder tree of size 12 with m = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The corresponding element in the monoid M11 is ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5ρ11ρ1ρ11ρ2ρ1ρ11ρ10ρ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T be a non-empty Schröder tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If T has a subtree T1 satisfying the three following properties: (1) The root r1 of T1 is not the root of T, hence it has a parent r0 which has at least two children, (2) The root r1 has exactly two children, (3) The right subtree of T1 is the empty tree with only one leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, we can construct another tree �T by contracting the edge r0 − r1, in other words by removing the root r1 of T1 and attaching the two subtrees of T1 to r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' See Figure 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We call such a transformation, or the inverse transformation, a local move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that, since r0 has at least two children in the configuration described above (see also the left picture in Figure 5), we get that r0 has at least three children in the configuration obtained after applying the local move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular, to apply a local move in the other direction, we need to have a Schröder tree �T with a subtree T1 satisfying : (1) The parent r0 of T1 (which is allowed to be the root of T) has at least three children, (2) The tree T1 is not the last child of r0, and is directly followed by an empty tree with only one leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' r0 Sk r1 r2 A1 Sk+2 ←→ r0 Sk r2 A1 Sk+2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Local move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 11 Formally, if the subtree of T with root r0 is S = � r0, (S1, · · · , Sk, T1, Sk+2, · · · , Sr) � and the subtree T1 is � r1, (A1, A2) � , then we obtain the tree �T by replacing S by � r0, (S1, · · · , Sk, A1, A2, Sk+2, · · · , Sr) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T and S be two Schröder trees with n leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then one can pass from the tree T to the tree S by applying a sequence of local moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is enough to show that T can be transformed into the Schröder bush δn–recall that this is the Schröder tree in which every child of the root is a leaf–by a sequence of local moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The Schröder tree S can then be transformed as well into δn, and hence T can be transformed into S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We argue by induction on the number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For n = 1 and n = 2 there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If T has a subtree S which is not of the form δk, then we can transform S into δk for some k by applying the induction hypothesis to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence we can assume that T = � r, (δn1, · · · , δnk) � with � ni = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If T is not equal to δn, then it has at least a non-empty subtree S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If S has only two leaves, then we can apply a local move to remove its root and to attach the two leaves to the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If it has more than 3 leaves, by induction there is a sequence of local moves from S to a left comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, by repeatedly applying a local move at the root of the left comb, we remove all the inner vertices of the left comb and attach all its leaves to the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Applying this to all subtrees S of T wich are not empty, we end up getting δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T be a Schröder tree with n leaves and m ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then Φm(T) is a word for ρn−1(ρm)n−1ρm−n+1 in Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular if m = n − 1, then it is a word for the Garside element of Mn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If T = δn is the Schröder tree with only one root and n leaves, then Φm(T) = ρn−1(ρm)n−1ρm−n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If T is another Schröder tree, then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6 there is a sequence of local moves from T to δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' To finish the proof it is enough to show that applying a local move to a Schröder tree T amounts to applying a relation of the monoid Mm to Φm(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This is easily obtained by staring at Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed, if T is the tree at the left of Figure 5, then the label of r2 is 1, the label of the leaf on its right is m and the label of r1 is a certain integer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since r1 is not the root of T, we have 1 ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Moreover, since r1 is not a leaf of T, we have ℓ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence in Φm(T) we have the factor ρ1ρmρℓ with 1 ≤ ℓ ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If �T denotes the right tree of Figure 5, then the label of r2 is ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed r2 is a leftmost child in �T if and only if r1 is a leftmost child in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In this case its label is the number of leaves of the forest in its right and in �T there is precisely one more leaf in this forest than in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In the other case, the label of r2 in �T is m − � w∈LD(r2) λ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The label of r1 is ℓ = m − 1 − � w∈LD(r2) λ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' So the label of r2 is ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The leaf on the right of r2 in �T is labelled by m, hence Φm( �T) is obtained by replacing ρ1ρmρℓ in Φm(T) by ρℓ+1ρm, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For m = n−1, the map Φm from the set of Schröder trees with n leaves to the set of words for ρn n−1 in Mn−1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have to show that to each word y for ρn n−1 ∈ Mn−1, we can attach a Schröder tree T with n leaves, in such a way that Φm(T) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The word y and the word ρn n−1 can be transformed into each other by applying a sequence of defining relations of Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We already know that the word ρn n−1 is in the image of Φm since it is the image of the Schröder bush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' To conclude the proof, we therefore need to show the following claim: given a Schröder tree S, if the corresponding labelling has a substring of the form 1mℓ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (ℓ + 1)m) with 1 ≤ ℓ ≤ m − 1, then we are necessarily in the configuration of the left 12 THOMAS GOBET AND BAPTISTE ROGNERUD picture in Figure 5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' the right picture), and hence we can apply a local move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed, as one can pass from the word ρn n−1 to the word y by a sequence of defining relations let y0 = ρn n−1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , yk = y be expressions of ρn n−1 such that yi is obtained from yi−1 by applying a single relation in Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Applying the relation on y0 = Φm(T) to get y1 corresponds to applying a local move on T to get a Schröder tree T1 and as seen in the proof of lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7, we get Φm(T1) = y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' To show the claim, assume that S is a Schröder tree with labelling having a substring of the form 1mℓ with 1 ≤ ℓ ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that m can only be the label of a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let v be the parent of that leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is a root of a family of trees, say (v, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk) and our leaf with label m corresponds to one of the trees Ti (which has to be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is clear that such a tree cannot be T1: indeed, as T1 is the leftmost child of v, in that case m = n − 1 would be the number of leafs in the forest T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk, which is at most n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As m = n − 1, the only possibility would be that v is the root of S, hence m would be the first label and therefore could not be preceded by a label 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence m labels one of the trees T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk, say Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that the label 1 preceding m is the label of the root of Ti−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If i = 2 then k = 2 as the label 1 is then the label of the leftmost child of v, meaning that there is only one leaf in the forest T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In that case, it only remains to show that v cannot be the root of S to match the configuration in the left picture of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' But this is clear for if v was the root of S, the last label would be m corresponding to T2, hence no ℓ could appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence v is not the root of S, and its label is ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now if i ̸= 2, then i − 1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The root v′ of Ti−1 is labelled by 1 and as v′ is not the leftmost child of v, we have 1 = λ(v′) = m−� w∈LD(v′) λ(w), yielding � w∈LD(v′) λ(w) = m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This means that there are m leaves in Ti−1, and as there is one leaf in Ti and m = n−1, the only possibility is that i − 1 = 1 and k = 2, contradicting i ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now, assume that S is a Schröder tree with labelling having a substring of the form (ℓ + 1)m with 1 ≤ ℓ ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Again, m can only label a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let v be the parent of that leaf as above, which is a root of a family T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk of trees with Ti corresponding to our leaf for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We need to show that i ̸= 1 and k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In the previous case we have seen that if i = 1, then m = n − 1 is the number of leaves in T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk, forcing v to be the root of S and m to be the first label in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If k = 2 (hence i = 2), then the root of T1 is labelled by 1 = ℓ + 1, contradicting 1 ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (1) Let T be a Schröder tree with n leaves labelled by m ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, the weight of T is nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (2) Let w be a vertex of T which is not a leaf and v its leftmost child, that is w is the root of a Schröder tree � w, (T1, · · · , Tk) � and v is the root of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then the weight of the forest F = (T2, · · · , Tk) attached to w is λ(v)m, and the labelling of of a vertex in a tree Ti for i ≥ 2 is the same as its labelling inside T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The first result is proved by induction on the number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If the tree has one leaf the result holds by definition of our labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T = (r, T1, · · · , Tk) be a Schröder tree, where Ti has ni leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By induction, the tree Ti has weight mni for i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Us- ing Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4, the sum of the labels of the vertices of the tree Ti (in T) is equal to mni for i ≥ 2 and the sum of the labels of the vertices of T1 and of the root of T is equal to mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence, the tree T has weight �k i=1 mni = mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For the second point, the number of leaves of the forest F is equal to λ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence by the first point, the forest F has weight λ(v)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let m ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then the map Φm from the set of Schröder trees with n leaves to the set of words for the element ρn−1(ρm)n−1ρm−n+1 ∈ Mm is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T = (r, T1, · · · , Tk) be a Schröder tree with n leaves labelled by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This proof is purely combinatorial and it only involves the word W in N obtained by reading the labels of the tree in post-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The first step of the proof is to remark that one can recover the decomposition ‘root and list of subtrees’ of a Schröder tree just by looking at W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We will illustrate the algorithm in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Precisely we want to split the word W into a certain number of factors W = W1 · · · Wk such that each subword Wi is equal to the word obtained by reading the labels of Lm(Ti) in post-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The first letter w1 of W is the label of the leftmost leaf r1 of T and by induction we will find the letters w2, w3, · · · , wi corresponding to the ancestors r2, r3, · · · , ri of r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since the labels of these vertices count a number of leaves of T, when �i j=1 wj = n − 1, then all the leaves of T have been counted so ri is the root of T1 and we stop the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If we have found the letter wk corresponding to rk ̸= r, then wk is the number of leaves of the right forest attached to the parent rk+1 of rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='9, the weight of F is m · wk, hence the word obtained by reading the vertices of F is wk+1 · · · wi where i is the smallest integer such that �i j=k+1 wj = mwk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' All these letters correspond to the vertices of F, hence the next letter is the label of the vertex read after F in the post-order traversal, which is the vertex rk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since the word W only contains strictly non-negative integers (except possibly the label of the root of T), at each step of the induction the value w1 +· · ·+wi strictly increases and the induction stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If wn1 is the letter corresponding to the leftmost child of the root of T, then the word w1 · · · wn1 is the word obtained by reading all the vertices of the subtree T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4, this is almost the word obtained by reading Lm(T1) we just need to ‘correct’ the label of the root of T1 by adding the label of the root of T which is the last letter wl of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' To conclude the word consisting of the labels of T1 is WT1 = w1 · · · wn1−1(wn1 + wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let � W be the word obtained by removing the letters w1, · · · , wn1 and wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We use the same procedure to extract the subwords corresponding to the other subtrees of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Due to the asymmetry of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4, there is a slight difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have found all the labels w1, w2, · · · , wt of the vertices r1, r2, · · · , rt of the left branch of Ti when �t j=1 wj = m and there is no need to ‘correct’ the word as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We are now ready to prove that Φm is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If the words of two trees T = (r, (T1, · · · , Tk)) and S = (s, (S1, · · · , Sl)) obtained by reading the labels of their vertices in post-order are equal, then by the discussion above we have k = l and for i ∈ {1, · · · , k}, the words obtained by reading the vertices of the subtrees Lm(Ti) and Lm(Si) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By induction on the number of leaves, we have Si = Ti for i = 1, · · · , k and we get that T = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We illustrate the decomposition involved in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='10 with the example of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We consider the leftmost subtree T1 of T with n = 7 leaves and which is labelled by m = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have Φ11(T1) = ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The first letter 1 tels us that the forest on the right of the leftmost leaf r1 has 1 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Its weight is m = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence ρ11 labels the only vertex of the forest and the next letter 5 corresponds to the parent r2 of r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since 1 + 5 = 6 we know that it is the leftmost child of the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence the word ρ1ρ11ρ5 is obtained by reading the vertices of the leftmost subtree S of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We apply the ‘correction’ and we get ρ1ρ11ρ10 = Φ11(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The rest of the word ρ1ρ11ρ10ρ2ρ11ρ11ρ9 corresponds to the other subtrees of T1 and it splits as ρ1ρ11ρ10 and ρ2ρ11ρ11ρ9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='10 we get our main result of the section: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For m = n − 1, the map Φm from the set of Schröder trees with n leaves to the set of words for ρn n−1 in Mn−1 is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 14 THOMAS GOBET AND BAPTISTE ROGNERUD Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The following two graphs are isomorphic under Φn−1: (1) The graph of words for ρn n−1 in Mn−1, where vertices are given by expressions of ρn n−1 and there is an edge between two expressions whenever they differ by applica- tion of a single relation, (2) The graph of Schröder trees with n leaves, where vertices are given by Schröder trees and there is an edge between two trees whenever they differ by application of a local move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The previous theorem gives the bijection between the sets of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7 shows that whenever one can apply a local move, one can apply a relation on the corresponding words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='8 shows that whenever one can apply a relation on words, a local move can be applied on the corresponding trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ We illustrate the situation for M3 in Figure 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The number of words for the Garside element of Mn is a little Schröder number A001003 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T = (r, S1, · · · , Sk) be a Schröder tree with n leaves labelled by m = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, the word obtained by reading all the labels of a subtree Sj is a word for ρljn where lj is the number of leaves of Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let us assume that the tree Sj has s + 1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7, the labels of the subtree Sj is a word for ρsρs n−1ρn−1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If s = 0, then we have a word for ρn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Otherwise, we can apply the relations [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5] with i = s and j = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Alternatively, using the Schröder trees, it is easy to see that these relations comes from the following modifications of the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The word ρsρs n−1ρn−1−s correspond to the case where the tree Sj is the Schröder bush with s + 1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Using our local moves, we can modify it to the left comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The corresponding word is now (ρ1ρs)sρn−1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Now we can inductively apply the local move to contract the edge between the root of T and the root left comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The result is s+1 empty trees attached to the root of T and the corresponding word is ρs+1 n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have the following isomorphisms of posets: (1) D0 n ∼= Div(∆n−1), Dn+1 n ∼= Div(∆0) = {•}, (2) For all 1 ≤ i ≤ n, Di n ∼= Div(∆n−i), where every set is ordered by the restriction of the left-divisibility order in the monoid Mk for suitable k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We begin by proving the second statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' An element x of Di n can be written in the form ρi nx′, where x′ is uniquely determined by cancellativity, and such that ρn is not a left- divisor of x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular, there is y a divisor of ∆n such that ρi nx′y = ρn+1 n , and y ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We associate a tree (or rather a family of trees) to x as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Write x′ as a product a1a2 · · · aj of elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Complete the word ρi na1a2 · · · aj to a word ρi na1a2 · · · ajb1b2 · · · bℓ for ∆n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=', choose a word b1b2 · · · bℓ for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' There are several possibilités for the bi’s, but the condition that x ∈ Di n ensures that, writing the corresponding Schröder tree in the form (r, T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti, S1, S2, · · · Sd), where the i first trees are empty trees with a single leaf, then a1a2 · · · aj has all its labels inside S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed, the labelling a1a2 · · · aj begins at the beginning (in the post-order convention) of the tree S1 since the trees T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti yield the label ρi n, and if another tree among S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Sd was partly labelled by the ai’s, then a power of ρn would left-divide x′, since the word obtained from S1 is a power of ρn (lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is then possible to reduce all the trees S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Sd to a single tree S still having the labelling a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , aj at the beginning, by first reducing S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Sd to a set of empty ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 15 ρ2ρ1ρ3ρ2ρ1ρ3 ρ3ρ1ρ3ρ2ρ3 ρ3ρ1ρ3ρ1ρ3ρ1 ρ3ρ2ρ3ρ3ρ1 ρ2ρ3ρ3ρ1ρ3 ρ3ρ3ρ3ρ3 ρ3ρ3ρ1ρ3ρ2 ρ3ρ2ρ1ρ3ρ2ρ1 ρ1ρ3ρ1ρ3ρ1ρ3 ρ1ρ3ρ2ρ3ρ3 ρ1ρ3ρ2ρ1ρ3ρ2 1 2 2 1 3 3 3 2 1 3 3 3 1 1 1 3 3 3 1 2 3 3 1 2 3 3 3 3 3 3 3 3 3 2 1 3 3 1 2 2 1 3 1 1 1 3 3 3 2 1 3 3 3 2 1 3 2 1 3 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Illustration of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='13 for n = 4: the graph of reduced words for ∆3 and the isomorphic graph of Schröder trees on 4 = 3 + 1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' trees �T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , �Td′ with single leafs, and then merging S1 and �T2 using a local move, then merging the resulting tree with �T3, and so on (see Figure 7 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In this way we associate to x a Schröder tree of the form (r, T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti, S), where the Tk’s are empty trees with a single leaf, and the labelling corresponding to the chosen word a1a2 · · · aj is an initial section of the tree S (in fact, in algebraic terms, what we did is modify the word for y to get a suitable one yielding a unique tree after the empty trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that by initial section we mean a prefix of the word obtained from the labelling of S read in post-order, where we exclude the label of the root, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=', if the root has a label, then the prefix is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We denote by Sn,i the set of such Schröder trees, that is, those Schröder trees on n leaves with i + 1 child of the root, and such that the i first child are leafs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that the tree that we attached to x depends on a choice of word for x, but applying a defining relation 16 THOMAS GOBET AND BAPTISTE ROGNERUD in the word x corresponds to applying a local move in the tree S, and this cannot make S split into several trees since the root of S is frozen (its label corresponds to the last letter of y ̸= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence we can apply all local moves with all labels in the (strict) initial section corresponding to a word for x, and we keep a Schröder tree on n−i+1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In this way, forgetting the i first empty trees, what we attached to x is an equivalence class of a (strict) initial section of a Schröder tree on n − i + 1 leaves under local moves, that is, a divisor of ∆n−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This mapping is injective since one can recover a word for x from the obtained Schröder tree on n − i + 1 leaves easily by mapping S to (r, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti, S), labelling such a tree, and reading the word obtained by reading the i first empty trees and then the initial section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It remains to show that it is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence consider an initial section of a Schröder tree S on n − i leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We must show that, in the tree (r, T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti, S), the initial section of S is a word a1a2 · · · aj which labels an element x′ of D0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Assume that ρn is a left-divisor of x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, using local moves only involving those labels in the initial section of S corresponding to a word for x′, one can transform (r, T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti, S) into a tree of the form (r, T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Ti, Ti+1, S′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , S′ e), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=', S can be split into several trees, the first one (corresponding to ρn) being an empty tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This is a contradiction: to split S into several trees, one would need to apply a local move involving the root of S, which is frozen since the initial section does not cover the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence x′ ∈ D0 n, and our mapping is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This completes the proof of the second point, as it is clear that our mappings preserve left-divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For the first point, we have Dn+1 n = {ρn+1 n }, hence there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' To show that D0 n ∼= Div(∆n−1), one proceeds in a similar way as in the proof of point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x ∈ D0 n and let y such that xy = ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Choose words for x and y, and consider the corresponding Schröder tree T = (r, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since x ∈ D0 n, the initial section of T corresponding to the word for x must be a proper initial section of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Using local moves on T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , Tk (which amounts to changing the word for y), we can find a Schröder tree that is equivalent to T under local moves, and that is of the form (r, �T1, �T2), where �T1 still has the chosen word for x as a proper initial section, and �T2 is the empty tree with only one leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In particular �T1 is a Schröder tree on n leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Applying defining relations to words for x amounts to applying local moves inside the first tree, and arguing as in the first point this establishes the isomorphism of posets between D0 n and Div(∆n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ r Ti S1 �T1 �T2 −→ r Ti r1 S1 �T1 �T2 −→ r Ti r2 r1 S1 �T1 �T2 = r Ti S Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Illustration for the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Enumerative results We have already seen (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='12) that the words for ρn+1 n are in bijection with Schröder trees on n+1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In this section, we give some additional enumerative results for several families of particular elements of Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Number of simple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let n ≥ 2, and let An := |Div(∆n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then An = 2A0 + 2An−1 + n−2 � i=1 Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1) It follows that An = F2n, where F0, F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' denotes the Fibonacci sequence 1, 2, 3, 5, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' inductively defined by F0 = 1, F1 = 2, and Fi = Fi−1 + Fi−2 for all i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The sequence of the Ans is referred as A001906 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1) follows immediately from the disjoint union Div(∆n) = � 0≤i≤n+1 Di n and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have A0 = F0, A1 = 3 = F2, and it is elementary to check that the inductive formula given by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1 is also satisfied by the sequence F2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This shows that An = F2n for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We call the lattice (Div(∆n), ≤) the even Fibonacci lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Number of left-divisors of the lcm of the atoms and odd Fibonacci lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The set DivL(ρn n) of left-divisors of ρn n also forms a lattice under the restriction of left-divisibility, since it is an order ideal in the lattice (Div(∆n), ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In terms of the Garside monoid Mn, the element ρn n is both the left- and right-lcm of the generators S = {ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' , ρn} (see [8, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' For n ≥ 1 we set Bn := |DivL(ρn n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have Bn = F2n−1 for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The sequence of the Bns is referred as A001519 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let x ∈ Div(∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We claim that x ∈ DivL(ρn n) if and only if ρnx ∈ Div(∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed, if x ≤ ρn n, there is y ∈ Mn such that xy = ρn n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We then have ρnxy = ρn+1 n = ∆n, hence ρnx is a left-divisor of ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Conversely, assume that ρnx ∈ Div(∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that there is y ∈ Div(∆n) such that ρnxy = ∆n = ρn+1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By cancellativity we get that xy = ρn n, hence x ∈ DivL(ρn n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that DivL(ρn n) is in bijection with the set {ρnx | x ∈ Div(∆n)} ∩ Div(∆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' But this set is nothing but � 1≤i≤n+1 Di n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It follows that Bn = |Div(∆n)| − |D0 n| = |Div(∆n)| − |Div(∆n−1)|, where the last equality follows from point (1) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='1 we thus get that Bn = An − An−1 = F2n − F2n−2 = F2n−1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We call the lattice (DivL(ρn n), ≤) the odd Fibonacci lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Both lattices for M3 are depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that the set of right-divisors of ρn n also has cardinality Bn: in fact, the two posets (DivL(ρn n), ≤L) and (DivR(ρn n), ≤R) are anti-isomorphic via x �→ x, where x is the element of Mn such that xx = ρn n (this element is unique by right-cancellativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 18 THOMAS GOBET AND BAPTISTE ROGNERUD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Number of words for the divisors of the Garside element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let T1 and T2 be two Schröder trees with n leaves labelled by m ≥ n − 1, and denote by m1 and m2 the corresponding words obtained by reading the labels in post-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If the words m1 and m2 have a common prefix x1x2 · · · xl, then xi labels a leftmost child in T1 if and only if it labels a leftmost child in T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We prove the result by induction on the number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If x1 · · · xl is obtained by reading all the vertices of T1 = (r, S1, · · · , Sk), then m1 = x1 · · · xl = m2 and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='10, we have T1 = T2, hence there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Otherwise, let Sj be the first subtree of T1 which is not covered by the word x1 · · · xl, similarly let Uk the first subtree of T2 = (r, U1, · · · Uv) which is not covered by x1 · · · xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Looking at the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='10, we see that the first subtrees S1, · · · , Sj−1 are completely determined by the word x1 · · · xl, hence we have j = k and Si = Ui for all i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let xs be the letter of x1 · · · xl labelling the first vertex of Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let m′ 1 be the subword of m1 and m′ 2 the subword of m2 starting at the xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As explained in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='10, we can determine the subword mj 1 of m′ 1 which correspond to Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The trees Uj and Sj do not need to have the same number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If one of the trees, say Uj, has less leaves, then one can apply local moves in the trees Uj, Uj+1, · · · Uv as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16 in order to obtain a tree ˜Uj with the same number of leaves as Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This will modify the word m′ 2, but not the prefix xs · · · xl, and xi labels a leftmost child in Uj if and only if it labels a leftmost child in ˜Uj (see Figure 7 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' After doing the modification, we consider the subword mj 2 corresponding to the tree ˜Uj and apply the induction hypothesis to mj 1 and mj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The set of words for the left-divisors of ρn+1 n is in bijection with the set of Schröder trees with n + 2 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let us denote by sk the number of Schröder trees with k + 1 leaves, and dk the number of words for the divisors of ρk+1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Recall that Div(∆n) = � 0≤i≤n+1 Di n, and let di n be the number of words for the elements of Di n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If i = n + 1, then ρn+1 n is the only element of Di n and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='12, there are sn words for this element, hence we have dn+1 n = sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let 0 ≤ i ≤ n and w = x1 · · · xl be a word for an element of Di n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The word w is a strict prefix of a Schröder tree T = (r, S1, · · · , Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5, w = w1w2 where w1 is a word for ρi n and ρn is not a left divisor of w2 (when i = 0 the word w1 is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let Sj be the last subtree of T which has a vertex labelled by a letter of w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We can apply a succession of defining relations to w1 in order to obtain ρi n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' These relations correspond to local move in the trees S1, · · · , Sj which collapse all the trees S1, · · · Sj to empty trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In order to reduce Sj to a list of empty trees we must use its root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since the root is always the last label of the tree in post-order, the word w1 covers all the first j trees which have in total i leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Since ρn does not divide w2, we see that w2 is a (possibly empty) strict prefix of Sj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' It is also possible to modify the trees Sj+2, · · · , Sk without changing the first j trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed, as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='16 we can reduce the trees Sj+2, · · · , Sk to empty trees and then merge them (until we can) to Sj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' When i = 0, after modification we obtain a tree ˜T = (r, ˜S, L) where L the empty tree, ˜S is a tree with n leaves and w = w2 is a strict prefix of ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' When 1 ≤ i ≤ n, we obtain a tree ˜T = (r, S1, · · · , Sj, �Sj+1) and w2 is a strict prefix of the tree �Sj+1 with n + 1 − i leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' In both cases, the tree �Sj+1 is obtained by possibly introducing new vertices to Sj+1, and as Figure 7 shows, these new vertices occur after the vertices of Sj+1, in post-order, ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID 19 hence w2 is still a strict prefix of �Sj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence, we see that in the decomposition w = w1w2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='5, the word w1 is obtained by reading all the vertices of a Schröder tree with i leaves and w2 is a strict prefix of a Schröder tree, denoted by ˜S, with li + 1 leaves where li = n − i leaves if i ̸= 0 and li = n − 1 if i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let w = w1w2 be a word of an element of Di n with w2 having t letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Let ˜S be a Schröder tree with li + 1 leaves having w2 as a strict prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Then, we construct a word γ(w) by first extracting ˜S, then labelling it accordingly to its number of leaves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=', with m = li) and finally taking its first t letters in post-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Algebraically, it is easy to see how the word γ(w) is obtained from w2: if wi is the label of a leftmost child in ˜S, we have γ(w)i = wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Otherwise, since the tree ˜Sj has li + 1 leaves, we have γ(w)i = wi − n + (li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' A priori γ(w) depends on the choice of a tree ˜S, but Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6 tells us that γ(w) only depends on w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The word γ(w) is a prefix of a Schröder tree with li + 1 leaves, hence it is a word for a divisor of ∆li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have obtained a map γ from the set of words for the elements of Di n to the set of words for the divisors of ∆li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Conversely, if z is a word of length k for a divisor of ∆li, it is a prefix (strict since the root is not contributing) of a Schröder tree S with li +1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We can view S as a subtree of a Schröder tree with n + 1 leaves by considering: T = (r, S, L) when i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' T = (r, δi, S) when i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Reading up to the first k letters of the subtree S produces a word w = w1w2 of an element of Di n such that γ(w) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Hence γ is surjective and we set ǫ(z) = w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' As before ǫ(z) only depends on z, not on the tree having z as a prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' When i = 0, the map γ is injective, indeed if w and z are two words such that γ(w) = γ(z), then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='6 the labels of the leftmost child in γ(w) and γ(z) are the same, hence w and z are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' This proves that d0 n = dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' When i ≥ 1, then γ is far from being injective, since it forgets the first part of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The set of words for the elements of Di n is the disjoint union of two sets E1 and E2 where E1 is the set of words w = w1w2 where w1 covers exactly one tree S1 and E2 is the set of words where w1 covers at least two trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Note that when i = 1, the set E2 is empty otherwise both sets are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Indeed E2 contains at least all the words of the form ρi nw2 and E1 contains at least the words of the form ρi−1ρi−1 n ρn−i−1w2 which correspond to the Schröder bush δi attached as the leftmost subtree of a Schröder tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If z is a word for a divisor of ∆li, we compute the cardinality of the preimage of z by γ by looking at γ−1(z) ∩ E1 and γ−1(z) ∩ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' If i = 1, we obviously only consider the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' The elements of γ−1(z) ∩ E1 are obtained by concatenation of the word of a single Schröder tree with i leaves and ǫ(z), and the elements of γ−1(z) ∩ E2 are concatenation of the words of a forest with i leaves made of at least two Schröder tree and ǫ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Such a forest is nothing but a Schröder tree with i-leaves from which the root has been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' So we have |γ−1(z) ∩ E1| = si−1 = |γ−1(z) ∩ E2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Taking the sum on all possible words z, we have d1 n = s0 · dn−1 and di n = 2 · si−1 · dn−i when n ≥ i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' We have obtained: d0 n = dn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' d1 n = s0 · dn−1 = dn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' and di n = 2 · si−1 · dn−i when n ≥ i ≥ 2 and dn+1 n = sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' 20 THOMAS GOBET AND BAPTISTE ROGNERUD By induction on the number of leaves, we have di = si+1, for every i ≤ n − 1, and dn = 2sn + 2 n � i=2 si−1sn−i+1 + sn = 3sn + 2 n−1 � i=1 sisn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Using generating functions, it is not difficult to check that this implies that dn = sn+1, see for example [13, Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' □ References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Bessis, The dual braid monoid, Ann.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Qi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Guo Some explicit and recursive formulas of the large and little Schröder numbers, Arab Journal of Mathematical Sciences Vol: 23, Issue: 2, Page: 141-147 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content=' Institut Denis Poisson, CNRS UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 TOURS, France Email address: thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='gobet@lmpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='univ-tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='fr Institut de Mathématiques de Jussieu, Paris Rive Gauche (IMJ-PRG), Campus des Grands Moulins, Université de Paris - Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 PARIS Cedex 13, France Email address: baptiste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='rognerud@imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQf1_mj/content/2301.00744v1.pdf'} diff --git a/39E2T4oBgHgl3EQfjgft/content/tmp_files/2301.03970v1.pdf.txt b/39E2T4oBgHgl3EQfjgft/content/tmp_files/2301.03970v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..be732cefaf83c2ab9702eca64321f91d1a79668f --- /dev/null +++ b/39E2T4oBgHgl3EQfjgft/content/tmp_files/2301.03970v1.pdf.txt @@ -0,0 +1,1472 @@ +arXiv:2301.03970v1 [math.GR] 10 Jan 2023 +Ulam stability of lamplighters and Thompson groups +Francesco Fournier-Facio and Bharatram Rangarajan +January 11, 2023 +Abstract +We show that a large family of groups is uniformly stable relative to unitary groups +equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten +p-norms. These include lamplighters Γ ≀ Λ where Λ is infinite and amenable, as well as +several groups of dynamical origin such as the classical Thompson groups F, F ′, T and +V . We prove this by means of vanishing results in asymptotic cohomology, a theory +introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable +for studying uniform stability. +Along the way, we prove some foundational results +in asymptotic cohomology, and use them to prove some hereditary features of Ulam +stability. We further discuss metric approximation properties of such groups, taking +values in unitary or symmetric groups. +1 +Introduction +Let Γ be a countable discrete group, and let U be a family of finite-dimensional unitary +groups. The problem of stability asks whether every almost-homomorphism Γ → U ∈ U +is close to a homomorphism. To formalize this we need to choose a norm, and a way to +interpret these approximate notions. We focus on the classical setting of uniform defects and +distances, with respect to submultiplicative norms. +Let U := {(U(k), ∥·∥)} be a family of finite-dimensional unitary groups equipped with bi- +invariant submultiplicative norms ∥·∥ (we allow U(k) to appear multiple times with different +norms). For instance ∥ · ∥ could be the operator norm - the most classical case - or more +generally a Schatten p-norm. Given a map φ : Γ → U(k), we define its defect to be +def(φ) := sup +g,h∈Γ +∥φ(gh) − φ(g)φ(h)∥. +Given another map ψ : Γ → U(k), we define the distance between them to be +dist(φ, ψ) := sup +g∈Γ +∥φ(g) − ψ(g)∥. +Definition 1.1. A uniform asymptotic homomorphism is a sequence of maps φn : Γ → U(kn) +such that def(φn) → 0. We denote this simply by φ : Γ → U. We say that φ, ψ : Γ → U are +uniformly asymptotically close if they have the same range degrees and dist(φn, ψn) → 0. +The group Γ is uniformly U-stable if every uniform asymptotic homomorphism is uni- +formly asymptotically close to a sequence of homomorphisms. +1 + +We can also talk quantitatively about stability, by asking how close a homomorphism we +can choose, in terms of the defect. This leads to the notion of stability with a linear estimate, +which will be relevant for us and which we define precisely in Section 2.1. +Early mentions of similar problems can be found in the works of von Neumann [vN29] +and Turing [Tur38]. In [Ula60, Chapter 6] Ulam discussed more general versions of stability, +which has since inspired a large body of work. Uniform U-stability has been studied mostly +when U is the family of unitary groups equipped with the operator norm, for which the +notion is typically referred to as Ulam stability. In this contest, Kazhdan proved stability of +amenable groups [Kaz82], while Burger, Ozawa and Thom proved stability of certain special +linear groups over S-integers, and instability of groups admitting non-trivial quasimorphisms +[BOT13]. +More recently, the second author, Glebsky, Lubotzky and Monod proved Ulam stabil- +ity of certain lattices in higher rank Lie groups, with respect to arbitrary submultiplicative +norms [GLMR23]. For the proof, they introduce a new cohomology theory, called asymptotic +cohomology, and prove that stability is implied by the vanishing of certain asymptotic co- +homology classes α ∈ H2 +a(Γ, V). We refer the reader to Section 2.2 for the relevant definitions. +The goal of this paper is to further the understanding of asymptotic cohomology, and +apply this to prove new stability results. +The main one is the stability of the classical +Thompson groups: +Theorem 1.2 (Section 5). Thompson’s groups F, F ′, T and V are uniformly U-stable, with +a linear estimate. +As remarked by Arzhantseva and P˘aunescu [AP15, Open problem], the analogous state- +ment for pointwise stability in permutation of F would imply that F is not sofic, thus proving +at once the existence of a non-sofic group and the non-amenability of F: two of the most +remarkable open problems in modern group theory. We will discuss these problems and their +relation to our results in Section 7. +Theorem 1.2 for F and F ′ will follow from a stability result for certain lamplighters. +Given groups Γ, Λ, the corresponding lamplighter (or restricted wreath product) is the group +Γ ≀ Λ = (⊕ΛΓ) ⋊ Λ, where Λ acts by shifting the coordinates. +Theorem 1.3. Let Γ, Λ be two countable groups, where Λ is infinite and amenable. Then +Γ ≀ Λ is uniformly U-stable, with a linear estimate. +By itself, Theorem 1.3 provides a plethora of examples of uniformly U-stable groups, to a +degree of flexibility that was not previously available. For instance, using classical embedding +results [HNN49] it immediately implies the following: +Corollary 1.4. Every countable group embeds into a 3-generated group which is uniformly +U-stable, with a linear estimate. +In particular, this gives a proof that there exist uncountably many finitely generated +uniformly U-stable groups, a fact which could also be obtained by applying Kazhdan’s The- +orem [Kaz82] to an infinite family of finitely generated amenable groups, such as the one +constructed by B. H. Neumann [Neu37]. +2 + +In order to obtain stability of F and F ′ from Theorem 1.3, we exploit coamenability. +Recall that a subgroup Λ ≤ Γ is coamenable if the coset space Γ/Λ admits a Γ-invariant +mean. It is well known that F ′ and F contain a coamenable lamplighter F ≀ Z. Therefore the +stability of F and F ′ (Corollary 5.8) follows from Theorem 1.3, and the following result: +Proposition 1.5. Let Λ ≤ Γ be coamenable. If Λ is uniformly U-stable with a linear estimate, +then so is Γ. +This can be seen as a relative version of the celebrated result of Kazhdan, stating that +amenable groups are uniformly U-stable [Kaz82]. To complete the picture, we also prove +another relative version of Kazhdan’s Theorem, which is sort of dual to Proposition 1.5: +Proposition 1.6. Let N ≤ Γ be an amenable normal subgroup. If Γ is uniformly U-stable +with a linear estimate, then so is Γ/N. +The fact that Theorem 1.2 follows from Theorem 1.3 and Proposition 1.5 is not special +to Thompson’s group F: this phenomenon is typical of several groups of piecewise linear and +piecewise projective homeomorphisms, which enjoy some kind of self-similarity properties +(Theorem 5.1 and Corollary 5.2). Stability of T and V then follow from these results, to- +gether with a bounded generation argument analogous to the one from [BOT13] (Corollaries +5.11 and 5.12). +As we mentioned above, the tool underlying the proofs of Theorem 1.3 and Proposition +1.5 is asymptotic cohomology, in particular the vanishing of certain classes in degree 2. In +this framework, Theorem 1.3 takes the following form: +Theorem 1.7. Let Γ, Λ be two countable groups, where Λ is infinite and amenable. Then +Hn +a(Γ ≀ Λ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V. +Here the word finitary refers to the fact that these modules arise from stability problems +with respect to finite-dimensional unitary representations. This hypothesis is crucial: see +Remark 6.1. Propositions 1.5 and 1.6 also follow from results in asymptotic cohomology, +that this time does not need the finitary assumption: +Proposition 1.8. Let Λ ≤ Γ be coamenable. Then the restriction map Hn +a(Γ, V) → Hn +a(Λ, V) +is injective, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V. +Proposition 1.9. Let N ≤ Γ be an amenable normal subgroup. Then the pullback map +Hn +a(Γ/N, V) → Hn +a(Γ, V) is an isomorphism, for all n ≥ 0 and all dual asymptotic Banach +∗(Γ/N)-modules V. +Despite the lack of a general theorem connecting the two theories, asymptotic cohomology +seems to be closely connected to bounded cohomology, a well-established cohomology theory +[Joh72, Gro82, Iva85, Mon01, Fri17] that has become a fundamental tool in rigidity theory. +The vanishing result for asymptotic cohomology of lattices leading to stability [GLMR23] +follows closely the vanishing result for bounded cohomology of high-rank lattices [BM99, +BM02, MS04]. Similarly, our proofs of Theorem 1.7 and Propositions 1.8 and 1.9 follow +closely the corresponding bounded-cohomological results: for Theorem 1.7 this was recently +proven by Monod [Mon22], while for Proposition 1.8 this is a foundational result in bounded +cohomology [Mon01, 8.6] (see also [MP03]), and Proposition 1.9 is an analogue of Gromov’s +3 + +Mapping Theorem [Gro82]. Note that the bounded cohomology of T and V has also been +recently computed [FFLM21, MN21, And22], but only with trivial real coefficients, and our +proofs are of a different nature. +We thus hope that the steps we undertake to prove our main results will be useful to +produce more computations in asymptotic cohomology, and therefore more examples of uni- +formly U-stable, and in particular Ulam stable, groups. +Our results have applications to the study of approximating properties of groups. While +questions on pointwise approximation, such as soficity, hyperlinearity, and matricial finite- +ness, are in some sense disjoint from the content of this paper, our stability results imply +that some of the groups considered in this paper are not uniformly approximable with respect +to the relevant families U (Corollary 7.6). We are also able to treat the case of symmetric +groups endowed with the Hamming distance, by a more direct argument (Proposition 7.7). +We end this introduction by proposing a question. There is a notion of strong Ulam stabil- +ity, where the approximations take values in unitary groups of possibly infinite-dimensional +Hilbert spaces, with the operator norm. It is a well-known open question whether strong +Ulam stability coincides with amenability. In this direction it is known that strong Ulam +stable groups have no non-abelian free subgroups [BOT13, Theorem 1.2], but there exist +groups without non-abelian free subgroups that are not strong Ulam stable [Alp20]. +On the other hand, our results also prove uniform U-stability stability of the piecewise +projective groups of Monod [Mon13] and Lodha–Moore [LM16], which are nonamenable and +without free subgroups (see Section 5.2). Therefore we ask the following: +Question 1.10. Let Γ be a countable group without non-abelian free subgroups. +Is Γ +uniformly U-stable (with a linear estimate)? Or at least Ulam stable? +In particular, are all countable torsion groups Ulam stable? +In other words: if Γ is not Ulam stable, must Γ contain a non-abelian free subgroup? To +our knowledge it is not every known if groups admitting non-trivial quasimorphisms must +contain non-abelian free subgroups: see [Man05] and [Cal10] for partial results in this direc- +tion. +Conventions: All groups are assumed to be discrete and countable. The set of natu- +ral numbers N starts at 0. A non-principal ultrafilter ω on N is fixed for the rest of the paper. +Outline: We start in Section 2 by reviewing the framework of asymptotic cohomology +and its applications to stability, as developed in [GLMR23]. In Section 3 we discuss hered- +itary properties for Ulam stability, and prove Propositions 1.5 and 1.6. We then move to +lamplighters and prove Theorem 1.3 in Section 4, then to Thompson groups proving Theorem +1.2 in Section 5. In Section 6 we provide examples showing that some of our results and some +of the results from [GLMR23] are sharp, and conclude in Section 7 by discussing applications +to the study of metric approximations of groups. +Acknowledgements: The authors are indebted to Alon Dogon, Lev Glebsky, Alexander +Lubotzky and Nicolas Monod for useful conversations. +4 + +2 +Uniform stability and asymptotic cohomology +In this section, we shall briefly summarize the notion of defect diminishing that allows us +to formulate the stability problem as a problem of lifting of homomorphisms with abelian +kernel, which in turn motivates the connection to second cohomology. For a more detailed +description, refer to Section 2 in [GLMR23]. +2.1 +Uniform stability and defect diminishing +We begin by reviewing some basic notions of ultraproducts and non-standard analysis, be- +fore formulating the stability problem as a homomorphism lifting problem. For this, it is +convenient to describe a uniform asymptotic homomorphism (which is a sequence of maps) +as one map of ultraproducts. This in turn allows us to perform a soft analysis to obtain +a (true) homomorphism to a quotient group. Recall that ω is a fixed non-principal ultra- +filter on N. The algebraic ultraproduct � +ω Xn of an indexed collection {Xn}n∈N of sets is +defined to be � +ω Xn := � +n∈N Xn/ ∼ where for {xn}n∈N, {yn}n∈N ∈ � +n∈N Xn, we define +{xn}n∈N ∼ {yn}n∈N if {n : xn = yn} ∈ ω. Ultraproducts can be made to inherit algebraic +structures of their building blocks. For instance, for a group Γ, the ultraproduct � +ω Γ, called +the ultrapower and denoted ∗Γ, is itself a group. Another important example we will use is +the field of hyperreals ∗R, the ultrapower of R. +Objects (sets, functions, etc.) that arise as ultraproducts of standard objects are referred +to as internal. Important examples of non-internal objects are the subsets ∗Rb of bounded +hyperreals, consisting of elements {xn}ω ∈ ∗R for which there exists S ∈ ω and C ∈ R≥0 such +that |xn| ≤ C for every n ∈ S, and the subset ∗Rinf of infinitesimals, consisting of elements +{xn}ω ∈ ∗R such that for every real ε > 0, there exists S ∈ ω such that |xn| < ε for every +n ∈ S. +For x, y ∈ ∗R, write x = Oω(y) if x/y ∈ ∗Rb, and write x = oω(y) if x/y ∈ ∗Rinf. In +particular, x ∈ ∗Rb is equivalent to x = Oω(1) while ε ∈ ∗Rinf is equivalent to ε = oω(1). The +subset ∗Rb forms a valuation ring with ∗Rinf being the unique maximal ideal, with quotient +∗Rb/∗Rinf ∼= R. The quotient map st : ∗Rb → R is known as the standard part map or limit +along the ultrafilter ω. The previous construction can also be replicated for Banach spaces. +Let {Wn}n∈N be a family of Banach spaces. Then W = � +ω Wn can be given the structure +of a ∗R-vector space. In fact, it also comes equipped with a ∗R-valued norm, allowing us to +define the external subsets Wb and Winf. The quotient ˜ +W := Wb/Winf is a real Banach space. +Given a uniform asymptotic homomorphism {φn : Γ → U(kn)}n∈N with def(φn) =: εn → +0, construct the internal map φ : ∗Γ → � +ω U(kn) where φ := � +ω φn, with (hyperreal) defect +ε := {εn}ω ∈ ∗Rinf. Then the question of uniform stability with a linear estimate can be +rephrased as asking whether there exists an internal homomorphism ψ : ∗Γ → � +ω U(kn) such +that their (hyperreal) distance satisfies dist(φ, ψ) := {dist(φn, ψn)}ω = Oω(ε). +The advantage of rephrasing the question in terms of internal maps is that an internal +map φ : ∗Γ → � +ω U(kn) with defect ε ∈ ∗Rinf induces a true homomorphism ˜φ : ∗Γ → +� +ω U(kn)/B(ε) where B(ε) is the (external) normal subgroup of � +ω U(kn) comprising ele- +ments that are at a distance Oω(ε) from the identity. In particular, the question of uniform +stability with a linear estimate can equivalently be rephrased as asking whether given such +an internal map φ, can the homomorphism ˜φ : ∗Γ → � +ω U(kn)/B(ε) be lifted to an internal +5 + +homomorphism ψ : ∗Γ → � +ω U(kn). +Reinterpreting uniform stability with a linear estimate as a homomorphism lifting problem +motivates a cohomological approach to capturing the obstruction. However, the obstacle here +is that the kernel B(ε) of the lifting problem is not abelian. This can be handled by lifting +in smaller steps so that each step involves an abelian kernel. Define a normal subgroup I(ε) +of B(ε) comprising elements that are at a distance of oω(ε) from the identity. Then we can +attempt to lift ˜φ : ∗Γ → � +ω U(kn)/B(ε) to an internal map ψ : ∗Γ → � +ω U(kn) that is a +homomorphism modulo I(ε). The problem is simpler from the cohomological point of view: +since the norms are submultiplicative, the kernel B(ε)/I(ε) of this lifting problem is abelian. +The group Γ is said to have the defect diminishing property with respect to U if such a lift +exists; more explicitly, Γ has the defect diminishing property if for every uniform asymptotic +homomorphism φ : Γ → U there exists a uniform asymptotic homomorphism ψ with the +same range such that dist(φ, ψ) = Oω(def(φ)) and def(ψ) = oω(def(φ)). +Theorem 2.1 ([GLMR23, Theorem 2.3.11]). Γ has the defect diminishing property with +respect to U if and only if Γ is uniformly U-stable with a linear estimate. +The obstruction to such a homomorphism lifting, with an abelian kernel B(ε)/I(ε), can +be carefully modeled using a cohomology H• +a(Γ, W) so that H2 +a(Γ, W) = 0 implies the defect +diminishing property (and consequently, uniform stability with a linear estimate). +Here +W = � +ω u(kn) is an internal Lie algebra of � +ω U(kn) equipped with an asymptotic action of +the ultrapower ∗Γ constructed from the uniform asymptotic homomorphism φ that we start +out with. The logarithm of the defect map +∗Γ × ∗Γ → +� +ω +U(kn) : (g1, g2) �→ φ(g1)φ(g2)φ(g1g2)−1 +would correspond to an asymptotic 2-cocycle in H2 +a(Γ, W). Such a cocycle is a coboundary +in this setting (that is, it represents the zero class in H2 +a(Γ, W)), if and only if the defect +diminishing property holds for the asymptotic homomorphism φ. +2.2 +Asymptotic cohomology +The reduction to a lifting problem with abelian kernel motivates a cohomology theory of Γ +with coefficients in the internal Lie algebra W = � +ω u(kn) of � +ω U(kn), equipped with an +asymptotic conjugation action of Γ. In this section we review the formal definition of this +cohomology, and state some results from [GLMR23] that we shall need to work with it. +Let (Vn)n≥1 be a sequence of Banach spaces, and let V := � +ω +Vn be their algebraic ultra- +product: we refer to such V as an internal Banach space. For v ∈ V we denote by ∥v∥ the +hyperreal (∥vn∥)ω ∈ ∗R. We then denote by +Vb := {v ∈ V : ∥v∥ ∈ ∗Rb}; +Vinf := {v ∈ V : ∥v∥ ∈ ∗Rinf}. +Then the quotient ˜V := Vb/Vinf is a real Banach space, whose norm is induced by the +ultralimit of ∥ · ∥ on Vb. For each Vn denote by V # +n its continuous dual, and let V# be the +corresponding algebraic ultraproduct. The pairing ⟨·, ·⟩n : V # +n × Vn → R induces a pairing +V# × V → ∗R which descends to ˜V# × ˜V → R. We call V# the internal dual of V. +6 + +Now let Γ be a countable discrete group, and let π : ∗Γ×V → V be an internal map which +preserves ∥ · ∥ and induces an isometric linear action ˜π : ∗Γ × ˜V → ˜V of ∗Γ. Such a map π +is referred to as an asymptotic ∗Γ-action on V. We then call (π, V), or V, if π is understood +from context, an asymptotic Banach ∗Γ-module. Given an internal Banach ∗Γ-module (π, V), +the contragradient on each coordinate induces an internal map π# : ∗Γ × V# → V# mak- +ing (π#, V#) into an asymptotic Banach ∗Γ-module. We call a module V a dual asymptotic +Banach ∗Γ-module if V is the dual of some asymptotic ∗Γ-module denoted V♭. We decorate +these definitions with the adjective finitary if each Vn is finite-dimensional. +Now for each m ≥ 0 define the internal Banach space L∞((∗Γ)m, V) := � +ω +ℓ∞(Γm, Vn) +(note that m is fixed and n runs through the natural numbers with respect to the ultrafilter +ω). Similarly to before, for f ∈ L∞((∗Γ)m, V) we denote ∥f∥ := (∥fn∥)ω ∈ ∗R and +L∞ +b ((∗Γ)m, V) := {f ∈ L∞((∗Γ)m, V) : ∥f∥ ∈ ∗Rb}; +L∞ +inf((∗Γ)m, V) := {f ∈ L∞((∗Γ)m, V) : ∥f∥ ∈ ∗Rinf}. +Given an asymptotic ∗Γ-action π on V, we can construct a natural asymptotic ∗Γ-action +ρm : ∗Γ × L∞((∗Γ)m, V) → L∞((∗Γ)m, V) given by +(ρm(g)(f))(g1, g2, . . . , gm) := πG(g)f(g−1g1, . . . , g−1gm) +(1) +Then the quotient +˜L∞((∗Γ)m, V) := L∞ +b ((∗Γ)m, V)/L∞ +inf((∗Γ)m, V) +is again a real Banach space equipped with an isometric ∗Γ-action induced coordinate-wise +by ρm, which defines the invariant subspaces ˜L∞((∗Γ)m, V) +∗Γ. +Now define the internal coboundary map +dm : L∞((∗Γ)m, V) → L∞((∗Γ)m+1, V); +dm(f)(g0, . . . , gm) := +m +� +j=0 +(−1)jf(g0, . . . , ˆgj, . . . , gm), +(2) +which descends to coboundary maps +˜ +dm : ˜L∞((∗Γ)m, V) → ˜L∞((∗Γ)m+1, V). +Since ˜ +dm is ∗Γ-equivariant, it defines the cochain complex: +0 +˜ +d0 +−→ ˜L∞(∗Γ, V) +∗Γ +˜ +d1 +−→ ˜L∞((∗Γ)2, V) +∗Γ +˜d2 +−→ ˜L∞((∗Γ)3, V) +∗Γ +˜ +d3 +−→ · · · +Definition 2.2 ([GLMR23, Definition 4.2.2]). The m-th asymptotic cohomology of Γ with +coefficients in V is +Hm +a (Γ, V) := ker( +˜ +dm+1)/ im( ˜ +dm). +7 + +Other resolutions may also be used to compute asymptotic cohomology. Recall ([Mon01, +5.3.2]) that a regular Γ-space S is said to be a Zimmer-amenable Γ-space if there exists a Γ- +equivariant conditional expectation m : L∞(Γ×S) → L∞(S). Let S be a regular Γ-space with +a Zimmer-amenable action of Γ, and let L∞((∗S)m, V) := � +ω +L∞ +w∗(Sm, Vn) (where L∞ +w∗(Sm, Vn) +is the space of bounded weak-∗ measurable function classes from Sm to Vn). Again, the +asymptotic ∗Γ-action on V gives rise to a natural asymptotic ∗Γ-action on L∞((∗Γ)m, V) as +in (1), making L∞((∗S)m, V) an asymptotic Banach ∗Γ-module. The coboundary maps too +can be defined just as in (2), to construct the cochain complex, and we have: +Theorem 2.3 ([GLMR23, Theorem 4.3.3]). Let S be a Zimmer-amenable Γ-space, and V +be a dual asymptotic Banach ∗Γ-module. Then H• +a(Γ, V) can be computed as the asymptotic +cohomology of the cochain complex +0 +˜ +d0 +−→ ˜L∞(∗S, V) +∗Γ +˜ +d1 +−→ ˜L∞((∗S)2, V) +∗Γ +˜ +d2 +−→ ˜L∞((∗S)3, V) +∗Γ +˜ +d3 +−→ · · · +In the context of uniform U-stability, the relevant asymptotic Banach ∗Γ-module we shall +be interested is the ultraproduct W = � +ω u(kn), where u(kn) is the Lie algebra of U(kn). Note +that we are only considering finite-dimensional unitary groups, so such a module is finitary. +Given a uniform asymptotic homomorphism φ : ∗Γ → � +ω U(n) with defect def(φ) ≤ω ε ∈ +∗Rinf, this can be used to construct the asymptotic action π : ∗Γ × W → W given by +π(g)v = φ(g)vφ(g)−1, making W an asymptotic Banach ∗Γ-module. We call such a module +an Ulam ∗Γ-module supported on U. +Also, consider the map α : ∗Γ × ∗Γ → W given by +α(g1, g2) = 1 +ε log(φ(g1)φ(g2)φ(g1g2)−1). +(3) +This map α induces an inhomogeneous 2-cocycle ˜α : ∗Γ × ∗Γ → +˜ +W, and thus defines +a class in H2 +a(Γ, W), under the usual correspondence between inhomogeneous cochains and +invariant homogeneous cochains [GLMR23, Theorem 4.2.4]. We call such a class an Ulam +class supported on U. This class vanishes, i.e. ˜α is a coboundary, precisely when φ has the +defect diminishing property. Thus Theorem 2.1 yields: +Theorem 2.4 ([GLMR23, Theorem 4.2.4]). Γ is uniformly U-stable with respect to U if and +only if all Ulam classes supported on U vanish. In particular, if H2 +a(Γ, W) = 0 for every Ulam +∗Γ-module supported on U, then Γ is uniformly U-stable, with a linear estimate. +3 +Hereditary properties +In this section, we first prove Proposition 1.8 and deduce Proposition 1.5 from it; then +analogously we prove Proposition 1.9 and deduce Proposition 1.6 from it. Both stability +statements are not symmetric, and in fact we will see in Section 6 that the converses do not +hold. +3.1 +More on Zimmer-amenability +For the proofs of Propositions 1.8 and 1.9, we will need a more precise version of Theorem 2.3 +in a special case. A regular Γ-space S is said to be discrete if it is a countable set equipped +8 + +with the counting measure. It follows from the equivalent characterizations in [AEG94] that +a discrete Γ-space is Zimmer-amenable precisely when each point stabilizer is amenable. In +particular: +1. If Λ ≤ Γ is a subgroup, then the action of Λ on Γ by left multiplication is free, so Γ is +a discrete Zimmer-amenable Λ-space. +2. If N ≤ Γ is an amenable subgroup, then the action of Γ on the coset space Γ/N has +stabilizers equal to conjugates of N, so Γ/N is a discrete Zimmer-amenable Γ-space. +For such spaces, we can provide an explicit chain map that implements the isomorphism +in cohomology from Theorem 2.3. Indeed, the proof of Theorem 2.3 works by starting with +a Γ-homotopy equivalence between the two complexes: +0 → L∞(Γ) → L∞(Γ2) → L∞(Γ3) → · · · +0 → L∞(S) → L∞(S2) → L∞(S3) → · · · +which is then extended internally to the asymptotic version of these complexes. The case +of dual asymptotic coefficients follows via some suitable identifications of the corresponding +complexes (see the paragraph preceding [GLMR23, Theorem 4.20]). In the case of a discrete +group Γ and a discrete Zimmer-amenable Γ-space, the homotopy equivalence above can be +chosen to be the orbit map +om +b : L∞(Sm) −→ L∞(Γm) +om +b (f)(g1, . . . , gm) = f(g1b, . . . , gmb); +where b ∈ S is some choice of basepoint [Fri17, Section 4.9]. Therefore in this case we obtain +the following more explicit version of Theorem 2.3: +Theorem 3.1. Let S be a discrete Zimmer-amenable Γ-space, with a basepoint b ∈ S, and +let V be a dual asymptotic Banach ∗Γ-module. Then the orbit map +om +b : L∞((∗S)m, V) −→ L∞((∗Γ)m, V) +om +b (f)(g1, . . . , gm) = f(g1b, . . . , gmb); +induces induces an isomorphism between H• +a(Γ, V) and the cohomology of the complex +0 +˜ +d0 +−→ ˜L∞(∗S, V) +∗Γ +˜ +d1 +−→ ˜L∞((∗S)2, V) +∗Γ +˜ +d2 +−→ ˜L∞((∗S)3, V) +∗Γ +˜ +d3 +−→ · · · +In the two basic examples of discrete Zimmer-amenable spaces from above, we obtain: +Corollary 3.2. Let Λ ≤ Γ be a subgroup, and let V be a dual asymptotic Banach ∗Γ- +module, which restricts to a dual asymptotic Banach ∗Λ-module. +Then the restriction of +cochains L∞((∗Γ)m, V) → L∞((∗Λ)m, V) induces an isomorphism between H• +a(Λ, V) and the +cohomology of the complex: +0 +˜ +d0 +−→ ˜L∞(∗Γ, V) +∗Λ +˜ +d1 +−→ ˜L∞((∗Γ)2, V) +∗Λ +˜ +d2 +−→ ˜L∞((∗Γ)3, V) +∗Λ +˜ +d3 +−→ · · · +Proof. Seeing Γ as a discrete Zimmer-amenable Λ-space, with basepoint 1 ∈ Γ, the orbit map +is nothing but the restriction of cochains, and we conclude by Theorem 3.1. +9 + +Corollary 3.3. Let N ≤ Γ be an amenable normal subgroup, and let V be a dual asymptotic +Banach ∗(Γ/N)-module, which pulls back to a dual asymptotic Banach ∗Γ-module. Then the +pullback of cochains L∞((∗(Γ/N))m, V) → L∞((∗Γ)m, V) induces an isomorphism between +H• +a(Γ, V) and H• +a(Γ/N, V). +Proof. Seeing Γ/N as a discrete Zimmer-amenable Γ-space, with basepoint the coset N, the +orbit map is nothing but the pullback of cochains. So Theorem 3.1 yields an isomorphism +between H• +a(Γ, V) and the cohomology of the complex: +0 +˜ +d0 +−→ ˜L∞(∗(Γ/N), V) +∗Γ +˜ +d1 +−→ ˜L∞((∗(Γ/N))2, V) +∗Γ +˜ +d2 +−→ ˜L∞((∗(Γ/N))3, V) +∗Γ +˜d3 +−→ · · · +But since the action of ∗Γ on both ∗(Γ/N) and V factors through ∗(Γ/N), the above complex +coincides with the standard one computing H• +a(Γ/N, V). +We will use these explicit isomorphisms in this section. Later, for the proof of Theorem +1.7, non-discrete Zimmer-amenable spaces will also appear, but in that case we will only need +the existence of an abstract isomorphism as in Theorem 2.3. +3.2 +Restrictions and coamenability +Let Λ ≤ Γ be a (not necessarily coamenable) subgroup, and V be a dual asymptotic Ba- +nach ∗Γ-module, which restricts to a dual asymptotic Banach ∗Λ-module. The restriction +˜L∞((∗Γ)•, V) +∗Γ → ˜L∞((∗Λ)•, V) +∗Λ induces a map in cohomology, called the restriction map, +and denoted +res• : H• +a(Γ, V) → H• +a(Λ, V). +This map behaves well with respect to Ulam classes: +Lemma 3.4. Let W be an Ulam ∗Γ-module supported on U. Then W is also an Ulam ∗Λ- +module supported on U, and the restriction map res2 : H2 +a(Γ, W) → H2 +a(Λ, W) sends Ulam +classes to Ulam classes. +Proof. Let φ : Γ → U be a uniform asymptotic homomorphism, and let W be the corre- +sponding Ulam ∗Γ-module. Then restricting φn to Λ for each n yields a uniform asymptotic +homomorphism φ|Λ : Λ → U, with def(φ|Λ) ≤ω def(φ) and endows W with an asymptotic +∗Λ-action making it into an Ulam ∗Λ-module supported on U. The cocycle corresponding to +φ is defined via the map +α : ∗Γ × ∗Γ → W : (g1, g2) �→ 1 +ε log(φ(g1)φ(g2)φ(g1g2)−1). +Since def(φ|Λ) ≤ω ε, restricting α to ∗Λ × ∗Λ yields a valid cocycle associated to the +uniform asymptotic homomorphism φΛ. It follows that the chain map ˜L∞((∗Γ)•, V) +∗Γ → +˜L∞((∗Λ)•, V) +∗Λ preserves the set of cocycles defined via uniform asymptotic homomorphisms, +and therefore preserves Ulam classes. +Now suppose that Λ ≤ Γ is coamenable. This means, by definition, that there exists a +Γ-invariant mean on Γ/Λ; that is, there exists a linear functional m : ℓ∞(Γ/Λ) → R such +that +1. m(1Γ/Λ) = 1, where 1Γ/Λ denotes the constant function. +10 + +2. |m(f)| ≤ ∥f∥ for all f ∈ ℓ∞(Γ/Λ). +3. m(g · f) = m(f) for all g ∈ Γ and all f ∈ ℓ∞(Γ/Λ). +As with the absolute case [GLMR23, Lemma 3.20], we have the following: +Lemma 3.5. Suppose that Λ ≤ Γ is coamenable, and let V be a dual asymptotic Banach +∗Γ-module. Then there exists an internal map m : L∞(∗Γ/∗Λ, V) → V which induces a map +˜m : ˜L∞(∗Γ/∗Λ, V) → ˜V with the following properties: +1. If ˜f is the constant function equal to ˜v ∈ ˜V, then ˜m( ˜f) = ˜v. +2. ∥ ˜m( ˜f)∥ ≤ ∥ ˜f∥ for all ˜f ∈ ˜L∞(∗Γ/∗Λ, V). +3. ˜m(g · ˜f) = ˜m( ˜f) for all g ∈ ∗Γ and all ˜f ∈ ˜L∞(∗Γ/∗Λ, V). +Proof. Consider f = {fn}ω ∈ L∞(∗Γ/∗Λ, V). Since V is a dual asymptotic ∗Γ-module with +predual V♭, for each λ ∈ V♭, we get an internal map +f λ : ∗Γ/∗Λ → ∗R : x �→ f(x)(λ). +Note that f λ being internal, it is of the form {f λ +n}ω where f λ +n ∈ ℓ∞(Γ/Λ). This allows us to +construct the internal map mλ +in : L∞(∗Γ/∗Λ, V) → ∗R as +mλ +in(f) = {m +� +f λ +n +� +}ω +and finally min : L∞(∗Γ/∗Λ, V) → V as +min(f)(λ) = mλ +in(f) +It is straightforward to check that min as defined induces a linear map ˜m : ˜L∞(∗Γ/∗Λ, V) → ˜V. +As for ∗Γ-equivariance, this follows from the observation that (g · f)λ(x) = π(g)f(g−1x)(λ) +while (g·f λ)(x) = f(g−1x)(λ). The conditions on ˜m follow from the definition and properties +of the Γ-invariant mean m on ℓ∞(Γ/Λ). +We are now ready to prove Proposition 1.8. The proof goes along the lines of [Mon01, +Proposition 8.6.2]. +Proposition (Proposition 1.8). Let Λ ≤ Γ be coamenable. Then the restriction map Hn +a(Γ, V) → +Hn +a(Λ, V) is injective, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V. +Proof. We implement the asymptotic cohomology of Λ using the complex ˜L∞((∗Γ)•, V) +∗Λ +from Corollary 3.2. Since the chain map that defines the restriction map factors through +this complex, and the chain map ˜L∞((∗Γ)•, V) +∗Λ → ˜L∞((∗Λ)•, V) +∗Λ induces an isomorphism +in cohomology (Corollary 3.2), it suffices to show that the chain inclusion ˜L∞((∗Γ)•, V) +∗Γ → +˜L∞((∗Γ)•, V) +∗Λ induces an injective map in cohomology. Henceforth, we will refer to this as +the restriction map. +Our goal is construct a transfer map, that is a linear map trans• : H• +a(Λ, V) → H• +a(Γ, V) +such that trans• ◦ res• is the identity on H• +a(Λ, V). Then it follows at once that res• must be +injective. By the above paragraph, we may do this by constructing an internal chain map +� +trans +• : ˜L∞((∗Γ)•, V) +∗Λ → ˜L∞((∗Λ)•, V) +∗Γ that restricts to the identity on ˜L∞((∗Λ)•, V) +∗Γ. +11 + +Let f ∈ L∞((∗Γ)k, V) be such that ˜f ∈ ˜L∞((∗Γ)k, V) +∗Λ. For each x ∈ (∗Γ)k, define +fx : ∗Γ → V +fx(g) := π(g)f(g−1x) +In other words, fx(g) is just (ρ1(g)f)(x) as in (1). Since ˜f ∈ ˜L∞((∗Γ)k, V) +∗Λ, for any γ ∈ ∗Λ +and g ∈ ∗Γ, +fx(gγ) − fx(g) ∈ Vinf +Let us choose representatives of left ∗Λ-cosets in ∗Γ and restrict fx to this set of repre- +sentatives so that we can regard fx as an internal map fx : ∗Γ/∗Λ → V. Moreover, since +fx ∈ L∞(∗Γ/∗Λ, V), we can apply the mean m constructed in Lemma 3.5 to define the internal +map transk(f) : L∞((∗Γ)k, V) → L∞((∗Γ)k, V) by +transk(f)(x) = m(fx) +Since ˜m is ∗Γ-invariant, this means that for g ∈ ∗Γ, m(fgx) − π(g)m(fx) ∈ Vinf, and implies +that +(transk(f))(gx) − π(g) transk(f)(x) ∈ Vinf. +This establishes that for f ∈ L∞((∗Γ)k, V) with ˜f ∈ ˜L∞((∗Γ)k, V) +∗Λ, we have +� +transk(f) ∈ +˜L∞((∗Γ)k, V) +∗Γ. Therefore trans• induces a chain map +˜ +trans +• : ˜L∞((∗Γ)•, V) +∗Λ → ˜L∞((∗Λ)•, V) +∗Γ. +Finally, if ˜f is already ∗Γ-invariant, then fx is constant up to infinitesimals, and thus m(fx) +is equal, up to an infinitesimal, to the value of that constant, which is f(x). This shows that +� +trans +k is the identity when restricted to ˜L∞((∗Λ)•, V) +∗Γ, and concludes the proof. +Proposition 1.5 is now an easy consequence. +Proposition (Proposition 1.5). Let Λ ≤ Γ be coamenable. If Λ is uniformly U-stable with a +linear estimate, then so is Γ. +Proof. Suppose that Λ is uniformly U-stable with a linear estimate, and let Γ be a coamenable +supergroup of Λ. We aim to show that Γ is also uniformly U-stable with a linear estimate. +By Theorem 2.4, it suffices to show that all Ulam classes supported on U vanish in H2 +a(Γ, W), +where W is an Ulam ∗Γ-module. Now by Proposition 1.8, it suffices to show that the images +of such classes under the restriction map res2 : H2 +a(Γ, W) → H2 +a(Λ, W) vanish, since the latter +is injective. By Lemma 3.4 these are Ulam classes of Λ. But since Λ is uniformly U-stable +with a linear estimate, by Theorem 2.4 again, all Ulam classes in H2 +a(Λ, W) vanish, and we +conclude. +3.3 +Pullbacks and amenable kernels +Let N ≤ Γ be an amenable normal subgroup, and let V be a dual asymptotic Banach ∗(Γ/N)- +module, which pulls back to a dual asymptotic Banach ∗Γ-module. Precomposing cochains by +the projection ∗Γ → ∗(Γ/N) defines the pullback p• : H• +a(Γ/N, V) → H• +a(Γ, V). The following +can be proven via a similar argument as in Lemma 3.4: +Lemma 3.6. Let W be an Ulam ∗(Γ/N)-module. Then W is also an Ulam ∗Γ-module, and +the pullback p2 : H2 +a(Γ/N, W) → H2 +a(Γ, W) sends Ulam classes to Ulam classes. +12 + +With this language, Proposition 1.9 is just a reformulation of Corollary 3.3: +Proposition (Proposition 1.9). Let N ≤ Γ be an amenable normal subgroup. Then the +pullback Hn +a(Γ/N, V) → Hn +a(Γ, V) is an isomorphism, for all n ≥ 0 and all dual asymptotic +Banach ∗Γ-modules V. +And we deduce Proposition 1.6 analogously: +Proposition (Proposition 1.6). Let N ≤ Γ be an amenable normal subgroup. If Γ is uni- +formly U-stable with a linear estimate, then so is Γ/N. +Proof. Suppose that Γ is uniformly U-stable with a linear estimate, and let N be an amenable +normal subgroup of Γ. We aim to show that Γ/N is also uniformly U-stable with a linear +estimate. By Theorem 2.4, it suffices to show that all Ulam classes supported on U vanish +in H2 +a(Γ/N, W), where W is an Ulam ∗(Γ/N)-module. Now by Proposition 1.9, it suffices +to show that the pullback of such classes under H2 +a(Γ/N, W) → H2 +a(Γ, W) vanish, since the +latter is an isomorphism. By Lemma 3.6 these are Ulam classes of Γ. But since Γ is uniformly +U-stable with a linear estimate, by Theorem 2.4 again, all Ulam classes in H2 +a(Γ, W) vanish, +and we conclude. +4 +Asymptotic cohomology of lamplighters +In this section we prove Theorem 1.7, which we recall for the reader’s convenience: +Theorem. Let Γ, Λ be two countable groups, where Λ is infinite and amenable. +Then +Hn +a(Γ ≀ Λ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V. +Remark 4.1. In fact, the theorem will hold for a larger class of coefficients, obtained as +ultraproducts of separable Banach spaces. This does not however lead to a stronger stability +result: see Remark 6.1. +We start by finding a suitable Zimmer-amenable Γ-space: +Lemma 4.2 ([Mon22, Corollary 8, Proposition 9]). Let Γ, Λ be two countable groups, where +Λ is amenable. Let µ0 be a distribution of full support on Γ, and let µ be the product measure +on S := ΓΛ. Then S is a Zimmer-amenable (Γ ≀ Λ)-space. +The reason why this space is useful for computations is that it is highly ergodic. Recall that +a Γ-space S is ergodic if every Γ-invariant function S → R is essentially constant. When S is +doubly ergodic, that is the diagonal action of Γ on S ×S is ergodic, we even obtain ergodicity +with separable coefficients, meaning that for every Γ-module E, every Γ-equivariant map +S → E is essentially constant [Mon22, 2.A, 4.B]. +Lemma 4.3 (Kolmogorov [Mon22, 2.A, 4.B]). Let Γ, Λ be two countable groups, where Λ is +infinite, and let S be as in Lemma 4.2. Then Sm is an ergodic (Γ≀Λ)-space, for every m ≥ 1. +For our purposes, we will need an approximate version of ergodicity (namely, almost +invariant functions are almost constant) and also the module E will only be endowed with +an approximate action of Γ. The ergodicity assumption still suffices to obtain this: +13 + +Lemma 4.4. Let S be a probability measure Γ-space, and suppose that the action of Γ on S +is ergodic. Then whenever f : S → R is a measurable function such that ∥g · f − f∥ < ε for +all g ∈ Γ, there exists a constant c ∈ R such that |f(s) − c| < ε for almost every s ∈ S. +Proof. We define F : S → R : s �→ esssupg∈Γf(g−1s). By construction, F is Γ-invariant, +and moreover ∥F − f∥ < ε. By ergodicity, F is essentially equal to a constant c, and thus +|f(s) − c| < ε for a.e. s ∈ S. +Lemma 4.5. Let S be a probability measure Γ-space, and suppose that the action of Γ on +S × S is ergodic. Suppose moreover, that E is a separable Banach space endowed with a map +Γ × E → E : v �→ g · v such that ∥g · v∥ = ∥v∥ for all g ∈ Γ, v ∈ E. +Then whenever f : S → E is a measurable function such that ∥g · f − f∥ < ε for all +g ∈ Γ, where (g · f)(s) = g · f(g−1s), there exists a vector v ∈ E such that ∥f(s) − v∥ < 3ε +for almost every s ∈ S. +Proof. We define F : S × S → R : (s, t) �→ ∥f(s) − f(t)∥. Then +∥g · F − F∥ = ess sup| ∥g · f(g−1s) − g · f(g−1t)∥ − ∥f(s) − f(t)∥ | +≤ ess sup∥g · f(g−1s) − g · f(g−1t) − (f(s) − f(t))∥ ≤ 2∥g · f − f∥ < 2ε. +By the previous lemma, there exists a constant c such that |F(s, t) − c| < 2ε for all ε > 0. If +c < ε, then |f(s) − f(t)| < 3ε for a.e. s, t ∈ S, which implies the statement. +Otherwise, ∥f(s) − f(t)∥ > ε for a.e. s, t ∈ S. Let D ⊂ E be a countable dense subset. +Then for each d ∈ D the set f −1(Bε/2(d)) is a measurable subset of S, and the union of +such sets covers S. Since D is countable, there must exist d ∈ D such that f −1(Bε/2(d)) has +positive measure. But for all s, t in this set, ∥f(s) − f(t)∥ < ε, a contradiction. +We thus obtain: +Proposition 4.6. Let S be a doubly ergodic Γ-space. Let (Vn)n≥1 be a sequence of separable +dual Banach spaces such that V = � +ω +Vn has the structure of a dual asymptotic Banach +Γ-module be the corresponding asymptotic ∗Γ-module. Then the natural inclusion ˜V +∗Γ → +˜L∞(∗S, V) +∗Γ is an isomorphism. +Proof. Let f ∈ L∞ +b (∗S, V) = � +ω +L∞(S, Vn) be a lift of an element ˜f ∈ ˜L∞(∗S, V) +∗Γ. We write +f = (fn)ω. Then fact that ˜f is ∗Γ-invariant means that for every sequence (gn)n∈N ⊂ Γ it holds +(gn·fn−fn)ω ∈ L∞ +inf(∗S, V). Since this holds for every sequence (gn)n∈N, a diagonal argument +implies that there exists ε ∈ ∗Rinf such that for every g ∈ Γ it holds (g · fn −fn)ω ∈ ∗Rinf. It +then follows from Lemma 4.5 that there exist (vn)ω ∈ V such that (fn − 1vn) ∈ L∞ +inf(∗S, V). +Therefore f represents the same element of ˜L∞(∗S, V) as the image of an element of V. Since +˜f is ∗Γ-invariant, the corresponding element is actually in ˜V +∗Γ. +We are finally ready to prove Theorem 1.7: +Proof of Theorem 1.7. Let Γ, Λ be countable groups, where Λ is infinite and amenable. By +Lemma 4.2, using the same notation, S is a Zimmer-amenable (Γ ≀ Λ)-space. Therefore we +can apply Theorem 2.3, and obtain that the following complex computes H∗ +a(Γ ≀ Λ; V): +0 +˜ +d0 +−→ ˜L∞(∗S, V) +∗Γ +˜ +d1 +−→ ˜L∞((∗S)2, V) +∗Γ +˜ +d2 +−→ ˜L∞((∗S)3, V) +∗Γ +˜ +d3 +−→ · · · +14 + +Now by Lemma 4.3, Sm is a doubly ergodic (Γ ≀ Λ)-space, for every m ≥ 1. Thus Proposition +4.6 applies, and the natural inclusion ˜V +∗Γ → ˜L∞((∗S)m, V) +∗Γ is an isomorphism for every +m ≥ 1. Thus the above complex is isomorphic to +0 +˜ +d0 +−→ ˜V +∗Γ +˜ +d1 +−→ ˜V +∗Γ +˜ +d2 +−→ ˜V +∗Γ +˜d3 +−→ · · · +Each differential ˜ +dm is an alternating sum of (m+1) terms all equal to each other. Therefore +˜ +dm is the identity whenever m is even, and it vanishes whenever m is odd. The conclusion +follows. +5 +Thompson groups +In this section we prove Theorem 1.2. The statement for F ′ will be a special case of a more +general result for a large family of self-similar groups. The most general statement is the +following: +Theorem 5.1. Let Γ be a group, Γ0 a subgroup with the following properties: +1. There exists g ∈ Γ such that the groups {giΓ0g−i : i ∈ Z} pairwise commute; +2. Every finite subset of Γ is contained in some conjugate of Γ0. +Then Hn +a(Γ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V. In +particular, Γ is uniformly U-stable, with a linear estimate. +The theorem applies to the following large family of groups of homeomorphisms of the +real line: +Corollary 5.2. Let Γ be a proximal, boundedly supported group of orientation-preserving +homeomorphisms of the line. Then Hn +a(Γ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic +Banach ∗Γ-modules V. In particular, Γ is uniformly U-stable, with a linear estimate. +Remark 5.3. The fact that such groups have no quasimorphisms is well-known: see e.g. +[GG17, FFL21, Mon22]. +We refer the reader to Section 5.2 for the relevant definitions. In Corollary 5.8 we will +apply Corollary 5.2 to Thompson’s group F ′; the result for Thompson’s group F will follow +from Proposition 1.5. We deduce the stability of Thompson’s group T and V from these +general criteria in Section 5.3. +5.1 +Self-similar groups +In this section we prove Theorem 5.1. This will be done in a series of lemmas: +Lemma 5.4. Let Γ be a group, and suppose that there exists g ∈ Γ and Γ0 ≤ Γ such that +{giΓ0g−i : i ∈ Z} pairwise commute. Then there exists an epimorphism Γ0 ≀ Z → ⟨Γ0, g⟩ with +amenable (in fact, metabelian) kernel. +This is well-known and stated without proof in [Mon22]. We include a proof for com- +pleteness. +15 + +Proof. To make a clear distinction, we denote by H the abstract group Γ0, and by Γ0 the +subgroup of Γ. So we want to construct an epimorphism H ≀Z → ⟨Γ0, g⟩ ≤ Γ with metabelian +kernel. We define naturally +ϕ((gi)i∈Z, p) = +�� +i∈Z +tigit−i +� +tp. +Note that this product is well-defined since there are only finitely many non-identity terms, +and the order does not matter since different conjugates commute. By construction ϕ is +injective on Hi, that is the copy of H supported on the i-th coordinate in H ≀ Z. +Let +K := ker ϕ ∩ � +i Hi, and note that K is the kernel of the retraction H ≀ Z → Z restricted to +ker ϕ. So it suffices to show that K is abelian. +Let g, h ∈ K and write them as (gi)i∈Z and (hi)i∈Z (we omit the Z-coordinate since it is +always 0). We need to show that g and h commute. We have +1Γ = ϕ(g) = +� +i∈Z +tigit−i +and thus +g0 = +� +i̸=0 +tigit−i ∈ Γ. +But now g0 belongs to a group generated by conjugates of Γ0 in Γ that commute with it. In +particular this implies that g0 and h0 commute in Γ. Since ϕ|H0 is injective, this shows that +g0 and h0 commute in H0. Running the same argument on the other coordinates, we obtain +that gi and hi commute in Hi, for all i ∈ Z, and thus g and h commute. +The next facts are all contained in the literature: +Lemma 5.5 ([Mon22, Proposition 10]). Suppose that Γ0 ≤ Γ is such that every finite subset +of Γ is contained in some Γ-conjugate of Γ0. Then Γ0 is coamenable in Γ. +Lemma 5.6 ([MP03]). Let K ≤ H ≤ Γ. +1. If K is coamenable in Γ, then H is coamenable in Γ; +2. If K is coamenable in H and H is coamenable in Γ, then K is coamenable in Γ. +Remark 5.7. We warn the reader that if K is coamenable in Γ, then K need not be +coamenable in H [MP03]. +We are now ready to prove Theorem 5.1: +Proof of Theorem 5.1. Let Γ, Γ0 and g be as in the statement. By Lemma 5.4, there exists +a map Γ0 ≀ Z → ⟨Γ0, g⟩ with metabelian kernel. By Theorem 1.7 and Proposition 1.9, we +have Hn +a(⟨Γ0, g⟩, V) for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V. Now +by Lemma 5.5, Γ0 is coamenable in Γ. Finally, by Lemma 5.6, ⟨Γ0, g⟩ is coameanble in Γ. +Proposition 1.8 allows to conclude. +5.2 +Groups of homeomorphisms of the line +Let Γ be a group acting by homeomorphisms on the real line. We say that the action is +proximal if for all reals a < b and c < d there exists g ∈ Γ such that g · a < c < d < g · b. +The support of g ∈ Γ is the set {x ∈ R : g · x ̸= x}. We say that Γ is boundedly supported if +every element has bounded support. Note that boundedly supported homeomorphisms are +automatically orientation-preserving. +16 + +Proof of Corollary 5.2. Let Γ be as in the statement. Let Γ0 be the subgroup of elements +whose support is contained in [0, 1]. Let g ∈ Γ be such that g(0) > 1: such an element exists +because the action of Γ is proximal. Then it follows by induction, and the fact that Γ is +orientation-preserving, that the intervals {gi[0, 1] : i ∈ Z} are pairwise disjoint. Therefore +the conjugates giΓ0g−i pairwise commute. +Since Γ is boundedly supported, for every finite subset A ⊂ Γ there exists n such that +the support of each element of A is contained in [−n, n]. By proximality, there exists h ∈ Γ +such that h(0) < −n and h(1) > n. Then hΓ0h−1 is the subgroup of elements whose support +is contained in [−n, n], in particular it contains A. +Thus Theorem 5.1 applies and we conclude. +Let us now show how to obtain the statements on F and F ′ from Theorem 1.2 from +Corollary 5.2 and Proposition 1.5. +We refer the reader to [CFP96] for more details on +Thompson’s groups. +Thompson’s group F is the group of orientation-preserving piecewise linear homeomor- +phisms of the interval, with breakpoints in Z[1/2] and slopes in 2Z. The derived subgroup F ′ +coincides with the subgroup of boundedly supported elements. The action of F ′ (and thus +F) on [0, 1] preserves Z[1/2] ∩ (0, 1), and acts highly transitively on it; that is, for every pair +of ordered n-tuples in Z[1/2] ∩ (0, 1) there exists an element of F ′ sending one to the other. +Corollary 5.8. Thompson’s groups F and F ′ are uniformly U-stable, with a linear estimate. +Proof. We identify (0, 1) with the real line. The group F ′ is boundedly supported, and it is +proximal, since it acts transitively on ordered pairs of a dense set. Therefore Corollary 5.2 +applies and F ′ is uniformly U-stable, with a linear estimate. +Since the quotient F/F ′ is abelian, thus amenable, we see that F ′ is coamenable in F, and +thus conclude from Proposition 1.5 that F ′ is uniformly U-stable, with a linear estimate. +Remark 5.9. We could also deduce the stability of F from the stability of F ′ more directly, +without appealing to Proposition 1.5. Indeed, since F ′ is uniformly U-stable, simple, and +not linear, every homomorphism F ′ → U(n) is trivial - something we will come back to in +the next section. Therefore uniform U-stability of F ′ implies that every uniform asymptotic +homomorphism F ′ → U is uniformly close to the trivial one. It follows that every uniform +asymptotic homomorphism F → U is uniformly asymptotically close to one that factors +through Z2. We conclude by the stability of amenable groups [Kaz82, GLMR23]. +Other groups to which these criteria apply include more piecewise linear groups [BS16], +such as the Stein–Thompson groups [Ste92], or the golden ratio Thompson group of Cleary +[Cle00, BNR21]. In such generality some more care is needed, since the commutator subgroup +is sometimes a proper subgroup of the boundedly supported subgroup. The criteria also apply +for the piecewise proejective groups of Monod [Mon13] and Lodha–Moore [LM16]. In this +case, further care is needed, since the role of the commutator subgroup in the proofs above +has to be taken by the double commutator subgroup [BLR18]. This ties back to Question +1.10 from the introduction. +5.3 +T and V +In this section, we show how our previous results allow to prove stability of groups of home- +omorphisms of the circle and of the Cantor set as well. For simplicity of the exposition, we +17 + +only focus on Thompson’s groups T and V , but the proofs generalize to some analogously +defined groups, with the appropriate modifications. Our proof will involve a bounded gen- +eration argument for stability that was pioneered in [BOT13]. We will only use it a simple +version thereof, closer to the one from [BC20]. Recall that Γ is said to be boundedly generated +by the collection of subgroups H if there exists k ≥ 1 such that the sets {H1 · · · Hk : Hi ∈ H} +cover Γ. +Lemma 5.10. Let Γ be a discrete group. Suppose that there exists a subgroup H ≤ Γ with +the following properties: +1. Every homomorphism H → U(n) is trivial; +2. H is uniformly U-stable (with a linear estimate); +3. Γ is boundedly generated by the conjugates of H. +Then Γ is uniformly U-stable (with a linear estimate). +Proof. Let φn : Γ → U(dn) be a uniform asymptotic homomorphism with def(φn) =: εn. +Then φn|H : H → U(dn) is a uniform asymptotic homomorphism of H, therefore it is δn- +close to a homomorphism, where δn → 0. But by assumption such a homomorphism must +be trivial, so ∥φn(h) − Ikn∥ ≤ δn for all n. The same holds for all conjugates of H, up to +replacing δn by δn + 2εn. +By bounded generation, there exists k ≥ 1 such that each g ∈ Γ can be written as +g = h1 · · · hk, where each hi belongs to a conjugate of H. We estimate: +∥φn(g) − Idn∥ = +�����φn +� k +� +i=1 +hi +� +− Idn +����� ≤ +�����φn +�k−1 +� +i=1 +hi +� +φn(hk) − Idn +����� + εn += +�����φn +�k−1 +� +i=1 +hi +� +− Idn +����� + ∥φn(hk) − Idn∥ + εn ≤ · · · +· · · ≤ +k +� +i=1 +∥φn(hi) − Idn∥ + kεn ≤ k(δn + εn). +Therefore φn is k(δn + εn)-close to the trivial homomorphism, and we conclude. +Thompson’s group T is the group of orientation-preserving piecewise linear homeomor- +phisms of the circle R/Z preserving Z[1/2]/Z, with breakpoints in Z[1/2]/Z, and slopes in +2Z. Given x ∈ Z[1/2]/Z, the stabilizer of x is naturally isomorphic to F. Moreover, the germ +stabilizer T(x) (i.e. the group consisting of elements that fix pointwise some neighbourhood +of x) is isomorphic to F ′. +Corollary 5.11. Thompson’s group T is uniformly U-stable with a linear estimate. +Proof. We claim that Lemma 5.10 applies with H = T(0) ∼= F ′. Item 1. follows from the +fact F ′ does not embed into U(n) (for instance because it contains F as a subgroup, which +is finitely generated and not residually finite, and so cannot be linear by Mal’cev’s Theorem +[Mal40]), and F ′ is simple [CFP96]. Also, F ′ is uniformly U-stable with a linear estimate, +by Corollary 5.8. Therefore we are left to show the bounded generation statement. We will +18 + +show that for every g ∈ T there exist x, y ∈ Z[1/2]/Z such that g ∈ T(x)T(y). This suffices +because T acts transitively on Z[1/2]/Z, so T(x) and T(y) are both conjugate to H = T(0). +Let 1 ̸= g ∈ T, and choose x ̸= y ∈ Z[1/2]/Z such that g(y) /∈ {x, y}. Let I be a small +dyadic arc around y such that x /∈ I and x, y /∈ g(I). Choose an element f ∈ T(x) such +that f(I) = g(I). Let h be an element supported on I such that h|I = f −1g|I. Since x /∈ I, +we also have h ∈ T(x). Moreover h−1f −1g|I = id|I, so h−1f −1g ∈ G(y). We conclude that +g = fh · h−1f −1g ∈ T(x)T(y). +Thompson’s group V can be described as a group of homeomorphisms of the dyadic Cantor +set X := 2N. A dyadic brick is a clopen subset of the form Xσ := σ × 2N>k, for some σ ∈ 2k, +and every two dyadic bricks are canonically homeomorphic via Xσ → Xτ : σ × x �→ τ × x. +An element g ∈ V is defined by two finite partitions of V of the same size into dyadic bricks, +that are sent to each other via canonical homeomorphisms. +Corollary 5.12. Thompson’s group V is uniformly U-stable, with a linear estimate. +The proof is very similar to the proof for T, so we only sketch it: +Sketch of proof. Let x ∈ 2N be a dyadic point, that is a sequence that is eventually all 0, +and let V (x) denote the subgroup of V consisting of elements that fix a neighbourhood of x +pointwise. The same argument as in the proof of Corollary 5.11 shows that V is boundedly +generated by conjugates of V (x). +Now V (x) is isomorphic to a directed union of copies of V , which is finitely generated +and simple [CFP96], so by Mal’cev’s Theorem every homomorphism V (x) → U(n) is trivial. +Finally, V (x) contains a copy V0 of V such that the pair (V (x), V0) satisfies the hypotheses of +Theorem 5.1 (see [And22, Proposition 4.3.4] and its proof). We conclude by Lemma 5.10. +6 +Sharpness of our results +In this section we point out certain ways in which our results are sharp, by providing explicit +counterexamples to generalizations and converses. +Remark 6.1. There is a notion of strong Ulam stability, where one takes U to include unitary +groups of infinite-dimensional Hilbert spaces as well, typically equipped with the operator +norm. It is shown in [BOT13] that a subgroup of a strongly Ulam stable group is Ulam +stable. Therefore it is clear that Theorem 1.3 does not hold for strong Ulam stability. Even +restricting to separable Hilbert spaces does not help: it follows from the construction in +[BOT13] that if a countable group contains a free subgroup, then separable Hilbert spaces +already witness the failure of strong Ulam stability. +The framework of stability via asymptotic cohomology can be developed in this general +setting as well, with dual asymptotic Banach modules that are not finitary. Therefore the +counterexample above shows that Theorem 1.7 really needs the finitary assumption. The fact +that we could obtain dual asymptotic Banach modules obtained as ultraproducts of separable +spaces, analogously to [Mon22], does not help, since the dual asymptotic Banach modules +arising from a stability problem over infinite-dimensional Hilbert spaces are not of this form, +even when the Hilbert space are separable. +19 + +Remark 6.2. We proved in Proposition 1.5 that if Λ is coamenable in Γ and Λ is uniformly +U-stable with a linear estimate, then so is Γ. The converse does not hold. Let Fn be a free +group of rank n ≥ 2. Then Λ := � +n≥1 Fn admits a non-trivial quasimorphism, so it is not +uniformly U(1)-stable [BOT13], in particular it is not uniformly U-stable. However, Λ is +coamenable in Fn ≀ Z, which is uniformly U-stable with a linear estimate by Theorem 1.3. +On the other hand, if we replace “coamenable” by “finite index”, then the converse does +hold. This follows from the induction procedure in [BOT13] for Ulam stability, as detailed +in [Gam11, Lemma II.22]; the same proof can be generalized to all submultiplicative norms +[GLMR23, Lemma 4.3.6]. +Remark 6.3. We proved in Proposition 1.6 that if N is an amenable normal subgroup of Γ, +and Γ is uniformly U-stable with a linear estimate, then so is Γ/N. The converse does not +hold. Let Γ be the lift of Thompson’s group T, that is, the group of orientation-preserving +homeomorphisms of R that commute with the group Z of integer translations and induce T +on the quotient R/Z. These groups fit into a central extension +1 → Z → Γ → T → 1. +Now T is uniformly U-stable with a linear estimate, by Corollary 5.11, however Γ is not: +it is not even uniformly U(1)-stable, by [BOT13], since it has a non-trivial quasimorphism +[GS87]. +The next two remarks show that some results from [GLMR23] are also sharp. +Remark 6.4. The fundamental result of [GLMR23] is that the vanishing of asymptotic +cohomology implies uniform U-stability. The converse does not hold. Indeed, since u(1) ∼= R +with trivial adjoint action (because U(1) is abelian), it follows that the implication of Theorem +2.4 specializes to: If H2 +a(Γ, ∗R) = 0, then Γ is uniformly U(1)-stable, where ∗R is seen as a +dual asymptotic ∗Γ-module with a trivial ∗Γ action. +Now, let again Γ be the lift of Thompson’s group T, so that Γ contains a central subgroup +Z with Γ/Z ∼= T. The fact that Γ is not uniformly U(1)-stable implies that H2 +a(Γ, ∗R) ̸= 0. +But Proposition 1.9 then shows that H2 +a(T, ∗R) ̸= 0 either. However, T is uniformly U-stable +with a linear estimate, by Corollary 5.11. Morally, this is due to the fact that H2 +b(Γ, R) ∼= +H2 +b(T, R) ∼= R, but the former is spanned by a quasimorphisms, while the latter is not (see +e.g. [Cal09, Chapter 5]). +Remark 6.5. In [BOT13] it is shown that groups admitting non-trivial quasimorphisms are +not uniformly U(1)-stable. In [GLMR23, Proposition 1.0.6] this result is sharpened: the +authors show that Γ is uniformly U(1)-stable if and only if the non-zero element in the image +of H2 +b(Γ, Z) in H2 +b(Γ, R) have Gromov norm ∥·∥ bounded away from 0. They use this to show +that a finitely presented group is uniformly U(1)-stable if and only if it admits no non-trivial +quasimorphism [GLMR23, Corollary 1.0.10]. +The hypothesis of finite presentability is necessary. Let Γn denote the lift of Thompson’s +group T to R/nZ. That is, Γn is the group of orientation-preserving homeomorphisms of the +topological circle R/nZ, which commute with the cyclic group of rotations Z/nZ and induce +T on the quotient R/Z. Now T has no unbounded quasimorphisms (see e.g. [Cal09, Chapter +5]), and so Γn also has no unbounded quasimorphisms (this follows from the left exactness +of the quasimorphism functor [Cal09, Remark 2.90]). Therefore the group Γ := � +n≥2 Γn has +no unbounded quasimorphisms. +20 + +However, we claim that Γ is not uniformly U(1)-stable. By [GLMR23, Proposition 1.0.6], +it suffices to show that there exist bounded cohomology classes 0 ̸= ρn ∈ im(H2 +b(Γ, Z) → +H2 +b(Γ, R) such that ∥ρn∥ → 0. We let ρn be the Euler class of the representation Γ → Γn → +Homeo+(R/nZ), which admits an integral representative and so lies in the image of H2 +b(Γ, Z) +(see [Ghy01] for more information about Euler classes of circle actions). Moreover, using the +terminology of [Bur11], the representation is minimal, unbounded, and has a centralizer of +order n. Therefore ∥ρn∥ = 1/2n by [Bur11, Corollary 1.6], and we conclude. +Note that Γ is countable but infinitely generated. It would be interesting to produce a +finitely generated example (which would necessarily be infinitely presented). +7 +Approximation properties +In this section we discuss open problems about approximation properties of the groups treated +in this paper, and their relation to our results. We recall the following notions: +Definition 7.1. Let G be a family of metric groups. We say that Γ is (pointwise, uniformly) +G-approximable if there exists a (pointwise, uniform) asymptotic homomorphism φn : Γ → +Gn ∈ G that is moreover asymptotically injective, meaning that for all g ∈ Γ, g ̸= 1 it holds +lim inf +n→∞ φn(g) > 0. +The above terminology is not standard: most of the literature only deals with the point- +wise notion, and refers to that as G-approximability. The notion of uniform approximability +appeared in [FF21] with the name of strong G-approximability. +Example 7.2. If G is the family of symmetric groups equipped with the normalized Hamming +distance, then pointwise G-approximable groups are called sofic [Gro99, Wei00]. +If G is the family of unitary groups equipped with the Hilbert–Schmidt distance, then +pointwise G-approximable groups are called hyperlinear [R˘08]. +All amenable and residually finite groups are sofic, and all sofic groups are hyperlinear. +It is a major open question to determine whether there exists a non-sofic group. +In our context of submultiplicative norms on unitary groups, the following two notions of +approximability have been studied: +Example 7.3. Let G be the family of unitary groups equipped with the operator norm. +Then pointwise G-approximable groups are called MF [CDE13]. All amenable groups are +MF [TWW17]. It is an open problem to determine whether there exists a non-MF group. +Let G be the family of unitary groups equipped with the Frobenius norm, or more generally +with a Schatten p-norm, for 1 < p < ∞. Groups that are not pointwise G-approximable have +been constructed in [DCGLT20, LO20]. This is one of the very few cases in which a non- +example for pointwise approximability is known. +The following observation is well-known, and due to Glebsky and Rivera [GR09] and +Arzhantseva and P˘aunescu in the pointwise symmetric case [AP15]. We give a general proof +for reference: +Proposition 7.4. Let G be a family of metric groups that are locally residually finite, and +let Γ be a finitely generated group. Suppose that Γ is (pointwise, uniformly) G-stable and +(pointwise, uniformly) G-approximable. Then Γ is residually finite. +21 + +The hypothesis on G covers all cases above. When the groups in G are finite, this is clear, +and when they are linear, this follows from Mal’cev’s Theorem [Mal40]. +Proof. We proceed with the proof without specifying the type of asymptotic homomorphisms, +closeness, and approximability: the reader should read everything as pointwise, or everything +as uniform. +Let φ : Γ → G be an asymptotically injective asymptotic homomorphism. By stability, +there exists a sequence of homomorphisms ψ : Γ → G which is asymptotically close to φ. +Since φ is asymptotically injective, for each g ∈ Γ there exists N such that φn(g) ≥ ρ for all +n ≥ N and some ρ = ρ(g) > 0. Up to taking a larger N, we also have that ψn(g) ≥ ρ/2, in +particular ψn(g) ̸= 1. Since ψn(Γ) is a finitely generated subgroup of Gn ∈ G, it is residually +finite by hypothesis, and so ψn(g) survives in some finite quotient of ψn(Γ). Since this is also +a finite quotient of Γ, we conclude that Γ is residually finite. +In the special case of pointwise stability and Thompson’s group F, we obtain the following +more general version of a remark of Arzhantseva and Paunescu [AP15, Open problem]: +Corollary 7.5. Let G be the family of symmetric groups with the normalized Hamming +distance, the family of unitary groups with the Hilbert–Schmidt norm, or the family of unitary +groups with the operatorn norm. If Thompson’s group F is pointwise G-stable, then it is not +pointwise G-approximable, and in particular it is non-amenable. +As we mentioned in the introduction, the amenability of Thompson’s group F is one of +the most outstanding open problems in modern group theory. +Proof. Thompson’s group F is not residually finite [CFP96]. So it follows from Proposition +7.4 that it cannot be simultaneously pointwise G-stable and pointwise G-approximable. The +last statement follows from the fact that amenable groups are sofic, hyperlinear, and MF. +On the other hand, our results allow to settle the uniform approximability of Thompson’s +groups, with respect to unitary groups and submultiplicative norms: +Corollary 7.6. As usual, let U be the family of unitary groups equipped with submultiplicative +norms. Then Thompson’s groups F, F ′, T and V are not uniformly U-approximable. The +same holds for Γ ≀ Λ, whenever Λ is infinite and amenable, and Γ is non-abelian. +We remark that Thompson’s groups T and V are generally regarded as good candidates +for counterexamples to approximability problems. +Proof. The statement for F, T and V follows from Theorem 1.2 and Proposition 7.4, together +with the fact that they are not residually finite, and the statement for F ′ (which is not finitely +generated) follows from the fact that F ′ contains a copy of F [CFP96]. The lamplighter case +follows from Theorem 1.3 and Proposition 7.4, together with the fact that such lamplighters +are not residually finite [Gru57]. +We do not know whether Thompson’s groups are uniformly G-approximable, when G is +the family of unitary groups equipped with the Hilbert–Schmidt norm, and we conjecture +that this is not the case. In the next section, we examine the case of symmetric groups via +a more direct argument. +22 + +7.1 +Approximations by symmetric groups +We end by proving, by a cohomology-free argument, that some of the groups studied in this +paper are not uniformly approximable by symmetric groups, in a strong sense. For the rest +of this section, we denote by S the family of symmetric groups equipped with the normalized +Hamming distance. Our main result is an analogue of Corollary 5.2 for this approximating +family (see Section 5.2 for the relevant definitions): +Proposition 7.7. Let Γ be a proximal, boundedly supported group of orientation-preserving +homeomorphisms of the line. Then every uniform asymptotic homomorphism φn : Γ′ → +Skn ∈ S is uniformly asymptotically close to the trivial one. In particular, Γ′ is uniformly +S-stable, and not uniformly S-approximable. +The non-approximability follows from the fact that Γ′ is non-trivial (see Lemma 7.8). +Note that for Γ as in the statement, Γ′ is simple [GG17, Theorem 1.1], so in particular every +homomorphism Γ′ → Skn is trivial. +The proof relies on known results on the flexible uniform stability of amenable groups +[BC20] and uniform perfection of groups with proximal actions [GG17]. The finiteness of the +groups in S will play a crucial role. We start with the following lemma: +Lemma 7.8. Let Γ be as in Proposition 7.7. Then Γ′ is non-trivial, and the action of Γ′ on +the line has no global fixpoints. +Proof. If Γ′ is trivial, then Γ is abelian. This contradicts that the action is proximal and +boundedly supported. Indeed, given g ∈ Γ, since g is centralized, the action of Γ on R must +preserve the support of g, which is a proper subset of R. But then the action cannot be +proximal. +Now the set of global fixpoints of Γ′ is a closed subset X ⊂ R. Since Γ′ is normal in Γ, +the action of Γ preserves X. But the action of Γ on R is proximal, in particular every orbit +is dense, and since X is closed we obtain X = R. That is, Γ′ acts trivially on R. Since Γ +is a subgroup of Homeo+(R), this implies that Γ′ is trivial, which contradicts the previous +paragraph. +We proceed with the proof: +Proof of Proposition 7.7. It follows from [GG17, Theorem 1.1] that Γ′ is 2-uniformly perfect; +that is, every element of Γ′ may be written as the product of at most 2 commutators (this +uses the proximality hypothesis). Therefore it suffices to show that there exists a constant C +such that for all g, h ∈ Γ′ it holds dkn(φn([g, h]), idkn) ≤ Cεn, where dkn denotes the Hamming +distance on Skn and εn := def(φn). We drop the subscript n on φ and ε for clarity. +Now let g, h ∈ Γ′, and let I, J ⊂ R be bounded intervals such that g is supported on I +and h is supported on J. Since Γ′ acts without global fixpoints by Lemma 7.8, there exists +t ∈ Γ′ such that t · inf(J) > sup(I). Since Γ′ is orientation-preserving, the same holds for +all powers of t. In particular [g, tiht−i] = 1 for all i ≥ 1. Next, we apply [BC20, Theorem +1.2] to the amenable group ⟨t⟩, to obtain an integer N such that kn ≤ N ≤ (1 + 1218ε)kn +and a permutation τ in SN such that dN(φ(t)i, τ i) ≤ 2039ε for all i ∈ Z. Here dN denotes +the normalized Hamming distance on the symmetric group SN, and φ is extended to a map +23 + +φ : Γ′ → SN with every φ(g) fixing each point in {kn + 1, . . . , N}. +We compute (using +τ N! = idN): +dkn(φ([g, h]), idkn) ≤ dN(φ([g, h]), idN) ≤ dN([φ(g), φ(h)], idN) + O(ε) += dN([φ(g), τ N!φ(h)τ −N!], idN) + O(ε) +≤ dN([φ(g), φ(tN!)φ(h)φ(t−N!)], idN) + O(ε) +≤ dN(φ([g, tN!ht−N!]), idN) + O(ε) += dN(φ(1), idN) + O(ε) ≤ O(ε). +Thus, there exists a constant C independent of g and h (C = 20000 suffices) such that +dkn(φ([g, h]), idkn) ≤ Cε, which concludes the proof. +Corollary 7.9. Consider the Thompson groups F ′, F, T. +1. Every asymptotic homomorphism φn : F ′ → Skn ∈ S is uniformly asymptotically close +to the trivial one. +2. Every asymptotic homomorphism φn : F → Skn ∈ S is uniformly asymptotically close +to one that factors through the abelianization. +3. Every asymptotic homomorphism φn : T → Skn ∈ S is uniformly asymptotically close +to the trivial one. +Proof. Item 1. is an instance of Proposition 7.7: indeed F ′ satisfies the hypotheses for Γ, +and F ′′ = F ′ since F ′ is simple. For Item 2., pick a section σ : Ab(F) → F, and define +ψn(g) := φn(σ(Ab(g))). Using that ψn|F ′ is uniformly asymptotically close to the sequence +of trivial maps, we obtain that φn and ψn are uniformly asymptotically close, and ψn factors +as F → Ab(F) +φn◦σ +−−−→ Skn. Finally, Item 3. follows again from Item 1. and the fact that every +element of T can be written as a product of two elements in isomorphic copies of F ′ (see the +proof of Corollary 5.11). +The corollary immediately implies that F, F ′ and T are not uniformly S-approximable, +and that F ′ and T are uniformly S-stable. Since F has infinite abelianization, it follows from +[BC20, Theorem 1.4] that it is not uniformly S-stable. 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Ergodic +theory and harmonic analysis (Mumbai, 1999). +Department of Mathematics, ETH Z¨urich, Switzerland +E-mail address: francesco.fournier@math.ethz.ch +Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel +E-mail address: bharatrm.rangarajan@mail.huji.ac.il +27 + diff --git a/39E2T4oBgHgl3EQfjgft/content/tmp_files/load_file.txt b/39E2T4oBgHgl3EQfjgft/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2057e588079b5bf53d907d73c67c56a75510cdb0 --- /dev/null +++ b/39E2T4oBgHgl3EQfjgft/content/tmp_files/load_file.txt @@ -0,0 +1,1358 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf,len=1357 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='03970v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='GR] 10 Jan 2023 Ulam stability of lamplighters and Thompson groups Francesco Fournier-Facio and Bharatram Rangarajan January 11, 2023 Abstract We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' These include lamplighters Γ ≀ Λ where Λ is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups F, F ′, T and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 1 Introduction Let Γ be a countable discrete group, and let U be a family of finite-dimensional unitary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The problem of stability asks whether every almost-homomorphism Γ → U ∈ U is close to a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' To formalize this we need to choose a norm, and a way to interpret these approximate notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We focus on the classical setting of uniform defects and distances, with respect to submultiplicative norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let U := {(U(k), ∥·∥)} be a family of finite-dimensional unitary groups equipped with bi- invariant submultiplicative norms ∥·∥ (we allow U(k) to appear multiple times with different norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For instance ∥ · ∥ could be the operator norm - the most classical case - or more generally a Schatten p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given a map φ : Γ → U(k), we define its defect to be def(φ) := sup g,h∈Γ ∥φ(gh) − φ(g)φ(h)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given another map ψ : Γ → U(k), we define the distance between them to be dist(φ, ψ) := sup g∈Γ ∥φ(g) − ψ(g)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' A uniform asymptotic homomorphism is a sequence of maps φn : Γ → U(kn) such that def(φn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We denote this simply by φ : Γ → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We say that φ, ψ : Γ → U are uniformly asymptotically close if they have the same range degrees and dist(φn, ψn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The group Γ is uniformly U-stable if every uniform asymptotic homomorphism is uni- formly asymptotically close to a sequence of homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 1 We can also talk quantitatively about stability, by asking how close a homomorphism we can choose, in terms of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This leads to the notion of stability with a linear estimate, which will be relevant for us and which we define precisely in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Early mentions of similar problems can be found in the works of von Neumann [vN29] and Turing [Tur38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In [Ula60, Chapter 6] Ulam discussed more general versions of stability, which has since inspired a large body of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Uniform U-stability has been studied mostly when U is the family of unitary groups equipped with the operator norm, for which the notion is typically referred to as Ulam stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In this contest, Kazhdan proved stability of amenable groups [Kaz82], while Burger, Ozawa and Thom proved stability of certain special linear groups over S-integers, and instability of groups admitting non-trivial quasimorphisms [BOT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' More recently, the second author, Glebsky, Lubotzky and Monod proved Ulam stabil- ity of certain lattices in higher rank Lie groups, with respect to arbitrary submultiplicative norms [GLMR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For the proof, they introduce a new cohomology theory, called asymptotic cohomology, and prove that stability is implied by the vanishing of certain asymptotic co- homology classes α ∈ H2 a(Γ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We refer the reader to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 for the relevant definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The goal of this paper is to further the understanding of asymptotic cohomology, and apply this to prove new stability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The main one is the stability of the classical Thompson groups: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s groups F, F ′, T and V are uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' As remarked by Arzhantseva and P˘aunescu [AP15, Open problem], the analogous state- ment for pointwise stability in permutation of F would imply that F is not sofic, thus proving at once the existence of a non-sofic group and the non-amenability of F: two of the most remarkable open problems in modern group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We will discuss these problems and their relation to our results in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 for F and F ′ will follow from a stability result for certain lamplighters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given groups Γ, Λ, the corresponding lamplighter (or restricted wreath product) is the group Γ ≀ Λ = (⊕ΛΓ) ⋊ Λ, where Λ acts by shifting the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Λ be two countable groups, where Λ is infinite and amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Γ ≀ Λ is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By itself, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 provides a plethora of examples of uniformly U-stable groups, to a degree of flexibility that was not previously available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For instance, using classical embedding results [HNN49] it immediately implies the following: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Every countable group embeds into a 3-generated group which is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, this gives a proof that there exist uncountably many finitely generated uniformly U-stable groups, a fact which could also be obtained by applying Kazhdan’s The- orem [Kaz82] to an infinite family of finitely generated amenable groups, such as the one constructed by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Neumann [Neu37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2 In order to obtain stability of F and F ′ from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3, we exploit coamenability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Recall that a subgroup Λ ≤ Γ is coamenable if the coset space Γ/Λ admits a Γ-invariant mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It is well known that F ′ and F contain a coamenable lamplighter F ≀ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore the stability of F and F ′ (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8) follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3, and the following result: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Λ ≤ Γ be coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Λ is uniformly U-stable with a linear estimate, then so is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This can be seen as a relative version of the celebrated result of Kazhdan, stating that amenable groups are uniformly U-stable [Kaz82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' To complete the picture, we also prove another relative version of Kazhdan’s Theorem, which is sort of dual to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let N ≤ Γ be an amenable normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Γ is uniformly U-stable with a linear estimate, then so is Γ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The fact that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 is not special to Thompson’s group F: this phenomenon is typical of several groups of piecewise linear and piecewise projective homeomorphisms, which enjoy some kind of self-similarity properties (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Stability of T and V then follow from these results, to- gether with a bounded generation argument analogous to the one from [BOT13] (Corollaries 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' As we mentioned above, the tool underlying the proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 is asymptotic cohomology, in particular the vanishing of certain classes in degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In this framework, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 takes the following form: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Λ be two countable groups, where Λ is infinite and amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Hn a(Γ ≀ Λ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Here the word finitary refers to the fact that these modules arise from stability problems with respect to finite-dimensional unitary representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This hypothesis is crucial: see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 also follow from results in asymptotic cohomology, that this time does not need the finitary assumption: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Λ ≤ Γ be coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the restriction map Hn a(Γ, V) → Hn a(Λ, V) is injective, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let N ≤ Γ be an amenable normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the pullback map Hn a(Γ/N, V) → Hn a(Γ, V) is an isomorphism, for all n ≥ 0 and all dual asymptotic Banach ∗(Γ/N)-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Despite the lack of a general theorem connecting the two theories, asymptotic cohomology seems to be closely connected to bounded cohomology, a well-established cohomology theory [Joh72, Gro82, Iva85, Mon01, Fri17] that has become a fundamental tool in rigidity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The vanishing result for asymptotic cohomology of lattices leading to stability [GLMR23] follows closely the vanishing result for bounded cohomology of high-rank lattices [BM99, BM02, MS04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Similarly, our proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7 and Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9 follow closely the corresponding bounded-cohomological results: for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7 this was recently proven by Monod [Mon22], while for Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8 this is a foundational result in bounded cohomology [Mon01, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6] (see also [MP03]), and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9 is an analogue of Gromov’s 3 Mapping Theorem [Gro82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that the bounded cohomology of T and V has also been recently computed [FFLM21, MN21, And22], but only with trivial real coefficients, and our proofs are of a different nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We thus hope that the steps we undertake to prove our main results will be useful to produce more computations in asymptotic cohomology, and therefore more examples of uni- formly U-stable, and in particular Ulam stable, groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Our results have applications to the study of approximating properties of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' While questions on pointwise approximation, such as soficity, hyperlinearity, and matricial finite- ness, are in some sense disjoint from the content of this paper, our stability results imply that some of the groups considered in this paper are not uniformly approximable with respect to the relevant families U (Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We are also able to treat the case of symmetric groups endowed with the Hamming distance, by a more direct argument (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We end this introduction by proposing a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' There is a notion of strong Ulam stabil- ity, where the approximations take values in unitary groups of possibly infinite-dimensional Hilbert spaces, with the operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It is a well-known open question whether strong Ulam stability coincides with amenability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In this direction it is known that strong Ulam stable groups have no non-abelian free subgroups [BOT13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2], but there exist groups without non-abelian free subgroups that are not strong Ulam stable [Alp20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' On the other hand, our results also prove uniform U-stability stability of the piecewise projective groups of Monod [Mon13] and Lodha–Moore [LM16], which are nonamenable and without free subgroups (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore we ask the following: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be a countable group without non-abelian free subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Is Γ uniformly U-stable (with a linear estimate)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Or at least Ulam stable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, are all countable torsion groups Ulam stable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In other words: if Γ is not Ulam stable, must Γ contain a non-abelian free subgroup?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' To our knowledge it is not every known if groups admitting non-trivial quasimorphisms must contain non-abelian free subgroups: see [Man05] and [Cal10] for partial results in this direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Conventions: All groups are assumed to be discrete and countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The set of natu- ral numbers N starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' A non-principal ultrafilter ω on N is fixed for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Outline: We start in Section 2 by reviewing the framework of asymptotic cohomology and its applications to stability, as developed in [GLMR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In Section 3 we discuss hered- itary properties for Ulam stability, and prove Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We then move to lamplighters and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 in Section 4, then to Thompson groups proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In Section 6 we provide examples showing that some of our results and some of the results from [GLMR23] are sharp, and conclude in Section 7 by discussing applications to the study of metric approximations of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Acknowledgements: The authors are indebted to Alon Dogon, Lev Glebsky, Alexander Lubotzky and Nicolas Monod for useful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 4 2 Uniform stability and asymptotic cohomology In this section, we shall briefly summarize the notion of defect diminishing that allows us to formulate the stability problem as a problem of lifting of homomorphisms with abelian kernel, which in turn motivates the connection to second cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For a more detailed description, refer to Section 2 in [GLMR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 Uniform stability and defect diminishing We begin by reviewing some basic notions of ultraproducts and non-standard analysis, be- fore formulating the stability problem as a homomorphism lifting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For this, it is convenient to describe a uniform asymptotic homomorphism (which is a sequence of maps) as one map of ultraproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This in turn allows us to perform a soft analysis to obtain a (true) homomorphism to a quotient group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Recall that ω is a fixed non-principal ultra- filter on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The algebraic ultraproduct � ω Xn of an indexed collection {Xn}n∈N of sets is defined to be � ω Xn := � n∈N Xn/ ∼ where for {xn}n∈N, {yn}n∈N ∈ � n∈N Xn, we define {xn}n∈N ∼ {yn}n∈N if {n : xn = yn} ∈ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Ultraproducts can be made to inherit algebraic structures of their building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For instance, for a group Γ, the ultraproduct � ω Γ, called the ultrapower and denoted ∗Γ, is itself a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Another important example we will use is the field of hyperreals ∗R, the ultrapower of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Objects (sets, functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=') that arise as ultraproducts of standard objects are referred to as internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Important examples of non-internal objects are the subsets ∗Rb of bounded hyperreals, consisting of elements {xn}ω ∈ ∗R for which there exists S ∈ ω and C ∈ R≥0 such that |xn| ≤ C for every n ∈ S, and the subset ∗Rinf of infinitesimals, consisting of elements {xn}ω ∈ ∗R such that for every real ε > 0, there exists S ∈ ω such that |xn| < ε for every n ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For x, y ∈ ∗R, write x = Oω(y) if x/y ∈ ∗Rb, and write x = oω(y) if x/y ∈ ∗Rinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, x ∈ ∗Rb is equivalent to x = Oω(1) while ε ∈ ∗Rinf is equivalent to ε = oω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The subset ∗Rb forms a valuation ring with ∗Rinf being the unique maximal ideal, with quotient ∗Rb/∗Rinf ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The quotient map st : ∗Rb → R is known as the standard part map or limit along the ultrafilter ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The previous construction can also be replicated for Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let {Wn}n∈N be a family of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then W = � ω Wn can be given the structure of a ∗R-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In fact, it also comes equipped with a ∗R-valued norm, allowing us to define the external subsets Wb and Winf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The quotient ˜ W := Wb/Winf is a real Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given a uniform asymptotic homomorphism {φn : Γ → U(kn)}n∈N with def(φn) =: εn → 0, construct the internal map φ : ∗Γ → � ω U(kn) where φ := � ω φn, with (hyperreal) defect ε := {εn}ω ∈ ∗Rinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the question of uniform stability with a linear estimate can be rephrased as asking whether there exists an internal homomorphism ψ : ∗Γ → � ω U(kn) such that their (hyperreal) distance satisfies dist(φ, ψ) := {dist(φn, ψn)}ω = Oω(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The advantage of rephrasing the question in terms of internal maps is that an internal map φ : ∗Γ → � ω U(kn) with defect ε ∈ ∗Rinf induces a true homomorphism ˜φ : ∗Γ → � ω U(kn)/B(ε) where B(ε) is the (external) normal subgroup of � ω U(kn) comprising ele- ments that are at a distance Oω(ε) from the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, the question of uniform stability with a linear estimate can equivalently be rephrased as asking whether given such an internal map φ, can the homomorphism ˜φ : ∗Γ → � ω U(kn)/B(ε) be lifted to an internal 5 homomorphism ψ : ∗Γ → � ω U(kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Reinterpreting uniform stability with a linear estimate as a homomorphism lifting problem motivates a cohomological approach to capturing the obstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' However, the obstacle here is that the kernel B(ε) of the lifting problem is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This can be handled by lifting in smaller steps so that each step involves an abelian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Define a normal subgroup I(ε) of B(ε) comprising elements that are at a distance of oω(ε) from the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then we can attempt to lift ˜φ : ∗Γ → � ω U(kn)/B(ε) to an internal map ψ : ∗Γ → � ω U(kn) that is a homomorphism modulo I(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The problem is simpler from the cohomological point of view: since the norms are submultiplicative, the kernel B(ε)/I(ε) of this lifting problem is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The group Γ is said to have the defect diminishing property with respect to U if such a lift exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' more explicitly, Γ has the defect diminishing property if for every uniform asymptotic homomorphism φ : Γ → U there exists a uniform asymptotic homomorphism ψ with the same range such that dist(φ, ψ) = Oω(def(φ)) and def(ψ) = oω(def(φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 ([GLMR23, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Γ has the defect diminishing property with respect to U if and only if Γ is uniformly U-stable with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The obstruction to such a homomorphism lifting, with an abelian kernel B(ε)/I(ε), can be carefully modeled using a cohomology H• a(Γ, W) so that H2 a(Γ, W) = 0 implies the defect diminishing property (and consequently, uniform stability with a linear estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Here W = � ω u(kn) is an internal Lie algebra of � ω U(kn) equipped with an asymptotic action of the ultrapower ∗Γ constructed from the uniform asymptotic homomorphism φ that we start out with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The logarithm of the defect map ∗Γ × ∗Γ → � ω U(kn) : (g1, g2) �→ φ(g1)φ(g2)φ(g1g2)−1 would correspond to an asymptotic 2-cocycle in H2 a(Γ, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Such a cocycle is a coboundary in this setting (that is, it represents the zero class in H2 a(Γ, W)), if and only if the defect diminishing property holds for the asymptotic homomorphism φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 Asymptotic cohomology The reduction to a lifting problem with abelian kernel motivates a cohomology theory of Γ with coefficients in the internal Lie algebra W = � ω u(kn) of � ω U(kn), equipped with an asymptotic conjugation action of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In this section we review the formal definition of this cohomology, and state some results from [GLMR23] that we shall need to work with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let (Vn)n≥1 be a sequence of Banach spaces, and let V := � ω Vn be their algebraic ultra- product: we refer to such V as an internal Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For v ∈ V we denote by ∥v∥ the hyperreal (∥vn∥)ω ∈ ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We then denote by Vb := {v ∈ V : ∥v∥ ∈ ∗Rb};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Vinf := {v ∈ V : ∥v∥ ∈ ∗Rinf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the quotient ˜V := Vb/Vinf is a real Banach space, whose norm is induced by the ultralimit of ∥ · ∥ on Vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For each Vn denote by V # n its continuous dual, and let V# be the corresponding algebraic ultraproduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The pairing ⟨·, ·⟩n : V # n × Vn → R induces a pairing V# × V → ∗R which descends to ˜V# × ˜V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We call V# the internal dual of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 6 Now let Γ be a countable discrete group, and let π : ∗Γ×V → V be an internal map which preserves ∥ · ∥ and induces an isometric linear action ˜π : ∗Γ × ˜V → ˜V of ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Such a map π is referred to as an asymptotic ∗Γ-action on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We then call (π, V), or V, if π is understood from context, an asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given an internal Banach ∗Γ-module (π, V), the contragradient on each coordinate induces an internal map π# : ∗Γ × V# → V# mak- ing (π#, V#) into an asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We call a module V a dual asymptotic Banach ∗Γ-module if V is the dual of some asymptotic ∗Γ-module denoted V♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We decorate these definitions with the adjective finitary if each Vn is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now for each m ≥ 0 define the internal Banach space L∞((∗Γ)m, V) := � ω ℓ∞(Γm, Vn) (note that m is fixed and n runs through the natural numbers with respect to the ultrafilter ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Similarly to before, for f ∈ L∞((∗Γ)m, V) we denote ∥f∥ := (∥fn∥)ω ∈ ∗R and L∞ b ((∗Γ)m, V) := {f ∈ L∞((∗Γ)m, V) : ∥f∥ ∈ ∗Rb};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' L∞ inf((∗Γ)m, V) := {f ∈ L∞((∗Γ)m, V) : ∥f∥ ∈ ∗Rinf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given an asymptotic ∗Γ-action π on V, we can construct a natural asymptotic ∗Γ-action ρm : ∗Γ × L∞((∗Γ)m, V) → L∞((∗Γ)m, V) given by (ρm(g)(f))(g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gm) := πG(g)f(g−1g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , g−1gm) (1) Then the quotient ˜L∞((∗Γ)m, V) := L∞ b ((∗Γ)m, V)/L∞ inf((∗Γ)m, V) is again a real Banach space equipped with an isometric ∗Γ-action induced coordinate-wise by ρm, which defines the invariant subspaces ˜L∞((∗Γ)m, V) ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now define the internal coboundary map dm : L∞((∗Γ)m, V) → L∞((∗Γ)m+1, V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' dm(f)(g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gm) := m � j=0 (−1)jf(g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , ˆgj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gm), (2) which descends to coboundary maps ˜ dm : ˜L∞((∗Γ)m, V) → ˜L∞((∗Γ)m+1, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since ˜ dm is ∗Γ-equivariant, it defines the cochain complex: 0 ˜ d0 −→ ˜L∞(∗Γ, V) ∗Γ ˜ d1 −→ ˜L∞((∗Γ)2, V) ∗Γ ˜d2 −→ ˜L∞((∗Γ)3, V) ∗Γ ˜ d3 −→ · · · Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 ([GLMR23, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The m-th asymptotic cohomology of Γ with coefficients in V is Hm a (Γ, V) := ker( ˜ dm+1)/ im( ˜ dm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 7 Other resolutions may also be used to compute asymptotic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Recall ([Mon01, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2]) that a regular Γ-space S is said to be a Zimmer-amenable Γ-space if there exists a Γ- equivariant conditional expectation m : L∞(Γ×S) → L∞(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let S be a regular Γ-space with a Zimmer-amenable action of Γ, and let L∞((∗S)m, V) := � ω L∞ w∗(Sm, Vn) (where L∞ w∗(Sm, Vn) is the space of bounded weak-∗ measurable function classes from Sm to Vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Again, the asymptotic ∗Γ-action on V gives rise to a natural asymptotic ∗Γ-action on L∞((∗Γ)m, V) as in (1), making L∞((∗S)m, V) an asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The coboundary maps too can be defined just as in (2), to construct the cochain complex, and we have: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 ([GLMR23, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let S be a Zimmer-amenable Γ-space, and V be a dual asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then H• a(Γ, V) can be computed as the asymptotic cohomology of the cochain complex 0 ˜ d0 −→ ˜L∞(∗S, V) ∗Γ ˜ d1 −→ ˜L∞((∗S)2, V) ∗Γ ˜ d2 −→ ˜L∞((∗S)3, V) ∗Γ ˜ d3 −→ · · · In the context of uniform U-stability, the relevant asymptotic Banach ∗Γ-module we shall be interested is the ultraproduct W = � ω u(kn), where u(kn) is the Lie algebra of U(kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that we are only considering finite-dimensional unitary groups, so such a module is finitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given a uniform asymptotic homomorphism φ : ∗Γ → � ω U(n) with defect def(φ) ≤ω ε ∈ ∗Rinf, this can be used to construct the asymptotic action π : ∗Γ × W → W given by π(g)v = φ(g)vφ(g)−1, making W an asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We call such a module an Ulam ∗Γ-module supported on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Also, consider the map α : ∗Γ × ∗Γ → W given by α(g1, g2) = 1 ε log(φ(g1)φ(g2)φ(g1g2)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' (3) This map α induces an inhomogeneous 2-cocycle ˜α : ∗Γ × ∗Γ → ˜ W, and thus defines a class in H2 a(Γ, W), under the usual correspondence between inhomogeneous cochains and invariant homogeneous cochains [GLMR23, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We call such a class an Ulam class supported on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This class vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' ˜α is a coboundary, precisely when φ has the defect diminishing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thus Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 yields: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4 ([GLMR23, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Γ is uniformly U-stable with respect to U if and only if all Ulam classes supported on U vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, if H2 a(Γ, W) = 0 for every Ulam ∗Γ-module supported on U, then Γ is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3 Hereditary properties In this section, we first prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8 and deduce Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 from it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' then analogously we prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9 and deduce Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Both stability statements are not symmetric, and in fact we will see in Section 6 that the converses do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 More on Zimmer-amenability For the proofs of Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9, we will need a more precise version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 in a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' A regular Γ-space S is said to be discrete if it is a countable set equipped 8 with the counting measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It follows from the equivalent characterizations in [AEG94] that a discrete Γ-space is Zimmer-amenable precisely when each point stabilizer is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Λ ≤ Γ is a subgroup, then the action of Λ on Γ by left multiplication is free, so Γ is a discrete Zimmer-amenable Λ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If N ≤ Γ is an amenable subgroup, then the action of Γ on the coset space Γ/N has stabilizers equal to conjugates of N, so Γ/N is a discrete Zimmer-amenable Γ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For such spaces, we can provide an explicit chain map that implements the isomorphism in cohomology from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Indeed, the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 works by starting with a Γ-homotopy equivalence between the two complexes: 0 → L∞(Γ) → L∞(Γ2) → L∞(Γ3) → · · · 0 → L∞(S) → L∞(S2) → L∞(S3) → · · · which is then extended internally to the asymptotic version of these complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The case of dual asymptotic coefficients follows via some suitable identifications of the corresponding complexes (see the paragraph preceding [GLMR23, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In the case of a discrete group Γ and a discrete Zimmer-amenable Γ-space, the homotopy equivalence above can be chosen to be the orbit map om b : L∞(Sm) −→ L∞(Γm) om b (f)(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gm) = f(g1b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gmb);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' where b ∈ S is some choice of basepoint [Fri17, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore in this case we obtain the following more explicit version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let S be a discrete Zimmer-amenable Γ-space, with a basepoint b ∈ S, and let V be a dual asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the orbit map om b : L∞((∗S)m, V) −→ L∞((∗Γ)m, V) om b (f)(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gm) = f(g1b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , gmb);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' induces induces an isomorphism between H• a(Γ, V) and the cohomology of the complex 0 ˜ d0 −→ ˜L∞(∗S, V) ∗Γ ˜ d1 −→ ˜L∞((∗S)2, V) ∗Γ ˜ d2 −→ ˜L∞((∗S)3, V) ∗Γ ˜ d3 −→ · · · In the two basic examples of discrete Zimmer-amenable spaces from above, we obtain: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Λ ≤ Γ be a subgroup, and let V be a dual asymptotic Banach ∗Γ- module, which restricts to a dual asymptotic Banach ∗Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the restriction of cochains L∞((∗Γ)m, V) → L∞((∗Λ)m, V) induces an isomorphism between H• a(Λ, V) and the cohomology of the complex: 0 ˜ d0 −→ ˜L∞(∗Γ, V) ∗Λ ˜ d1 −→ ˜L∞((∗Γ)2, V) ∗Λ ˜ d2 −→ ˜L∞((∗Γ)3, V) ∗Λ ˜ d3 −→ · · · Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Seeing Γ as a discrete Zimmer-amenable Λ-space, with basepoint 1 ∈ Γ, the orbit map is nothing but the restriction of cochains, and we conclude by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 9 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let N ≤ Γ be an amenable normal subgroup, and let V be a dual asymptotic Banach ∗(Γ/N)-module, which pulls back to a dual asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the pullback of cochains L∞((∗(Γ/N))m, V) → L∞((∗Γ)m, V) induces an isomorphism between H• a(Γ, V) and H• a(Γ/N, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Seeing Γ/N as a discrete Zimmer-amenable Γ-space, with basepoint the coset N, the orbit map is nothing but the pullback of cochains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' So Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 yields an isomorphism between H• a(Γ, V) and the cohomology of the complex: 0 ˜ d0 −→ ˜L∞(∗(Γ/N), V) ∗Γ ˜ d1 −→ ˜L∞((∗(Γ/N))2, V) ∗Γ ˜ d2 −→ ˜L∞((∗(Γ/N))3, V) ∗Γ ˜d3 −→ · · · But since the action of ∗Γ on both ∗(Γ/N) and V factors through ∗(Γ/N), the above complex coincides with the standard one computing H• a(Γ/N, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We will use these explicit isomorphisms in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Later, for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7, non-discrete Zimmer-amenable spaces will also appear, but in that case we will only need the existence of an abstract isomorphism as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 Restrictions and coamenability Let Λ ≤ Γ be a (not necessarily coamenable) subgroup, and V be a dual asymptotic Ba- nach ∗Γ-module, which restricts to a dual asymptotic Banach ∗Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The restriction ˜L∞((∗Γ)•, V) ∗Γ → ˜L∞((∗Λ)•, V) ∗Λ induces a map in cohomology, called the restriction map, and denoted res• : H• a(Γ, V) → H• a(Λ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This map behaves well with respect to Ulam classes: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let W be an Ulam ∗Γ-module supported on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then W is also an Ulam ∗Λ- module supported on U, and the restriction map res2 : H2 a(Γ, W) → H2 a(Λ, W) sends Ulam classes to Ulam classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let φ : Γ → U be a uniform asymptotic homomorphism, and let W be the corre- sponding Ulam ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then restricting φn to Λ for each n yields a uniform asymptotic homomorphism φ|Λ : Λ → U, with def(φ|Λ) ≤ω def(φ) and endows W with an asymptotic ∗Λ-action making it into an Ulam ∗Λ-module supported on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The cocycle corresponding to φ is defined via the map α : ∗Γ × ∗Γ → W : (g1, g2) �→ 1 ε log(φ(g1)φ(g2)φ(g1g2)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since def(φ|Λ) ≤ω ε, restricting α to ∗Λ × ∗Λ yields a valid cocycle associated to the uniform asymptotic homomorphism φΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It follows that the chain map ˜L∞((∗Γ)•, V) ∗Γ → ˜L∞((∗Λ)•, V) ∗Λ preserves the set of cocycles defined via uniform asymptotic homomorphisms, and therefore preserves Ulam classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now suppose that Λ ≤ Γ is coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This means, by definition, that there exists a Γ-invariant mean on Γ/Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' that is, there exists a linear functional m : ℓ∞(Γ/Λ) → R such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' m(1Γ/Λ) = 1, where 1Γ/Λ denotes the constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' |m(f)| ≤ ∥f∥ for all f ∈ ℓ∞(Γ/Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' m(g · f) = m(f) for all g ∈ Γ and all f ∈ ℓ∞(Γ/Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' As with the absolute case [GLMR23, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='20], we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose that Λ ≤ Γ is coamenable, and let V be a dual asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then there exists an internal map m : L∞(∗Γ/∗Λ, V) → V which induces a map ˜m : ˜L∞(∗Γ/∗Λ, V) → ˜V with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If ˜f is the constant function equal to ˜v ∈ ˜V, then ˜m( ˜f) = ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' ∥ ˜m( ˜f)∥ ≤ ∥ ˜f∥ for all ˜f ∈ ˜L∞(∗Γ/∗Λ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' ˜m(g · ˜f) = ˜m( ˜f) for all g ∈ ∗Γ and all ˜f ∈ ˜L∞(∗Γ/∗Λ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Consider f = {fn}ω ∈ L∞(∗Γ/∗Λ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since V is a dual asymptotic ∗Γ-module with predual V♭, for each λ ∈ V♭, we get an internal map f λ : ∗Γ/∗Λ → ∗R : x �→ f(x)(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that f λ being internal, it is of the form {f λ n}ω where f λ n ∈ ℓ∞(Γ/Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This allows us to construct the internal map mλ in : L∞(∗Γ/∗Λ, V) → ∗R as mλ in(f) = {m � f λ n � }ω and finally min : L∞(∗Γ/∗Λ, V) → V as min(f)(λ) = mλ in(f) It is straightforward to check that min as defined induces a linear map ˜m : ˜L∞(∗Γ/∗Λ, V) → ˜V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' As for ∗Γ-equivariance, this follows from the observation that (g · f)λ(x) = π(g)f(g−1x)(λ) while (g·f λ)(x) = f(g−1x)(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The conditions on ˜m follow from the definition and properties of the Γ-invariant mean m on ℓ∞(Γ/Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We are now ready to prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The proof goes along the lines of [Mon01, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proposition (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Λ ≤ Γ be coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the restriction map Hn a(Γ, V) → Hn a(Λ, V) is injective, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We implement the asymptotic cohomology of Λ using the complex ˜L∞((∗Γ)•, V) ∗Λ from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since the chain map that defines the restriction map factors through this complex, and the chain map ˜L∞((∗Γ)•, V) ∗Λ → ˜L∞((∗Λ)•, V) ∗Λ induces an isomorphism in cohomology (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2), it suffices to show that the chain inclusion ˜L∞((∗Γ)•, V) ∗Γ → ˜L∞((∗Γ)•, V) ∗Λ induces an injective map in cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Henceforth, we will refer to this as the restriction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Our goal is construct a transfer map, that is a linear map trans• : H• a(Λ, V) → H• a(Γ, V) such that trans• ◦ res• is the identity on H• a(Λ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then it follows at once that res• must be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By the above paragraph, we may do this by constructing an internal chain map � trans : ˜L∞((∗Γ)•, V) ∗Λ → ˜L∞((∗Λ)•, V) ∗Γ that restricts to the identity on ˜L∞((∗Λ)•, V) ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 11 Let f ∈ L∞((∗Γ)k, V) be such that ˜f ∈ ˜L∞((∗Γ)k, V) ∗Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For each x ∈ (∗Γ)k, define fx : ∗Γ → V fx(g) := π(g)f(g−1x) In other words, fx(g) is just (ρ1(g)f)(x) as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since ˜f ∈ ˜L∞((∗Γ)k, V) ∗Λ, for any γ ∈ ∗Λ and g ∈ ∗Γ, fx(gγ) − fx(g) ∈ Vinf Let us choose representatives of left ∗Λ-cosets in ∗Γ and restrict fx to this set of repre- sentatives so that we can regard fx as an internal map fx : ∗Γ/∗Λ → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Moreover, since fx ∈ L∞(∗Γ/∗Λ, V), we can apply the mean m constructed in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 to define the internal map transk(f) : L∞((∗Γ)k, V) → L∞((∗Γ)k, V) by transk(f)(x) = m(fx) Since ˜m is ∗Γ-invariant, this means that for g ∈ ∗Γ, m(fgx) − π(g)m(fx) ∈ Vinf, and implies that (transk(f))(gx) − π(g) transk(f)(x) ∈ Vinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This establishes that for f ∈ L∞((∗Γ)k, V) with ˜f ∈ ˜L∞((∗Γ)k, V) ∗Λ, we have � transk(f) ∈ ˜L∞((∗Γ)k, V) ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore trans• induces a chain map ˜ trans : ˜L∞((∗Γ)•, V) ∗Λ → ˜L∞((∗Λ)•, V) ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Finally, if ˜f is already ∗Γ-invariant, then fx is constant up to infinitesimals, and thus m(fx) is equal, up to an infinitesimal, to the value of that constant, which is f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This shows that � trans k is the identity when restricted to ˜L∞((∗Λ)•, V) ∗Γ, and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 is now an easy consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proposition (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Λ ≤ Γ be coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Λ is uniformly U-stable with a linear estimate, then so is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose that Λ is uniformly U-stable with a linear estimate, and let Γ be a coamenable supergroup of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We aim to show that Γ is also uniformly U-stable with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4, it suffices to show that all Ulam classes supported on U vanish in H2 a(Γ, W), where W is an Ulam ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8, it suffices to show that the images of such classes under the restriction map res2 : H2 a(Γ, W) → H2 a(Λ, W) vanish, since the latter is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4 these are Ulam classes of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But since Λ is uniformly U-stable with a linear estimate, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4 again, all Ulam classes in H2 a(Λ, W) vanish, and we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 Pullbacks and amenable kernels Let N ≤ Γ be an amenable normal subgroup, and let V be a dual asymptotic Banach ∗(Γ/N)- module, which pulls back to a dual asymptotic Banach ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Precomposing cochains by the projection ∗Γ → ∗(Γ/N) defines the pullback p• : H• a(Γ/N, V) → H• a(Γ, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The following can be proven via a similar argument as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let W be an Ulam ∗(Γ/N)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then W is also an Ulam ∗Γ-module, and the pullback p2 : H2 a(Γ/N, W) → H2 a(Γ, W) sends Ulam classes to Ulam classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 12 With this language, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9 is just a reformulation of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3: Proposition (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let N ≤ Γ be an amenable normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the pullback Hn a(Γ/N, V) → Hn a(Γ, V) is an isomorphism, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' And we deduce Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 analogously: Proposition (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let N ≤ Γ be an amenable normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Γ is uni- formly U-stable with a linear estimate, then so is Γ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose that Γ is uniformly U-stable with a linear estimate, and let N be an amenable normal subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We aim to show that Γ/N is also uniformly U-stable with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4, it suffices to show that all Ulam classes supported on U vanish in H2 a(Γ/N, W), where W is an Ulam ∗(Γ/N)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9, it suffices to show that the pullback of such classes under H2 a(Γ/N, W) → H2 a(Γ, W) vanish, since the latter is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 these are Ulam classes of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But since Γ is uniformly U-stable with a linear estimate, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4 again, all Ulam classes in H2 a(Γ, W) vanish, and we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 4 Asymptotic cohomology of lamplighters In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7, which we recall for the reader’s convenience: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Λ be two countable groups, where Λ is infinite and amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Hn a(Γ ≀ Λ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In fact, the theorem will hold for a larger class of coefficients, obtained as ultraproducts of separable Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This does not however lead to a stronger stability result: see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We start by finding a suitable Zimmer-amenable Γ-space: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 ([Mon22, Corollary 8, Proposition 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Λ be two countable groups, where Λ is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let µ0 be a distribution of full support on Γ, and let µ be the product measure on S := ΓΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then S is a Zimmer-amenable (Γ ≀ Λ)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The reason why this space is useful for computations is that it is highly ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Recall that a Γ-space S is ergodic if every Γ-invariant function S → R is essentially constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' When S is doubly ergodic, that is the diagonal action of Γ on S ×S is ergodic, we even obtain ergodicity with separable coefficients, meaning that for every Γ-module E, every Γ-equivariant map S → E is essentially constant [Mon22, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='A, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 (Kolmogorov [Mon22, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='A, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Λ be two countable groups, where Λ is infinite, and let S be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Sm is an ergodic (Γ≀Λ)-space, for every m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For our purposes, we will need an approximate version of ergodicity (namely, almost invariant functions are almost constant) and also the module E will only be endowed with an approximate action of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The ergodicity assumption still suffices to obtain this: 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let S be a probability measure Γ-space, and suppose that the action of Γ on S is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then whenever f : S → R is a measurable function such that ∥g · f − f∥ < ε for all g ∈ Γ, there exists a constant c ∈ R such that |f(s) − c| < ε for almost every s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We define F : S → R : s �→ esssupg∈Γf(g−1s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By construction, F is Γ-invariant, and moreover ∥F − f∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By ergodicity, F is essentially equal to a constant c, and thus |f(s) − c| < ε for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let S be a probability measure Γ-space, and suppose that the action of Γ on S × S is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose moreover, that E is a separable Banach space endowed with a map Γ × E → E : v �→ g · v such that ∥g · v∥ = ∥v∥ for all g ∈ Γ, v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then whenever f : S → E is a measurable function such that ∥g · f − f∥ < ε for all g ∈ Γ, where (g · f)(s) = g · f(g−1s), there exists a vector v ∈ E such that ∥f(s) − v∥ < 3ε for almost every s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We define F : S × S → R : (s, t) �→ ∥f(s) − f(t)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then ∥g · F − F∥ = ess sup| ∥g · f(g−1s) − g · f(g−1t)∥ − ∥f(s) − f(t)∥ | ≤ ess sup∥g · f(g−1s) − g · f(g−1t) − (f(s) − f(t))∥ ≤ 2∥g · f − f∥ < 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By the previous lemma, there exists a constant c such that |F(s, t) − c| < 2ε for all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If c < ε, then |f(s) − f(t)| < 3ε for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' s, t ∈ S, which implies the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Otherwise, ∥f(s) − f(t)∥ > ε for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' s, t ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let D ⊂ E be a countable dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then for each d ∈ D the set f −1(Bε/2(d)) is a measurable subset of S, and the union of such sets covers S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since D is countable, there must exist d ∈ D such that f −1(Bε/2(d)) has positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But for all s, t in this set, ∥f(s) − f(t)∥ < ε, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We thus obtain: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let S be a doubly ergodic Γ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let (Vn)n≥1 be a sequence of separable dual Banach spaces such that V = � ω Vn has the structure of a dual asymptotic Banach Γ-module be the corresponding asymptotic ∗Γ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then the natural inclusion ˜V ∗Γ → ˜L∞(∗S, V) ∗Γ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let f ∈ L∞ b (∗S, V) = � ω L∞(S, Vn) be a lift of an element ˜f ∈ ˜L∞(∗S, V) ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We write f = (fn)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then fact that ˜f is ∗Γ-invariant means that for every sequence (gn)n∈N ⊂ Γ it holds (gn·fn−fn)ω ∈ L∞ inf(∗S, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since this holds for every sequence (gn)n∈N, a diagonal argument implies that there exists ε ∈ ∗Rinf such that for every g ∈ Γ it holds (g · fn −fn)ω ∈ ∗Rinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It then follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 that there exist (vn)ω ∈ V such that (fn − 1vn) ∈ L∞ inf(∗S, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore f represents the same element of ˜L∞(∗S, V) as the image of an element of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since ˜f is ∗Γ-invariant, the corresponding element is actually in ˜V ∗Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We are finally ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Λ be countable groups, where Λ is infinite and amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2, using the same notation, S is a Zimmer-amenable (Γ ≀ Λ)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3, and obtain that the following complex computes H∗ a(Γ ≀ Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' V): 0 ˜ d0 −→ ˜L∞(∗S, V) ∗Γ ˜ d1 −→ ˜L∞((∗S)2, V) ∗Γ ˜ d2 −→ ˜L∞((∗S)3, V) ∗Γ ˜ d3 −→ · · · 14 Now by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3, Sm is a doubly ergodic (Γ ≀ Λ)-space, for every m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thus Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 applies, and the natural inclusion ˜V ∗Γ → ˜L∞((∗S)m, V) ∗Γ is an isomorphism for every m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thus the above complex is isomorphic to 0 ˜ d0 −→ ˜V ∗Γ ˜ d1 −→ ˜V ∗Γ ˜ d2 −→ ˜V ∗Γ ˜d3 −→ · · · Each differential ˜ dm is an alternating sum of (m+1) terms all equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore ˜ dm is the identity whenever m is even, and it vanishes whenever m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 5 Thompson groups In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The statement for F ′ will be a special case of a more general result for a large family of self-similar groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The most general statement is the following: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be a group, Γ0 a subgroup with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' There exists g ∈ Γ such that the groups {giΓ0g−i : i ∈ Z} pairwise commute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Every finite subset of Γ is contained in some conjugate of Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Hn a(Γ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, Γ is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The theorem applies to the following large family of groups of homeomorphisms of the real line: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be a proximal, boundedly supported group of orientation-preserving homeomorphisms of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Hn a(Γ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, Γ is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The fact that such groups have no quasimorphisms is well-known: see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' [GG17, FFL21, Mon22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We refer the reader to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 for the relevant definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8 we will apply Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 to Thompson’s group F ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' the result for Thompson’s group F will follow from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We deduce the stability of Thompson’s group T and V from these general criteria in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 Self-similar groups In this section we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This will be done in a series of lemmas: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be a group, and suppose that there exists g ∈ Γ and Γ0 ≤ Γ such that {giΓ0g−i : i ∈ Z} pairwise commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then there exists an epimorphism Γ0 ≀ Z → ⟨Γ0, g⟩ with amenable (in fact, metabelian) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This is well-known and stated without proof in [Mon22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We include a proof for com- pleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' To make a clear distinction, we denote by H the abstract group Γ0, and by Γ0 the subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' So we want to construct an epimorphism H ≀Z → ⟨Γ0, g⟩ ≤ Γ with metabelian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We define naturally ϕ((gi)i∈Z, p) = �� i∈Z tigit−i � tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that this product is well-defined since there are only finitely many non-identity terms, and the order does not matter since different conjugates commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By construction ϕ is injective on Hi, that is the copy of H supported on the i-th coordinate in H ≀ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let K := ker ϕ ∩ � i Hi, and note that K is the kernel of the retraction H ≀ Z → Z restricted to ker ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' So it suffices to show that K is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let g, h ∈ K and write them as (gi)i∈Z and (hi)i∈Z (we omit the Z-coordinate since it is always 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We need to show that g and h commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We have 1Γ = ϕ(g) = � i∈Z tigit−i and thus g0 = � i̸=0 tigit−i ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But now g0 belongs to a group generated by conjugates of Γ0 in Γ that commute with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular this implies that g0 and h0 commute in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since ϕ|H0 is injective, this shows that g0 and h0 commute in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Running the same argument on the other coordinates, we obtain that gi and hi commute in Hi, for all i ∈ Z, and thus g and h commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The next facts are all contained in the literature: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 ([Mon22, Proposition 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose that Γ0 ≤ Γ is such that every finite subset of Γ is contained in some Γ-conjugate of Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Γ0 is coamenable in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 ([MP03]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let K ≤ H ≤ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If K is coamenable in Γ, then H is coamenable in Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If K is coamenable in H and H is coamenable in Γ, then K is coamenable in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We warn the reader that if K is coamenable in Γ, then K need not be coamenable in H [MP03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We are now ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1: Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ, Γ0 and g be as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4, there exists a map Γ0 ≀ Z → ⟨Γ0, g⟩ with metabelian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9, we have Hn a(⟨Γ0, g⟩, V) for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5, Γ0 is coamenable in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Finally, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6, ⟨Γ0, g⟩ is coameanble in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8 allows to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 Groups of homeomorphisms of the line Let Γ be a group acting by homeomorphisms on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We say that the action is proximal if for all reals a < b and c < d there exists g ∈ Γ such that g · a < c < d < g · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The support of g ∈ Γ is the set {x ∈ R : g · x ̸= x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We say that Γ is boundedly supported if every element has bounded support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that boundedly supported homeomorphisms are automatically orientation-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 16 Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ0 be the subgroup of elements whose support is contained in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let g ∈ Γ be such that g(0) > 1: such an element exists because the action of Γ is proximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then it follows by induction, and the fact that Γ is orientation-preserving, that the intervals {gi[0, 1] : i ∈ Z} are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore the conjugates giΓ0g−i pairwise commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since Γ is boundedly supported, for every finite subset A ⊂ Γ there exists n such that the support of each element of A is contained in [−n, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By proximality, there exists h ∈ Γ such that h(0) < −n and h(1) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then hΓ0h−1 is the subgroup of elements whose support is contained in [−n, n], in particular it contains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thus Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 applies and we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let us now show how to obtain the statements on F and F ′ from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We refer the reader to [CFP96] for more details on Thompson’s groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s group F is the group of orientation-preserving piecewise linear homeomor- phisms of the interval, with breakpoints in Z[1/2] and slopes in 2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The derived subgroup F ′ coincides with the subgroup of boundedly supported elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The action of F ′ (and thus F) on [0, 1] preserves Z[1/2] ∩ (0, 1), and acts highly transitively on it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' that is, for every pair of ordered n-tuples in Z[1/2] ∩ (0, 1) there exists an element of F ′ sending one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s groups F and F ′ are uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We identify (0, 1) with the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The group F ′ is boundedly supported, and it is proximal, since it acts transitively on ordered pairs of a dense set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 applies and F ′ is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since the quotient F/F ′ is abelian, thus amenable, we see that F ′ is coamenable in F, and thus conclude from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 that F ′ is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We could also deduce the stability of F from the stability of F ′ more directly, without appealing to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Indeed, since F ′ is uniformly U-stable, simple, and not linear, every homomorphism F ′ → U(n) is trivial - something we will come back to in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore uniform U-stability of F ′ implies that every uniform asymptotic homomorphism F ′ → U is uniformly close to the trivial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It follows that every uniform asymptotic homomorphism F → U is uniformly asymptotically close to one that factors through Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We conclude by the stability of amenable groups [Kaz82, GLMR23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Other groups to which these criteria apply include more piecewise linear groups [BS16], such as the Stein–Thompson groups [Ste92], or the golden ratio Thompson group of Cleary [Cle00, BNR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In such generality some more care is needed, since the commutator subgroup is sometimes a proper subgroup of the boundedly supported subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The criteria also apply for the piecewise proejective groups of Monod [Mon13] and Lodha–Moore [LM16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In this case, further care is needed, since the role of the commutator subgroup in the proofs above has to be taken by the double commutator subgroup [BLR18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This ties back to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='10 from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 T and V In this section, we show how our previous results allow to prove stability of groups of home- omorphisms of the circle and of the Cantor set as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For simplicity of the exposition, we 17 only focus on Thompson’s groups T and V , but the proofs generalize to some analogously defined groups, with the appropriate modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Our proof will involve a bounded gen- eration argument for stability that was pioneered in [BOT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We will only use it a simple version thereof, closer to the one from [BC20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Recall that Γ is said to be boundedly generated by the collection of subgroups H if there exists k ≥ 1 such that the sets {H1 · · · Hk : Hi ∈ H} cover Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be a discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose that there exists a subgroup H ≤ Γ with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Every homomorphism H → U(n) is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' H is uniformly U-stable (with a linear estimate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Γ is boundedly generated by the conjugates of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Γ is uniformly U-stable (with a linear estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let φn : Γ → U(dn) be a uniform asymptotic homomorphism with def(φn) =: εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then φn|H : H → U(dn) is a uniform asymptotic homomorphism of H, therefore it is δn- close to a homomorphism, where δn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But by assumption such a homomorphism must be trivial, so ∥φn(h) − Ikn∥ ≤ δn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The same holds for all conjugates of H, up to replacing δn by δn + 2εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By bounded generation, there exists k ≥ 1 such that each g ∈ Γ can be written as g = h1 · · · hk, where each hi belongs to a conjugate of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We estimate: ∥φn(g) − Idn∥ = �����φn � k � i=1 hi � − Idn ����� ≤ �����φn �k−1 � i=1 hi � φn(hk) − Idn ����� + εn = �����φn �k−1 � i=1 hi � − Idn ����� + ∥φn(hk) − Idn∥ + εn ≤ · · · · · ≤ k � i=1 ∥φn(hi) − Idn∥ + kεn ≤ k(δn + εn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore φn is k(δn + εn)-close to the trivial homomorphism, and we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s group T is the group of orientation-preserving piecewise linear homeomor- phisms of the circle R/Z preserving Z[1/2]/Z, with breakpoints in Z[1/2]/Z, and slopes in 2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Given x ∈ Z[1/2]/Z, the stabilizer of x is naturally isomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Moreover, the germ stabilizer T(x) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' the group consisting of elements that fix pointwise some neighbourhood of x) is isomorphic to F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s group T is uniformly U-stable with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We claim that Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='10 applies with H = T(0) ∼= F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' follows from the fact F ′ does not embed into U(n) (for instance because it contains F as a subgroup, which is finitely generated and not residually finite, and so cannot be linear by Mal’cev’s Theorem [Mal40]), and F ′ is simple [CFP96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Also, F ′ is uniformly U-stable with a linear estimate, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore we are left to show the bounded generation statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We will 18 show that for every g ∈ T there exist x, y ∈ Z[1/2]/Z such that g ∈ T(x)T(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This suffices because T acts transitively on Z[1/2]/Z, so T(x) and T(y) are both conjugate to H = T(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let 1 ̸= g ∈ T, and choose x ̸= y ∈ Z[1/2]/Z such that g(y) /∈ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let I be a small dyadic arc around y such that x /∈ I and x, y /∈ g(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Choose an element f ∈ T(x) such that f(I) = g(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let h be an element supported on I such that h|I = f −1g|I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since x /∈ I, we also have h ∈ T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Moreover h−1f −1g|I = id|I, so h−1f −1g ∈ G(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We conclude that g = fh · h−1f −1g ∈ T(x)T(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s group V can be described as a group of homeomorphisms of the dyadic Cantor set X := 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' A dyadic brick is a clopen subset of the form Xσ := σ × 2N>k, for some σ ∈ 2k, and every two dyadic bricks are canonically homeomorphic via Xσ → Xτ : σ × x �→ τ × x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' An element g ∈ V is defined by two finite partitions of V of the same size into dyadic bricks, that are sent to each other via canonical homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s group V is uniformly U-stable, with a linear estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The proof is very similar to the proof for T, so we only sketch it: Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let x ∈ 2N be a dyadic point, that is a sequence that is eventually all 0, and let V (x) denote the subgroup of V consisting of elements that fix a neighbourhood of x pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The same argument as in the proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11 shows that V is boundedly generated by conjugates of V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now V (x) is isomorphic to a directed union of copies of V , which is finitely generated and simple [CFP96], so by Mal’cev’s Theorem every homomorphism V (x) → U(n) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Finally, V (x) contains a copy V0 of V such that the pair (V (x), V0) satisfies the hypotheses of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 (see [And22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4] and its proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We conclude by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 6 Sharpness of our results In this section we point out certain ways in which our results are sharp, by providing explicit counterexamples to generalizations and converses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' There is a notion of strong Ulam stability, where one takes U to include unitary groups of infinite-dimensional Hilbert spaces as well, typically equipped with the operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It is shown in [BOT13] that a subgroup of a strongly Ulam stable group is Ulam stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore it is clear that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 does not hold for strong Ulam stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Even restricting to separable Hilbert spaces does not help: it follows from the construction in [BOT13] that if a countable group contains a free subgroup, then separable Hilbert spaces already witness the failure of strong Ulam stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The framework of stability via asymptotic cohomology can be developed in this general setting as well, with dual asymptotic Banach modules that are not finitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore the counterexample above shows that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7 really needs the finitary assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The fact that we could obtain dual asymptotic Banach modules obtained as ultraproducts of separable spaces, analogously to [Mon22], does not help, since the dual asymptotic Banach modules arising from a stability problem over infinite-dimensional Hilbert spaces are not of this form, even when the Hilbert space are separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 19 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We proved in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5 that if Λ is coamenable in Γ and Λ is uniformly U-stable with a linear estimate, then so is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The converse does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Fn be a free group of rank n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Λ := � n≥1 Fn admits a non-trivial quasimorphism, so it is not uniformly U(1)-stable [BOT13], in particular it is not uniformly U-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' However, Λ is coamenable in Fn ≀ Z, which is uniformly U-stable with a linear estimate by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' On the other hand, if we replace “coamenable” by “finite index”, then the converse does hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This follows from the induction procedure in [BOT13] for Ulam stability, as detailed in [Gam11, Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' the same proof can be generalized to all submultiplicative norms [GLMR23, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We proved in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6 that if N is an amenable normal subgroup of Γ, and Γ is uniformly U-stable with a linear estimate, then so is Γ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The converse does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be the lift of Thompson’s group T, that is, the group of orientation-preserving homeomorphisms of R that commute with the group Z of integer translations and induce T on the quotient R/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' These groups fit into a central extension 1 → Z → Γ → T → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now T is uniformly U-stable with a linear estimate, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11, however Γ is not: it is not even uniformly U(1)-stable, by [BOT13], since it has a non-trivial quasimorphism [GS87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The next two remarks show that some results from [GLMR23] are also sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The fundamental result of [GLMR23] is that the vanishing of asymptotic cohomology implies uniform U-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The converse does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Indeed, since u(1) ∼= R with trivial adjoint action (because U(1) is abelian), it follows that the implication of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4 specializes to: If H2 a(Γ, ∗R) = 0, then Γ is uniformly U(1)-stable, where ∗R is seen as a dual asymptotic ∗Γ-module with a trivial ∗Γ action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now, let again Γ be the lift of Thompson’s group T, so that Γ contains a central subgroup Z with Γ/Z ∼= T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The fact that Γ is not uniformly U(1)-stable implies that H2 a(Γ, ∗R) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9 then shows that H2 a(T, ∗R) ̸= 0 either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' However, T is uniformly U-stable with a linear estimate, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Morally, this is due to the fact that H2 b(Γ, R) ∼= H2 b(T, R) ∼= R, but the former is spanned by a quasimorphisms, while the latter is not (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' [Cal09, Chapter 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In [BOT13] it is shown that groups admitting non-trivial quasimorphisms are not uniformly U(1)-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In [GLMR23, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6] this result is sharpened: the authors show that Γ is uniformly U(1)-stable if and only if the non-zero element in the image of H2 b(Γ, Z) in H2 b(Γ, R) have Gromov norm ∥·∥ bounded away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' They use this to show that a finitely presented group is uniformly U(1)-stable if and only if it admits no non-trivial quasimorphism [GLMR23, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The hypothesis of finite presentability is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γn denote the lift of Thompson’s group T to R/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' That is, Γn is the group of orientation-preserving homeomorphisms of the topological circle R/nZ, which commute with the cyclic group of rotations Z/nZ and induce T on the quotient R/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now T has no unbounded quasimorphisms (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' [Cal09, Chapter 5]), and so Γn also has no unbounded quasimorphisms (this follows from the left exactness of the quasimorphism functor [Cal09, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore the group Γ := � n≥2 Γn has no unbounded quasimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 20 However, we claim that Γ is not uniformly U(1)-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By [GLMR23, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6], it suffices to show that there exist bounded cohomology classes 0 ̸= ρn ∈ im(H2 b(Γ, Z) → H2 b(Γ, R) such that ∥ρn∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We let ρn be the Euler class of the representation Γ → Γn → Homeo+(R/nZ), which admits an integral representative and so lies in the image of H2 b(Γ, Z) (see [Ghy01] for more information about Euler classes of circle actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Moreover, using the terminology of [Bur11], the representation is minimal, unbounded, and has a centralizer of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore ∥ρn∥ = 1/2n by [Bur11, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6], and we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that Γ is countable but infinitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It would be interesting to produce a finitely generated example (which would necessarily be infinitely presented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 7 Approximation properties In this section we discuss open problems about approximation properties of the groups treated in this paper, and their relation to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We recall the following notions: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let G be a family of metric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We say that Γ is (pointwise, uniformly) G-approximable if there exists a (pointwise, uniform) asymptotic homomorphism φn : Γ → Gn ∈ G that is moreover asymptotically injective, meaning that for all g ∈ Γ, g ̸= 1 it holds lim inf n→∞ φn(g) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The above terminology is not standard: most of the literature only deals with the point- wise notion, and refers to that as G-approximability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The notion of uniform approximability appeared in [FF21] with the name of strong G-approximability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If G is the family of symmetric groups equipped with the normalized Hamming distance, then pointwise G-approximable groups are called sofic [Gro99, Wei00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If G is the family of unitary groups equipped with the Hilbert–Schmidt distance, then pointwise G-approximable groups are called hyperlinear [R˘08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' All amenable and residually finite groups are sofic, and all sofic groups are hyperlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It is a major open question to determine whether there exists a non-sofic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In our context of submultiplicative norms on unitary groups, the following two notions of approximability have been studied: Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let G be the family of unitary groups equipped with the operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then pointwise G-approximable groups are called MF [CDE13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' All amenable groups are MF [TWW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It is an open problem to determine whether there exists a non-MF group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let G be the family of unitary groups equipped with the Frobenius norm, or more generally with a Schatten p-norm, for 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Groups that are not pointwise G-approximable have been constructed in [DCGLT20, LO20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This is one of the very few cases in which a non- example for pointwise approximability is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The following observation is well-known, and due to Glebsky and Rivera [GR09] and Arzhantseva and P˘aunescu in the pointwise symmetric case [AP15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We give a general proof for reference: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let G be a family of metric groups that are locally residually finite, and let Γ be a finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Suppose that Γ is (pointwise, uniformly) G-stable and (pointwise, uniformly) G-approximable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Γ is residually finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 21 The hypothesis on G covers all cases above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' When the groups in G are finite, this is clear, and when they are linear, this follows from Mal’cev’s Theorem [Mal40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We proceed with the proof without specifying the type of asymptotic homomorphisms, closeness, and approximability: the reader should read everything as pointwise, or everything as uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let φ : Γ → G be an asymptotically injective asymptotic homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' By stability, there exists a sequence of homomorphisms ψ : Γ → G which is asymptotically close to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since φ is asymptotically injective, for each g ∈ Γ there exists N such that φn(g) ≥ ρ for all n ≥ N and some ρ = ρ(g) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Up to taking a larger N, we also have that ψn(g) ≥ ρ/2, in particular ψn(g) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since ψn(Γ) is a finitely generated subgroup of Gn ∈ G, it is residually finite by hypothesis, and so ψn(g) survives in some finite quotient of ψn(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since this is also a finite quotient of Γ, we conclude that Γ is residually finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In the special case of pointwise stability and Thompson’s group F, we obtain the following more general version of a remark of Arzhantseva and Paunescu [AP15, Open problem]: Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let G be the family of symmetric groups with the normalized Hamming distance, the family of unitary groups with the Hilbert–Schmidt norm, or the family of unitary groups with the operatorn norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Thompson’s group F is pointwise G-stable, then it is not pointwise G-approximable, and in particular it is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' As we mentioned in the introduction, the amenability of Thompson’s group F is one of the most outstanding open problems in modern group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thompson’s group F is not residually finite [CFP96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' So it follows from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4 that it cannot be simultaneously pointwise G-stable and pointwise G-approximable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The last statement follows from the fact that amenable groups are sofic, hyperlinear, and MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' On the other hand, our results allow to settle the uniform approximability of Thompson’s groups, with respect to unitary groups and submultiplicative norms: Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' As usual, let U be the family of unitary groups equipped with submultiplicative norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Thompson’s groups F, F ′, T and V are not uniformly U-approximable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The same holds for Γ ≀ Λ, whenever Λ is infinite and amenable, and Γ is non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We remark that Thompson’s groups T and V are generally regarded as good candidates for counterexamples to approximability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The statement for F, T and V follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4, together with the fact that they are not residually finite, and the statement for F ′ (which is not finitely generated) follows from the fact that F ′ contains a copy of F [CFP96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The lamplighter case follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='3 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4, together with the fact that such lamplighters are not residually finite [Gru57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We do not know whether Thompson’s groups are uniformly G-approximable, when G is the family of unitary groups equipped with the Hilbert–Schmidt norm, and we conjecture that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In the next section, we examine the case of symmetric groups via a more direct argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 22 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1 Approximations by symmetric groups We end by proving, by a cohomology-free argument, that some of the groups studied in this paper are not uniformly approximable by symmetric groups, in a strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For the rest of this section, we denote by S the family of symmetric groups equipped with the normalized Hamming distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Our main result is an analogue of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 for this approximating family (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2 for the relevant definitions): Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be a proximal, boundedly supported group of orientation-preserving homeomorphisms of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then every uniform asymptotic homomorphism φn : Γ′ → Skn ∈ S is uniformly asymptotically close to the trivial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular, Γ′ is uniformly S-stable, and not uniformly S-approximable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The non-approximability follows from the fact that Γ′ is non-trivial (see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Note that for Γ as in the statement, Γ′ is simple [GG17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1], so in particular every homomorphism Γ′ → Skn is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The proof relies on known results on the flexible uniform stability of amenable groups [BC20] and uniform perfection of groups with proximal actions [GG17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The finiteness of the groups in S will play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We start with the following lemma: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Let Γ be as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Then Γ′ is non-trivial, and the action of Γ′ on the line has no global fixpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' If Γ′ is trivial, then Γ is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' This contradicts that the action is proximal and boundedly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Indeed, given g ∈ Γ, since g is centralized, the action of Γ on R must preserve the support of g, which is a proper subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But then the action cannot be proximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now the set of global fixpoints of Γ′ is a closed subset X ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since Γ′ is normal in Γ, the action of Γ preserves X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' But the action of Γ on R is proximal, in particular every orbit is dense, and since X is closed we obtain X = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' That is, Γ′ acts trivially on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since Γ is a subgroup of Homeo+(R), this implies that Γ′ is trivial, which contradicts the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We proceed with the proof: Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' It follows from [GG17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='1] that Γ′ is 2-uniformly perfect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' that is, every element of Γ′ may be written as the product of at most 2 commutators (this uses the proximality hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Therefore it suffices to show that there exists a constant C such that for all g, h ∈ Γ′ it holds dkn(φn([g, h]), idkn) ≤ Cεn, where dkn denotes the Hamming distance on Skn and εn := def(φn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We drop the subscript n on φ and ε for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Now let g, h ∈ Γ′, and let I, J ⊂ R be bounded intervals such that g is supported on I and h is supported on J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since Γ′ acts without global fixpoints by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='8, there exists t ∈ Γ′ such that t · inf(J) > sup(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since Γ′ is orientation-preserving, the same holds for all powers of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' In particular [g, tiht−i] = 1 for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Next, we apply [BC20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2] to the amenable group ⟨t⟩, to obtain an integer N such that kn ≤ N ≤ (1 + 1218ε)kn and a permutation τ in SN such that dN(φ(t)i, τ i) ≤ 2039ε for all i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Here dN denotes the normalized Hamming distance on the symmetric group SN, and φ is extended to a map 23 φ : Γ′ → SN with every φ(g) fixing each point in {kn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' We compute (using τ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' = idN): dkn(φ([g, h]), idkn) ≤ dN(φ([g, h]), idN) ≤ dN([φ(g), φ(h)], idN) + O(ε) = dN([φ(g), τ N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='φ(h)τ −N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' ], idN) + O(ε) ≤ dN([φ(g), φ(tN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=')φ(h)φ(t−N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' )], idN) + O(ε) ≤ dN(φ([g, tN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='ht−N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' ]), idN) + O(ε) = dN(φ(1), idN) + O(ε) ≤ O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Thus, there exists a constant C independent of g and h (C = 20000 suffices) such that dkn(φ([g, h]), idkn) ≤ Cε, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Consider the Thompson groups F ′, F, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Every asymptotic homomorphism φn : F ′ → Skn ∈ S is uniformly asymptotically close to the trivial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Every asymptotic homomorphism φn : F → Skn ∈ S is uniformly asymptotically close to one that factors through the abelianization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Every asymptotic homomorphism φn : T → Skn ∈ S is uniformly asymptotically close to the trivial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' is an instance of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='7: indeed F ′ satisfies the hypotheses for Γ, and F ′′ = F ′ since F ′ is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' For Item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=', pick a section σ : Ab(F) → F, and define ψn(g) := φn(σ(Ab(g))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Using that ψn|F ′ is uniformly asymptotically close to the sequence of trivial maps, we obtain that φn and ψn are uniformly asymptotically close, and ψn factors as F → Ab(F) φn◦σ −−−→ Skn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Finally, Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' follows again from Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' and the fact that every element of T can be written as a product of two elements in isomorphic copies of F ′ (see the proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The corollary immediately implies that F, F ′ and T are not uniformly S-approximable, and that F ′ and T are uniformly S-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Since F has infinite abelianization, it follows from [BC20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='4] that it is not uniformly S-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' However the corollary together with [BC20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='2] implies that it is flexibly uniformly S-stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' that is, every uniform asymptotic homomorphism is uniformly close to a sequence of homomorphisms taking values in a symmetric group of slightly larger degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' The case of Thompson’s group V can also be treated analogously (see the sketch of proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' References [AEG94] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Adams, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Elliott, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Giordano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content=' Amenable actions 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='rangarajan@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} +page_content='il 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfjgft/content/2301.03970v1.pdf'} diff --git a/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf b/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7e0a0fbce14327160301807b0bf5efa2e418a6d7 --- /dev/null +++ b/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf @@ -0,0 +1,3 @@ +version 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a/3NFAT4oBgHgl3EQflB25/content/tmp_files/2301.08615v1.pdf.txt b/3NFAT4oBgHgl3EQflB25/content/tmp_files/2301.08615v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..183b67d51aec34492e74e69f8a1ca5e0997af3e1 --- /dev/null +++ b/3NFAT4oBgHgl3EQflB25/content/tmp_files/2301.08615v1.pdf.txt @@ -0,0 +1,865 @@ +arXiv:2301.08615v1 [hep-ph] 20 Jan 2023 +Photo-production of lowest Σ∗ +1/2− state within the Regge-effective Lagrangian approach +Yun-He Lyu,1 Han Zhang,1 Neng-Chang Wei,2 Bai-Cian Ke,1 En Wang,1 and Ju-Jun Xie3, 2, 4 +1School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China +2School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China +3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +4Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, +Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China +(Dated: January 23, 2023) +Since the lowest Σ∗ state, with quantum numbers spin-parity JP = 1/2−, is far from estab- +lished experimentally and theoretically, we have performed a theoretical study on the Σ∗ +1/2− photo- +production within the Regge-effective Lagrangian approach. Taking into account that the Σ∗ +1/2− +couples to the ¯KN channel, we have considered the contributions from the t-channel K exchange +diagram. Moreover, these contributions from t-channel K∗ exchange, s-channel nucleon pole, u- +channel Σ exchange, and the contact term, are considered. The differential and total cross sections +of the process γn → K+Σ∗− +1/2− are predicted with our model parameters. The results should be +helpful to search for the Σ∗ +1/2− state experimentally in future. +PACS numbers: +I. +INTRODUCTION +The study of the low-lying excited Λ∗ and Σ∗ hyperon +resonances is one of the most important issues in hadron +physics. +Especially, since the Λ(1405) was discovered +experimentally [1, 2], its nature has called many atten- +tions [3–8], and one explanation for Λ(1405) is that it +is a ¯KN hadronic molecular state [9–14]. In addition, +the isospin I = 1 partner of the Λ(1405), the lowest +Σ∗ +1/2− is crucial to understand the light baryon spec- +tra. At present, there is a Σ∗(1620) with JP = 1/2− +listed in the latest version of Review of Particle Physics +(RPP) [15]. It should be stressed that the Σ∗(1620) state +is a one-star baryon resonance, and many studies indicate +that the lowest Σ∗ +1/2− resonance is still far from estab- +lished, and its mass was predicted to lie in the range of +1380 ∼ 1500 MeV [13, 16–19]. Thus, searching for the +lowest Σ∗ +1/2− is helpful to understand the low-lying ex- +cited baryons with JP = 1/2− and the light flavor baryon +spectra. +The analyses of the relevant data of the process +K−p → Λπ+π− suggest that there may exist a Σ∗ +1/2− +resonance with mass about 1380 MeV [16, 17], which +is consistent with the predictions of the unquenched +quark models [20]. +The analyses of the K∗Σ photo- +production also indicate that the Σ∗ +1/2− is possibly buried +under the Σ∗(1385) peak with mass of 1380 MeV [21], +and it is proposed to search for the Σ∗ +1/2− in the pro- +cess Λc → ηπ+Λ [22]. A more delicate analysis of the +CLAS data on the process γp → KΣπ [23] suggests that +the Σ∗ +1/2− peak should be around 1430 MeV [13]. +In +Refs. [24, 25], we suggest to search for such state in the +processes of χc0(1P) → ¯ΣΣπ and χc0(1P) → ¯ΛΣπ. In +addition, Ref. [26] has found one Σ∗ +1/2− state with mass +around 1400 MeV by solving coupled channel scattering +equations, and Ref. [27] suggests to search for this state +in the photo-production process γp → K+Σ∗0 +1/2−. +It’s worth mentioning that a Σ∗(1480) resonance with +JP = 1/2− has been listed on the previous version of +RPP [28]. +As early as 1970, the Σ∗(1480) resonance +was reported in the Λπ+, Σπ, and p ¯K0 channels of the +π+p scattering in the Princeton-Pennsylvania Accelera- +tor 15-in.∼hydrogen bubble chamber [29, 30]. In 2004, +a bump structure around 1480 MeV was observed in the +K0 +Sp(¯p) invariant mass spectrum of the inclusive deep +inelastic ep scattering by the ZEUS Collaboration [31]. +Furthermore, a signal for a resonance at 1480 ± 15 MeV +with width of 60 ± 15 MeV was observed in the process +pp → K+pY ∗0 [32]. Theoretically, the Σ∗(1480) was in- +vestigated within different models [33–36]. In Ref. [36], +the S-wave meson-baryon interactions with strangeness +S = −1 were studied within the unitary chiral approach, +and one narrow pole with pole position of 1468−i 13 MeV +was found in the second Riemann sheet, which could be +associated with the Σ∗(1480) resonance. However, the +Σ∗(1480) signals are insignificant, and the existence of +this state still needs to be confirmed within more precise +experimental measurements. +As we known, the photo-production reactions have +been used to study the excited hyperon states Σ∗ and Λ∗, +and the Crystal Ball [37–39], LEPS [40], and CLAS [23] +Collaborations have accumulated lots of relevant exper- +imental data. +For instance, with these data, we have +analyzed the process γp → KΛ∗(1405) to deepen the un- +derstanding of the Λ∗(1405) nature in Ref. [41]. In order +to confirm the existence of the Σ∗(1480), we propose to +investigate the process γN → KΣ∗(1480) 1 within the +1 Here after, we denote Σ∗(1480) as the lowest Σ∗ +1/2− state unless +otherwise stated. + +2 +Regge-effective Lagrange approach. +Considering the Σ∗(1480) signal was first observed +in the π+Λ invariant mass distribution of the process +π+p → π+K+Λ, and the significance is about 3 ∼ +4σ [30], we search for the charged Σ∗(1480) in the process +γn → K+Σ∗− +1/2−, which could also avoid the contribu- +tions of possible excited Λ∗ states. We will consider the +t-, s-, u-channels diagrams in the Born approximation +by employing the effective Lagrangian approach, and the +t-channel K/K∗ exchanges terms within Regge model. +Then we will calculate the differential and total cross +sections of the process γn → K+Σ∗− +1/2− reaction, which +are helpful to search for Σ∗ +1/2− experimentally. +This paper is organized as follows. In Sec. II, the the- +oretical formalism for studying the γn → K+Σ∗−(1480) +reactions are presented. The numerical results of total +and differential cross sections and discussion are shown +in Sec. III. Finally, a brief summary is given in the last +section. +II. +FORMALISM +The reaction mechanisms of the Σ∗(1480) (≡ Σ∗) +photo-production process are depicted in the Fig. 1, +where we have taken into account the contributions from +the t-channel K and K∗ exchange term, s-channel nu- +cleon pole term, u-channel Σ exchange term, and the +contact term, respectively. +γ(k1) +K(k2) +N(p1) +Σ∗(p2) +K, K∗ +γ +K +N +Σ∗ +N +γ +Σ∗ +N +K +γ +K +N +Σ∗ +Σ +(a) +(b) +(c) +(d) +FIG. 1: The mechanisms of the γn → K+Σ∗− +1/2− process. (a) +t-channel K/K∗ exchange terms, (b) s-channel nuclear term, +(c) u-channel Σ exchange term, and (d) contact term. The +k1, k2, p1, and p2 stand for the four-momenta of the initial +photon, kaon, neutron, and Σ∗(1480), respectively. +To compute the scattering amplitudes of the Feynman +diagrams shown in Fig. 1 within the effective Lagrangian +approach, we use the Lagrangian densities for the elec- +tromagnetic and strong interaction vertices as used in +Refs. [27, 42–46] +LγKK = −ie +� +K† (∂µK) − +� +∂µK†� +K +� +Aµ, +(1) +LγKK∗ = gγKK∗ǫµναβ∂µAν∂αK∗ +βK, +(2) +LγNN = −e ¯N +� +γµˆe − +ˆκN +2MN +σµν∂ν +� +AµN, +(3) +LγΣΣ∗ = eµΣΣ∗ +2MN +¯Σγ5σµν∂νAµΣ∗ + h.c., +(4) +LKNΣ = −igKNΣ ¯Nγ5ΣK + h.c., +(5) +LK∗NΣ∗ = igK∗NΣ∗ +√ +3 +¯K∗µ ¯Σ∗γµγ5N + h.c. +(6) +LKNΣ∗ = gKNΣ∗ ¯K ¯ +Σ∗N + h.c., +(7) +where e(= +√ +4πα) is the elementary charge unit, Aµ is the +photon filed, and ˆe ≡ (1+τ3)/2 denotes the charge opera- +tor acting on the nucleon field. ˆκN ≡ κpˆe+κn(1−ˆe) is the +anomalous magnetic moment, and we take κn = −1.913 +for neutron [15]. MN and MΣ denote the masses of nu- +cleon and the ground-state of Σ hyperon, respectively. +The strong coupling gKNΣ is taken to be 4.09 from +Ref. [47]. +The gγKK∗ = 0.254 GeV−1 is determined +from the experimental data of ΓK∗→K+γ [15] and the +value of gK∗NΣ∗ = −3.26 − i0.06 is taken from Ref [26]. +In addition, the coupling gKNΣ∗ = 8.74 GeV is taken +from Ref. [36], and the transition magnetic moment +µΣΣ∗ = 1.28 is taken from Ref. [27] +With the effective interaction Lagrangian densities +given above, the invariant scattering amplitudes are de- +fined as +M = ¯uΣ∗(p2, sΣ∗)Mµ +huN(k2, sp)ǫµ(k1, λ), +(8) +where uΣ∗ and uN stand for the Dirac spinors, respec- +tively, while ǫµ(k1, λ) is the photon polarization vector +and the sub-indice h corresponds to different diagrams +of Fig. 1. The reduced amplitudes Mµ +h are written as +Mµ +K∗ = +egγKK∗gK∗NΣ∗ +√ +3MK∗(t − M 2 +K∗)ǫαβµνk1αk2βγνγ5, +(9) +Mµ +K− = −2iegKNΣ∗ +t − M 2 +K +kµ +2 , +(10) +Mµ +Σ− = −i +eµΣΣ∗gKNΣ +2Mn(u − M 2 +Σ∗)(q/u − MΣ)σµνk1ν, (11) +Mµ +n = +κngKNΣ∗ +2Mn(s − M 2n)σµνk1ν(q/s + Mn). +(12) +In order to keep the full photoproduction amplitudes +considered here gauge invariant, we adopt the amplitude +of the contact term +Mµ +c = −iegKNΣ∗ +pµ +2 +p2 · k1 +, +(13) +for γn → K+Σ∗− +1/2−. + +3 +It is known that the Reggeon exchange mechanism +plays a crucial role at high energies and forward an- +gles [48–51], thus we will adopt Regge model for mod- +eling the t-channel K and K∗ contributions by replacing +the usual pole-like Feynman propagator with the corre- +sponding Regge propagators as follows, +1 +t − M 2 +K +→ FRegge +K += +� s +sK +0 +�αK(t) +πα′ +K +sin(παK(t))Γ(1 + αK(t)),(14) +1 +t − M 2 +K∗ +→ FRegge +K∗ += +� s +sK∗ +0 +�αK∗(t) +πα′ +K∗ +sin(παK∗(t))Γ(αK∗(t)),(15) +with αK(t) = 0.7 GeV−2 × (t − M 2 +K) and αK∗(t) = 1 + +0.83 Gev−2 × (t − M 2 +K∗) the linear Reggeon trajectory. +The constants sK +0 and sK∗ +0 +are determined to be 3.0 GeV2 +and 1.5 GeV2, respectively [52]. Here, the α′ +K and α′ +K∗ +are the Regge-slopes. +Then, the full photo-production amplitudes for γn → +K+Σ∗− +1/2− reaction can be expressed as +Mµ = +� +Mµ +K− + Mµ +c +� � +t − M 2 +K− +� +FRegge +K ++ Mµ +Σ−fu ++ Mµ +K∗ +� +t − M 2 +K∗ +� +FRegge +K∗ ++ Mµ +nfs, +(16) +While FRegge +K +and FRegge +K∗ +stand for the Regge propaga- +tors. The form factors fs and fu are included to suppress +the large momentum transfer of the intermediate par- +ticles and describe their off-shell behavior, because the +intermediate hadrons are not point-like particles. +For +s-channel and u-channel baryon exchanges, we use the +following form factors [42, 53] +fi(q2 +i ) = +� +Λ4 +i +Λ4 +i + (q2 +i − M 2 +i )2 +�2 +, i = s, u +(17) +with Mi and qi being the masses and four-momenta of +the intermediate baryons, and the Λi is the cut-off values +for baryon exchange diagrams. +In this work, we take +Λs = Λu = 1.5 GeV, and will discuss the results with +different cut-off. +Finally, the unpolarized differential cross section in the +center of mass (c.m.) frame for the γn → KΣ∗− +1/2− reac- +tion reads +dσ +dΩ = MNMΣ∗|⃗kc.m. +1 +||⃗pc.m. +1 +| +8π2(s − M 2 +N)2 +� +λ,sp,sΣ∗ +|M|2, +(18) +where s denotes the invariant mass square of the center +of mass (c.m.) frame for Σ∗ +1/2− photo-production. Here +⃗kc.m. +1 +and ⃗pc.m. +1 +are the three-momenta of the photon and +K meson in the c.m. +frame, while dΩ = 2πdcosθc.m., +with θc.m. the polar outgoing K scattering angle. +III. +NUMERICAL RESULTS AND +DISCUSSIONS +In this section, we show our numerical results of the dif- +ferential and total cross sections for the γn → K+Σ∗− +1/2− +reaction. +The masses of the mesons and baryons are +taken from RPP [15], as given in Table I. In addition, the +mass and width of the Σ(1480) are M = 1480 ± 15 GeV +and Γ = 60 ± 15 GeV, respectively [28]. +TABLE I: Particle masses used in this work. +Particle +Mass (MeV) +n +939.565 +Σ− +1197.449 +K+ +493.677 +K− +493.677 +K∗ +891.66 +First we show the angle dependence of the differential +cross sections for the γn → K+Σ∗− +1/2− reaction in Fig. 2, +where the the center-of-mass energies W = √s varies +from 2.0 to 2.8 GeV. The black curves labeled as ‘Total’ +show the results of all the contributions from the t-, s-, +u-channels, and contact term. The blue-dot curves and +red-dashed curves stand for the contributions from the +u-channel Σ exchange and t-channel K exchange mecha- +nism, respectively. The magenta-dot-dashed curves and +the green-dot curves correspond to the contributions +from the s-channel and t-channel K∗ exchange diagrams, +respectively, while the cyan-dot-dashed curves represent +the contribution from the contact term. According to the +differential cross sections, one can find that the t-channel +K meson exchange term plays an important role at for- +ward angles for the process γn → K+Σ∗− +1/2−, mainly due +to the Regge effects of the t-change K exchange. The +K-Reggeon exchange shows steadily increasing behavior +with cosθc.m. and falls off drastically at very forward an- +gles. In addition, the u-channel Σ exchange term mainly +contribute to the backward angles for both processes. +It should be stressed that the contribution from the t- +channel K∗ exchange term is very small and could be +safely neglected for the process γn → K+Σ∗− +1/2−, which +is consistent with the results of Ref. [27]. +In addition to the the differential cross sections, we +have also calculated the total cross section of the γn → +K+Σ∗− +1/2− reaction as a function of the initial photon en- +ergy. The results are shown in Fig. 3. The black curve +labeled as ‘Total’ shows the results of all the contribu- +tions, including t-, s-, u- channels and contact term. The +blue-dot and red-dashed curves stand for the contribu- +tions from the u- channel Σ exchange and t- channel +K exchange mechanism, respectively. The magenta-dot- +dashed and the green-dot curves show the contribution of +s-channel and t-channel K∗ exchange diagrams, respec- +tively, while the cyan-dot-dashed curve represents the + +4 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 +dσ/dcosθc.m. (µb) +cosθc.m. +W=2.0 GeV +K-t +K*-t +s-channel +u-channel +contact term +Total +W=2.1 GeV +W=2.2 GeV + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 +W=2.3 GeV +W=2.4 GeV +W=2.5 GeV + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 +-1 +-0.5 +0 +0.5 +1 +W=2.6 GeV +-1 +-0.5 +0 +0.5 +1 +W=2.7 GeV +-1 +-0.5 +0 +0.5 +1 +W=2.8 GeV +FIG. 2: (Color online) γn → K+Σ∗− +1/2− differential cross sections as a function of cosθc.m. are plotted for γn-invariant mass +intervals (in GeV units). The black curve labeled as ‘Total’ shows the results of all the contributions, including t-, s-, u- channels +and contact term. The blue-dot and red-dashed curves stand for the contributions from the effective Lagrangian approach u- +channel Σ exchange and t- channel K exchange mechanism, respectively. The magenta-dot-dashed and the green-dot-dashed +curves show the contribution of s-channel and t-channel K∗ exchange diagrams, respectively, while the cyan-dot-dashed curve +represent the contribution of the contact term. +contribution of the contact term. For the γn → K+Σ∗− +1/2− +reaction its total cross section attains a maximum value +of about 4.3 µb at Eγ = 2.3 GeV. It is expected that the +Σ∗(1480) could be observed by future experiments in the +process γn → K+Σ∗− (1480) → Σ−π0/Σ0π−/Σ−γ. +Finally, we also show the total cross section for γn → +K+Σ∗− +1/2− with the cut-off Λs/u = 1.2, 1.5, and 1.8 GeV +in Fig. 4, where one can find the total cross sections are +weakly dependence on the value of the cut-off. Since the +precise couplings of the Σ(1480) are still unknown, the + +5 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 + 5.5 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +σ (µb) +Eγ (GeV) +K-t +K*-t +s-channel +u-channel +contact term +Total +FIG. 3: +(Color online) Total cross section for γn +→ +K+Σ∗ +1/2− is plotted as a function of the lab energy Eγ. The +black curve labeled as ‘Total’ shows the results of all the con- +tributions, including t-,s-,u- channels and contact term. The +blue-dot and red-dashed curves stand for the contributions +from the effective Lagrangian approach u- channel Σ exchange +and t- channel K exchange mechanism, respectively. +The +magenta-dot-dashed and the green-dot curves show the con- +tribution of s-channel and t-channel K∗ exchange diagrams, +respectively, while the cyan-dot-dashed curve represents the +contribution of the contact term. + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 + 5.5 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +σ (µb) +Eγ (GeV) +Λs,u = 1.2 GeV +Λs,u = 1.5 GeV +Λs,u = 1.8 GeV +FIG. 4: +(Color online) Total cross section for γn +→ +K+Σ∗ +1/2− with the cut-off Λs/u = 1.2, 1.5, and 1.8 GeV. +future experiment would be helpful to constrain these +couplings if the state Σ(1480) is confirmed. +IV. +SUMMARY +The lowest Σ∗− +1/2− is far from established, and its ex- +istence is important to understand the low-lying excited +baryon with JP = 1/2−. There are many experimen- +tal hints of the Σ∗(1480), which has been listed in the +previous version of the Review of Particle Physics. We +propose to search for this state in the photoproduction +process to confirm its existence. +Assuming that the JP += +1/2− low lying state +Σ∗ (1480) has a sizeable coupling to the ¯KN according +the study of Ref. [36], we have phenomenologically inves- +tigated the γn → K+Σ∗− +1/2− reaction by considering the +contributions from the t-channel K/K∗ exchange term, +s-channel nucleon term, u-channel Σ exchange term, and +contact term within the Regge-effective Lagrange ap- +proach. +The differential cross sections and total cross +sections for these processes are calculated with our model +parameters. The total cross section of γn → K+Σ∗− +1/2− +is about 4.3 µb around Eγ = 2.3 GeV. We encourage +our experimental colleagues to measure γn → K+Σ∗− +1/2− +process. +Acknowledgements +This +work +is +supported +by +the +National +Natu- +ral Science Foundation of China under Grant Nos. +12192263, 12075288, 11735003, and 11961141012, the +Natural Science Foundation of Henan under Grand No. +222300420554. +It is also supported by the Project of +Youth Backbone Teachers of Colleges and Universities +of Henan Province (2020GGJS017), the Youth Talent +Support Project of Henan (2021HYTP002), the Open +Project of Guangxi Key Laboratory of Nuclear Physics +and Nuclear Technology, No.NLK2021-08, the Youth In- +novation Promotion Association CAS. +[1] R. H. Dalitz and S. F. Tuan, A possible resonant state +in pion-hyperon scattering, Phys. Rev. Lett. 2 (1959), +425-428. +[2] M. H. Alston, L. W. Alvarez, P. Eberhard, M. L. Good, +W. Graziano, H. K. Ticho and S. G. Wojcicki, Study of +Resonances of the Σπ System, Phys. Rev. Lett. 6 (1961), +698-702. +[3] J. A. Oller and U. G. 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C 87 (2013), 054004. + diff --git a/3NFAT4oBgHgl3EQflB25/content/tmp_files/load_file.txt b/3NFAT4oBgHgl3EQflB25/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8e73e6fb3c5bb56d076906a01285953f89d6809 --- /dev/null +++ b/3NFAT4oBgHgl3EQflB25/content/tmp_files/load_file.txt @@ -0,0 +1,708 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf,len=707 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='08615v1 [hep-ph] 20 Jan 2023 Photo-production of lowest Σ∗ 1/2− state within the Regge-effective Lagrangian approach Yun-He Lyu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='1 Han Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='1 Neng-Chang Wei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='2 Bai-Cian Ke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='1 En Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='1 and Ju-Jun Xie3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 4 1School of Physics and Microelectronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Zhengzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Zhengzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Henan 450001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' China 2School of Nuclear Sciences and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Beijing 101408,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' China 3Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' China 4Southern Center for Nuclear-Science Theory (SCNT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Huizhou 516000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Guangdong Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' China (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 2023) Since the lowest Σ∗ state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' with quantum numbers spin-parity JP = 1/2−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' is far from estab- lished experimentally and theoretically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' we have performed a theoretical study on the Σ∗ 1/2− photo- production within the Regge-effective Lagrangian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Taking into account that the Σ∗ 1/2− couples to the ¯KN channel, we have considered the contributions from the t-channel K exchange diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Moreover, these contributions from t-channel K∗ exchange, s-channel nucleon pole, u- channel Σ exchange, and the contact term, are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The differential and total cross sections of the process γn → K+Σ∗− 1/2− are predicted with our model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The results should be helpful to search for the Σ∗ 1/2− state experimentally in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' PACS numbers: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' INTRODUCTION The study of the low-lying excited Λ∗ and Σ∗ hyperon resonances is one of the most important issues in hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Especially, since the Λ(1405) was discovered experimentally [1, 2], its nature has called many atten- tions [3–8], and one explanation for Λ(1405) is that it is a ¯KN hadronic molecular state [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In addition, the isospin I = 1 partner of the Λ(1405), the lowest Σ∗ 1/2− is crucial to understand the light baryon spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' At present, there is a Σ∗(1620) with JP = 1/2− listed in the latest version of Review of Particle Physics (RPP) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' It should be stressed that the Σ∗(1620) state is a one-star baryon resonance, and many studies indicate that the lowest Σ∗ 1/2− resonance is still far from estab- lished, and its mass was predicted to lie in the range of 1380 ∼ 1500 MeV [13, 16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Thus, searching for the lowest Σ∗ 1/2− is helpful to understand the low-lying ex- cited baryons with JP = 1/2− and the light flavor baryon spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The analyses of the relevant data of the process K−p → Λπ+π− suggest that there may exist a Σ∗ 1/2− resonance with mass about 1380 MeV [16, 17], which is consistent with the predictions of the unquenched quark models [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The analyses of the K∗Σ photo- production also indicate that the Σ∗ 1/2− is possibly buried under the Σ∗(1385) peak with mass of 1380 MeV [21], and it is proposed to search for the Σ∗ 1/2− in the pro- cess Λc → ηπ+Λ [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' A more delicate analysis of the CLAS data on the process γp → KΣπ [23] suggests that the Σ∗ 1/2− peak should be around 1430 MeV [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [24, 25], we suggest to search for such state in the processes of χc0(1P) → ¯ΣΣπ and χc0(1P) → ¯ΛΣπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In addition, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [26] has found one Σ∗ 1/2− state with mass around 1400 MeV by solving coupled channel scattering equations, and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [27] suggests to search for this state in the photo-production process γp → K+Σ∗0 1/2−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' It’s worth mentioning that a Σ∗(1480) resonance with JP = 1/2− has been listed on the previous version of RPP [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' As early as 1970, the Σ∗(1480) resonance was reported in the Λπ+, Σπ, and p ¯K0 channels of the π+p scattering in the Princeton-Pennsylvania Accelera- tor 15-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='∼hydrogen bubble chamber [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In 2004, a bump structure around 1480 MeV was observed in the K0 Sp(¯p) invariant mass spectrum of the inclusive deep inelastic ep scattering by the ZEUS Collaboration [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Furthermore, a signal for a resonance at 1480 ± 15 MeV with width of 60 ± 15 MeV was observed in the process pp → K+pY ∗0 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Theoretically, the Σ∗(1480) was in- vestigated within different models [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [36], the S-wave meson-baryon interactions with strangeness S = −1 were studied within the unitary chiral approach, and one narrow pole with pole position of 1468−i 13 MeV was found in the second Riemann sheet, which could be associated with the Σ∗(1480) resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' However, the Σ∗(1480) signals are insignificant, and the existence of this state still needs to be confirmed within more precise experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' As we known, the photo-production reactions have been used to study the excited hyperon states Σ∗ and Λ∗, and the Crystal Ball [37–39], LEPS [40], and CLAS [23] Collaborations have accumulated lots of relevant exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' For instance, with these data, we have analyzed the process γp → KΛ∗(1405) to deepen the un- derstanding of the Λ∗(1405) nature in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In order to confirm the existence of the Σ∗(1480), we propose to investigate the process γN → KΣ∗(1480) 1 within the 1 Here after, we denote Σ∗(1480) as the lowest Σ∗ 1/2− state unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 2 Regge-effective Lagrange approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Considering the Σ∗(1480) signal was first observed in the π+Λ invariant mass distribution of the process π+p → π+K+Λ, and the significance is about 3 ∼ 4σ [30], we search for the charged Σ∗(1480) in the process γn → K+Σ∗− 1/2−, which could also avoid the contribu- tions of possible excited Λ∗ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' We will consider the t-, s-, u-channels diagrams in the Born approximation by employing the effective Lagrangian approach, and the t-channel K/K∗ exchanges terms within Regge model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Then we will calculate the differential and total cross sections of the process γn → K+Σ∗− 1/2− reaction, which are helpful to search for Σ∗ 1/2− experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' II, the the- oretical formalism for studying the γn → K+Σ∗−(1480) reactions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The numerical results of total and differential cross sections and discussion are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Finally, a brief summary is given in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' FORMALISM The reaction mechanisms of the Σ∗(1480) (≡ Σ∗) photo-production process are depicted in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1, where we have taken into account the contributions from the t-channel K and K∗ exchange term, s-channel nu- cleon pole term, u-channel Σ exchange term, and the contact term, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' γ(k1) K(k2) N(p1) Σ∗(p2) K, K∗ γ K N Σ∗ N γ Σ∗ N K γ K N Σ∗ Σ (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1: The mechanisms of the γn → K+Σ∗− 1/2− process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' (a) t-channel K/K∗ exchange terms, (b) s-channel nuclear term, (c) u-channel Σ exchange term, and (d) contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The k1, k2, p1, and p2 stand for the four-momenta of the initial photon, kaon, neutron, and Σ∗(1480), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' To compute the scattering amplitudes of the Feynman diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1 within the effective Lagrangian approach, we use the Lagrangian densities for the elec- tromagnetic and strong interaction vertices as used in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [27, 42–46] LγKK = −ie � K† (∂µK) − � ∂µK†� K � Aµ, (1) LγKK∗ = gγKK∗ǫµναβ∂µAν∂αK∗ βK, (2) LγNN = −e ¯N � γµˆe − ˆκN 2MN σµν∂ν � AµN, (3) LγΣΣ∗ = eµΣΣ∗ 2MN ¯Σγ5σµν∂νAµΣ∗ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=', (4) LKNΣ = −igKNΣ ¯Nγ5ΣK + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=', (5) LK∗NΣ∗ = igK∗NΣ∗ √ 3 ¯K∗µ ¯Σ∗γµγ5N + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' (6) LKNΣ∗ = gKNΣ∗ ¯K ¯ Σ∗N + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=', (7) where e(= √ 4πα) is the elementary charge unit, Aµ is the photon filed, and ˆe ≡ (1+τ3)/2 denotes the charge opera- tor acting on the nucleon field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' ˆκN ≡ κpˆe+κn(1−ˆe) is the anomalous magnetic moment, and we take κn = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='913 for neutron [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' MN and MΣ denote the masses of nu- cleon and the ground-state of Σ hyperon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The strong coupling gKNΣ is taken to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='09 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The gγKK∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='254 GeV−1 is determined from the experimental data of ΓK∗→K+γ [15] and the value of gK∗NΣ∗ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='26 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='06 is taken from Ref [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In addition, the coupling gKNΣ∗ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='74 GeV is taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [36], and the transition magnetic moment µΣΣ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='28 is taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [27] With the effective interaction Lagrangian densities given above, the invariant scattering amplitudes are de- fined as M = ¯uΣ∗(p2, sΣ∗)Mµ huN(k2, sp)ǫµ(k1, λ), (8) where uΣ∗ and uN stand for the Dirac spinors, respec- tively, while ǫµ(k1, λ) is the photon polarization vector and the sub-indice h corresponds to different diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The reduced amplitudes Mµ h are written as Mµ K∗ = egγKK∗gK∗NΣ∗ √ 3MK∗(t − M 2 K∗)ǫαβµνk1αk2βγνγ5, (9) Mµ K− = −2iegKNΣ∗ t − M 2 K kµ 2 , (10) Mµ Σ− = −i eµΣΣ∗gKNΣ 2Mn(u − M 2 Σ∗)(q/u − MΣ)σµνk1ν, (11) Mµ n = κngKNΣ∗ 2Mn(s − M 2n)σµνk1ν(q/s + Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' (12) In order to keep the full photoproduction amplitudes considered here gauge invariant, we adopt the amplitude of the contact term Mµ c = −iegKNΣ∗ pµ 2 p2 · k1 , (13) for γn → K+Σ∗− 1/2−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 3 It is known that the Reggeon exchange mechanism plays a crucial role at high energies and forward an- gles [48–51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' thus we will adopt Regge model for mod- eling the t-channel K and K∗ contributions by replacing the usual pole-like Feynman propagator with the corre- sponding Regge propagators as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1 t − M 2 K → FRegge K = � s sK 0 �αK(t) πα′ K sin(παK(t))Γ(1 + αK(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='(14) 1 t − M 2 K∗ → FRegge K∗ = � s sK∗ 0 �αK∗(t) πα′ K∗ sin(παK∗(t))Γ(αK∗(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='(15) with αK(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='7 GeV−2 × (t − M 2 K) and αK∗(t) = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='83 Gev−2 × (t − M 2 K∗) the linear Reggeon trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The constants sK 0 and sK∗ 0 are determined to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='0 GeV2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 GeV2, respectively [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Here, the α′ K and α′ K∗ are the Regge-slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Then, the full photo-production amplitudes for γn → K+Σ∗− 1/2− reaction can be expressed as Mµ = � Mµ K− + Mµ c � � t − M 2 K− � FRegge K + Mµ Σ−fu + Mµ K∗ � t − M 2 K∗ � FRegge K∗ + Mµ nfs, (16) While FRegge K and FRegge K∗ stand for the Regge propaga- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The form factors fs and fu are included to suppress the large momentum transfer of the intermediate par- ticles and describe their off-shell behavior, because the intermediate hadrons are not point-like particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' For s-channel and u-channel baryon exchanges, we use the following form factors [42, 53] fi(q2 i ) = � Λ4 i Λ4 i + (q2 i − M 2 i )2 �2 , i = s, u (17) with Mi and qi being the masses and four-momenta of the intermediate baryons, and the Λi is the cut-off values for baryon exchange diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In this work, we take Λs = Λu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 GeV, and will discuss the results with different cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Finally, the unpolarized differential cross section in the center of mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=') frame for the γn → KΣ∗− 1/2− reac- tion reads dσ dΩ = MNMΣ∗|⃗kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1 ||⃗pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1 | 8π2(s − M 2 N)2 � λ,sp,sΣ∗ |M|2, (18) where s denotes the invariant mass square of the center of mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=') frame for Σ∗ 1/2− photo-production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Here ⃗kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1 and ⃗pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 1 are the three-momenta of the photon and K meson in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' frame, while dΩ = 2πdcosθc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=', with θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' the polar outgoing K scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' NUMERICAL RESULTS AND DISCUSSIONS In this section, we show our numerical results of the dif- ferential and total cross sections for the γn → K+Σ∗− 1/2− reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The masses of the mesons and baryons are taken from RPP [15], as given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In addition, the mass and width of the Σ(1480) are M = 1480 ± 15 GeV and Γ = 60 ± 15 GeV, respectively [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' TABLE I: Particle masses used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Particle Mass (MeV) n 939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='565 Σ− 1197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='449 K+ 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='677 K− 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='677 K∗ 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='66 First we show the angle dependence of the differential cross sections for the γn → K+Σ∗− 1/2− reaction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 2, where the the center-of-mass energies W = √s varies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='8 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The black curves labeled as ‘Total’ show the results of all the contributions from the t-, s-, u-channels, and contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The blue-dot curves and red-dashed curves stand for the contributions from the u-channel Σ exchange and t-channel K exchange mecha- nism, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The magenta-dot-dashed curves and the green-dot curves correspond to the contributions from the s-channel and t-channel K∗ exchange diagrams, respectively, while the cyan-dot-dashed curves represent the contribution from the contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' According to the differential cross sections, one can find that the t-channel K meson exchange term plays an important role at for- ward angles for the process γn → K+Σ∗− 1/2−, mainly due to the Regge effects of the t-change K exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The K-Reggeon exchange shows steadily increasing behavior with cosθc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' and falls off drastically at very forward an- gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In addition, the u-channel Σ exchange term mainly contribute to the backward angles for both processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' It should be stressed that the contribution from the t- channel K∗ exchange term is very small and could be safely neglected for the process γn → K+Σ∗− 1/2−, which is consistent with the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' In addition to the the differential cross sections, we have also calculated the total cross section of the γn → K+Σ∗− 1/2− reaction as a function of the initial photon en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The black curve labeled as ‘Total’ shows the results of all the contribu- tions, including t-, s-, u- channels and contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The blue-dot and red-dashed curves stand for the contribu- tions from the u- channel Σ exchange and t- channel K exchange mechanism, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The magenta-dot- dashed and the green-dot curves show the contribution of s-channel and t-channel K∗ exchange diagrams, respec- tively, while the cyan-dot-dashed curve represents the 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 dσ/dcosθc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' (µb) cosθc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='0 GeV K-t K*-t s-channel u-channel contact term Total W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='1 GeV W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='2 GeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='3 GeV W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='4 GeV W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 GeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='6 GeV 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='7 GeV 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 W=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='8 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 2: (Color online) γn → K+Σ∗− 1/2− differential cross sections as a function of cosθc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' are plotted for γn-invariant mass intervals (in GeV units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The black curve labeled as ‘Total’ shows the results of all the contributions, including t-, s-, u- channels and contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The blue-dot and red-dashed curves stand for the contributions from the effective Lagrangian approach u- channel Σ exchange and t- channel K exchange mechanism, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The magenta-dot-dashed and the green-dot-dashed curves show the contribution of s-channel and t-channel K∗ exchange diagrams, respectively, while the cyan-dot-dashed curve represent the contribution of the contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' contribution of the contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' For the γn → K+Σ∗− 1/2− reaction its total cross section attains a maximum value of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='3 µb at Eγ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' It is expected that the Σ∗(1480) could be observed by future experiments in the process γn → K+Σ∗− (1480) → Σ−π0/Σ0π−/Σ−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Finally, we also show the total cross section for γn → K+Σ∗− 1/2− with the cut-off Λs/u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='8 GeV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 4, where one can find the total cross sections are weakly dependence on the value of the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Since the precise couplings of the Σ(1480) are still unknown, the 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 σ (µb) Eγ (GeV) K-t K*-t s-channel u-channel contact term Total FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 3: (Color online) Total cross section for γn → K+Σ∗ 1/2− is plotted as a function of the lab energy Eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The black curve labeled as ‘Total’ shows the results of all the con- tributions, including t-,s-,u- channels and contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The blue-dot and red-dashed curves stand for the contributions from the effective Lagrangian approach u- channel Σ exchange and t- channel K exchange mechanism, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The magenta-dot-dashed and the green-dot curves show the con- tribution of s-channel and t-channel K∗ exchange diagrams, respectively, while the cyan-dot-dashed curve represents the contribution of the contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 4 σ (µb) Eγ (GeV) Λs,u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='2 GeV Λs,u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5 GeV Λs,u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='8 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 4: (Color online) Total cross section for γn → K+Σ∗ 1/2− with the cut-off Λs/u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='8 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' future experiment would be helpful to constrain these couplings if the state Σ(1480) is confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' SUMMARY The lowest Σ∗− 1/2− is far from established, and its ex- istence is important to understand the low-lying excited baryon with JP = 1/2−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' There are many experimen- tal hints of the Σ∗(1480), which has been listed in the previous version of the Review of Particle Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' We propose to search for this state in the photoproduction process to confirm its existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Assuming that the JP = 1/2− low lying state Σ∗ (1480) has a sizeable coupling to the ¯KN according the study of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' [36], we have phenomenologically inves- tigated the γn → K+Σ∗− 1/2− reaction by considering the contributions from the t-channel K/K∗ exchange term, s-channel nucleon term, u-channel Σ exchange term, and contact term within the Regge-effective Lagrange ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The differential cross sections and total cross sections for these processes are calculated with our model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' The total cross section of γn → K+Σ∗− 1/2− is about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='3 µb around Eγ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content='3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' We encourage our experimental colleagues to measure γn → K+Σ∗− 1/2− process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' Acknowledgements This work is supported by the National Natu- ral Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 12192263, 12075288, 11735003, and 11961141012, the Natural Science Foundation of Henan under Grand No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' 222300420554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFAT4oBgHgl3EQflB25/content/2301.08615v1.pdf'} +page_content=' It is also supported by the Project of Youth 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='03520v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='FA] 9 Jan 2023 CLASSIFYING WEAK PHASE RETRIEVAL P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' AKRAMI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We will give several surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' These results give a complete understanding of the dif- ference between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames {xi}m i=1 in Rn are not dense in the family of m-element sets of vectors in Rn for all m ≥ 2n − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (2) We show that any frame {xi}2n−2 i=1 containing one or more canonical basis vectors in Rn cannot do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We provide numerous examples to show that the obtained results are best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Introduction The concept of frames in a separable Hilbert space was originally introduced by Duffin and Schaeffer in the context of non-harmonic Fourier series [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Frames are a more flexible tool than bases because of the redundancy property that make them more applicable than bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Phase retrieval is an old problem of recovering a signal from the absolute value of linear measurement coefficients called intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Phase retrieval and norm retrieval have become very active areas of research in applied mathematics, computer science, engineering, and more today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Phase retrieval has been defined for both vectors and subspaces (projections) in all separable Hilbert spaces, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', [3], [4], [5], [6], [9], [10] and [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The concept of weak phase retrieval weakened the notion of phase retrieval and it has been first defined for vectors in ([8] and [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The rest of the paper is organized as follows: In Section 2, we give the basic definitions and certain preliminary results to be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Weak phase retrieval by vectors is introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' In section 4 we show that any family of vectors {xi}2n−2 i=1 doing weak phase retrieval cannot contain a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' In section 5, we show that the weak phase retrievable frames are not dense in all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' And in section 6 we give several surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' These results give a complete understanding of the difference between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' preliminaries First we give the background material needed for the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let H be a finite or infinite dimensional real Hilbert space and B(H) the class of all bounded linear operators defined on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The natural numbers and real numbers are denoted by “N” and “R”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We use [m] instead of the set {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', m} and use [{xi}i∈I] instead of span{xi}i∈I, where I is a finite or countable subset of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 42C15, 42C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Real Hilbert frames, Full spark, Phase retrieval, Weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The first author was supported by NSF DMS 1609760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' AKRAMI denote by Rn a n dimensional real Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We start with the definition of a real Hilbert space frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family of vectors {xi}i∈I in a finite or infinite dimensional separable real Hilbert space H is a frame if there are constants 0 < A ≤ B < ∞ so that A∥x∥2 ≤ � i∈I |⟨x, xi⟩|2 ≤ B∥x∥2, for all f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The constants A and B are called the lower and upper frame bounds for {xi}i∈I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If an upper frame bound exists, then {xi}i∈I is called a B-Bessel seqiemce or simply Bessel when the constant is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If A = B, it is called an A-tight frame and in case A = B = 1, it is called a Parseval frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The values {⟨x, xi⟩}∞ i=1 are called the frame coefficients of the vector x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It is immediate that a frame must span the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We will need to work with Riesz sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family X = {xi}i∈I in a finite or infinite dimensional real Hilbert space H is a Riesz sequence if there are constants 0 < A ≤ B < ∞ satisfying A � i∈I |ci|2 ≤ ∥ � i∈I cixi∥2 ≤ B � i∈I |ci|2 for all sequences of scalars {ci}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If it is complete in H, we call X a Riesz basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For an introduction to frame theory we recommend [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Throughout the paper the orthogonal projection or simply projection will be a self- adjoint positive projection and {ei}∞ i=1 will be used to denote the canonical basis for the real space Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', a basis for which ⟨ei, ej⟩ = δi,j = � 1 if i = j, 0 if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family of vectors {xi}i∈I in a real Hilbert space H does phase (norm) retrieval if whenever x, y ∈ H, satisfy |⟨x, xi⟩| = |⟨y, xi⟩| for all i ∈ I, then x = ±y (∥x∥ = ∥y∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Phase retrieval was introduced in reference [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' See reference [1] for an introduc- tion to norm retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Note that if {xi}i∈I does phase (norm) retrieval, then so does {aixi}i∈I for any 0 < ai < ∞ for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' But in the case where |I| = ∞, we have to be careful to maintain frame bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' This always works if 0 < infi∈I ai ≤ supi∈Iai < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' But this is not necessary in general [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The complement property is an essential issue here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family of vectors {xi}i∈I in a finite or infinite dimensional real Hilbert space H has the complement property if for any subset J ⊂ I, either span{xi}i∈J = H or span{xi}i∈Jc = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Fundamental to this area is the following for which the finite dimensional case appeared in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' WEAK PHASE RETRIEVAL 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family of vectors {xi}i∈I does phase retrieval in Rn if and only if it has the complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We recall: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family of vectors {xi}m i=1 in Rn is full spark if for every I ⊂ [m] with |I| = n , {xi}i∈I is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If {xi}m i=1 does phase retrieval in Rn, then m ≥ 2n− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If m = 2n− 1, {xi}m i=1 does phase retrieval if and only if it is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We rely heavily on a significant result from [2]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If {xi}2n−2 i=1 does weak phase retrieval in Rn then for every I ⊂ [2n−2], if x ⊥ span{xi}i∈I and y ⊥ {xi}i∈Ic then x ∥x∥ + y ∥y∥ and x ∥x∥ − y ∥y∥ are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' In particular, if ∥x∥ = ∥y∥ = 1, then x + y and x − y are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Hence, if x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , an) then y = (ǫ1a1, ǫ2a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , ǫnan), where ǫi = ±1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The above theorem may fail if ∥x∥ ̸= ∥y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For example, consider the weak phase retrievable frame in R3: \uf8ee \uf8ef\uf8ef\uf8f0 1 1 1 −1 1 1 1 −1 1 1 1 −1 \uf8f9 \uf8fa\uf8fa\uf8fb Also, x = (0, 1, −1) is perpendicular to rows 1 and 2 and y = (0, 1 2, 1 2) is orthogonal to rows 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' But x + y = (0, 3 2, 1 2) and x − y = (0, −1 2 , −3 2 ) and these are not disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' But if we let them have the same norm we get x = (0, 1, −1) and y = (0, 1, 1) so x + y = (0, 1, 0) and x − y = (0, 0, 1) and these are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Weak phase retrieval The notion of “Weak phase retrieval by vectors” in Rn was introduced in [8] and was developed further in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' One limitation of current methods used for retrieving the phase of a signal is computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Recall that a generic family of (2n − 1)- vectors in Rn satisfies phaseless reconstruction, however no set of (2n − 2)-vectors can (See [7] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' By generic we are referring to an open dense set in the set of (2n − 1)-element frames in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Two vectors x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , bn) in Rn weakly have the same phase if there is a |θ| = 1 so that phase(ai) = θphase(bi) for all i ∈ [n], for which ai ̸= 0 ̸= bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If θ = 1, we say x and y weakly have the same signs and if θ = −1, they weakly have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Therefore with above definition the zero vector in Rn weakly has the same phase with all vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For x ∈ R, sgn(x) = 1 if x > 0 and sgn(x) = −1 if x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A family of vectors {xi}m i=1 does weak phase retrieval in Rn if for any x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , bn) in Rn with |⟨x, xi⟩| = |⟨y, xi⟩| for all i ∈ [m], then x and y weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' A fundamental result here is 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' AKRAMI Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' [8] Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , bn) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The following are equivalent: (1) We have sgn(aiaj) = sgn(bibj), for all 1 ≤ i ̸= j ≤ n (2) Either x, y have weakly the same sign or they have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It is clear that if {xi}m i=1 does weak phase retrieval in Rn, then {cixi}m i=1 does weak phase retrieval as long as ci > 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The following appears in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If X = {xi}m i=1 does weak phase retrieval in Rn, then m ≥ 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Finally, we have: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' [7] If a frame X = {xi}2n−2 i=1 does weak phase retrieval in Rn, then X is a full spark frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Clearly the converse of above theorem is not hold, for example {(1, 0), (0, 1)} is full spark frame that fails weak phase retrieval in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If {xi}i∈I does phase retrieval and R is an invertible operator on the space then {Rxi}i∈I does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' This follows easily since |⟨x, Rxi⟩| = |⟨y, Rxi⟩| implies |⟨R∗x, xi⟩| = |⟨R∗y, xi⟩|, and so R∗x = θR∗y for |θ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since R is invertible, x = θy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' This result fails badly for weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For example, let e1 = (1, 0), e2 = (0, 1), x1 = ( 1 √ 2, 1 √ 2, x2 = ( 1 √ 2, −1 √ 2) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then {e1, e2} fails weak phase retrieval, {x1, x2} does weak phase retrieval and Uei = xi is a unitary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Frames Containing Unit Vectors Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Any frame {xi}2n−2 i=1 whith one or more canonical basis vectors in Rn cannot do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We proceed by way of contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Recall that {xi}2n−2 i=1 must be full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {ei}n i=1 be the canonical orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Assume I ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', 2n−2} with |I| = n − 1 and assume x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , bn) with ∥x∥ = ∥y∥ = 1 and x ⊥ X = span{xi}i∈I and y ⊥ span{xi}2n−2 i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' After reindexing {ei}n i=1 and {xi}2n−2 i=1 }, we assume x1 = e1, I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', n−1 and Ic = {n, n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , 2n− 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since ⟨x, x1⟩ = a1 = 0, by Theorem 2, b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let P be the projection on span{ei}n i=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So {Pxi}2n−2 i=n is (n − 1)-vectors in an (n − 1)-dimensional space and y is orthogonal to all these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So there exist {ci}2n−2 i=n not all zero so that 2n−2 � i=n ciPxi = 0 and so 2n−1 � i=n cixi(1)x1 − 2n−2 � i=n cixi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' That is, our vectors are not full spark, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The fact that there are (2n− 2) vectors in the theorem is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For example, e1, e2, e1 + e2 is full spark in R2, so it does phase retrieval - and hence weak phase retrieval - despite the fact that it contains both basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The converse of Theorem 5 is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Consider the full spark frame X = {(1, 2, 3), (0, 1, 0), (0, −2, 3), (1, −2, −3)} in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Every set of its two same coordinates, {(1, 2), (0, 1), (0, −2), (1, −2)}, {(1, 3), (0, 0), (0, 3), (1, −3)}, and WEAK PHASE RETRIEVAL 5 {(2, 3), (1, 0), (−2, 3), (−2, −3)} do weak phase retrieval in R2, but by Theorem 5, X cannot do weak phase retrieval in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Weak Phase Retrievable Frames are not Dense in all Frames If m ≥ 2n − 1 and {xi}m i=1 is full spark then it has complement property and hence does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since the full spark frames are dense in all frames, it follows that the frames doing phase retrieval are dense in all frames with ≥ 2n − 1 vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We will now show that this result fails for weak phase retrievable frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The easiest way to get very general frames failing weak phase retrieval is: Choose x, y ∈ Rn so that x + y, x − y do not have the same or opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let X1 = x⊥ and Y1 = y⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then span{X1, X2} = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Choose {xi}n−1 i=1 vectors spanning X1 and {xi}2n−2 i=n be vectors spanning X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then {xi}2n−2 i=1 is a frame for Rn with x ⊥ xi, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', n − 1 and y ⊥ xi, for all i = n, n + 1, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It follows that |⟨x + y, xi⟩| = |⟨x − y, xi⟩|, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , n, but x, y do not have the same or opposite signs and so {xi}2n−2 i=1 fails weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If X is a subspace of Rn, we define the sphere of X as SX = {x ∈ X : ∥x∥ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If X, Y are subspaces of Rn, we define the distance between X and Y as d(X, Y ) = supx∈SXinfy∈SY ∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It follows that if d(X, Y ) < ǫ then for any x ∈ X there is a z ∈ SY so that ∥ x ∥x∥ − z∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Letting y = ∥x∥z we have that ∥y∥ = ∥x∥ and ∥x − y∥ < ǫ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let X, Y be hyperplanes in Rn and unit vectors x ⊥ X, y ⊥ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If d(X, Y ) < ǫ then min{∥x − y∥, ∥x + y∥} < 6ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since span{y, Y } = Rn, x = ay + z for some z ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' By replacing y by −y if necessary, we may assume 0 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' By assumption, there is some w ∈ X with ∥w∥ = ∥z∥ so that ∥w − z∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Now a = a∥y∥ = ∥ay∥ = ∥x − z∥ ≥ ∥x − w∥ − ∥w − z∥ ≥ ∥x∥ − ǫ = 1 − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So, 1 − a < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Also, 1 = ∥x∥2 = a2 + ∥w∥2 implies a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 0 < 1 − a < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 1 = ∥x∥2 = ∥ay + z∥2 = a2∥y∥2 + ∥z∥2 = a2 + ∥z∥2 ≥ (1 − ǫ)2 + ∥z∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So ∥z∥2 ≤ 1 − (1 − ǫ)2 = 2ǫ − ǫ2 ≤ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Finally, ∥x − y∥2 = ∥(ay + z) − y∥2 ≤ (∥(1 − a)y∥ + ∥z∥)2 ≤ (1 − a)2∥y∥2 + ∥z∥2 + 2(1 − a)∥y∥∥z∥ < ǫ2 + 2ǫ + 2 √ 2ǫ2 < 6ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' AKRAMI □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let X, Y be hyperplanes in Rn, {xi}n−1 i=1 be a unit norm basis for X and {yi}n−1 i=1 be a unit norm basis for Y with basis bounds B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If �n−1 i=1 ∥xi − yi∥ < ǫ then d(X, Y ) < 2ǫB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let 0 < A ≤ B < ∞ be upper and lower basis bounds for the two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Given a unit vector x = �n−1 i=1 aixi ∈ X, let y = �n−1 i=1 aiyi ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We have that sup1≤i≤n−1|ai| ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We compute: ∥x − y∥ = ∥ n−1 � i=1 ai(xi − yi)∥ ≤ n−1 � i=1 |ai|∥xi − yi∥ ≤ (sup1≤i≤n−1|ai|) n−1 � i=1 ∥xi − yi∥ ≤ Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So ∥y∥ ≥ ∥x∥ − ∥x − y∥ ≥ 1 − Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' ����x − y ∥y∥ ���� ≤ ∥x − y∥ + ����y − y ∥y∥ ���� ≤ Bǫ + 1 ∥y∥∥(1 − ∥y∥)y∥ = Bǫ + (1 − ∥y∥) ≤ 2Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It follows that d(X, Y ) < 2Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {xi}n i=1 be a basis for Rn with unconditional basis constant B and assume yi ∈ Rn satisfies �n i=1 ∥xi − yi∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then {yi}n i=1 is a basis for Rn which is 1 + ǫB-equivalent to {xi}n i=1 and has unconditional basis constant B(1 + ǫB)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Fix {ai}n i=1 and compute ∥ n � i=1 aiyi∥ ≤ ∥ n � i=1 aixi∥ + ∥ n � i=1 |ai|(xi − yi)∥ ≤ ∥ n � i=1 aixi∥ + (sup1≤i≤n|ai|) n � i=1 ∥xi − yi∥ ≤ ∥ n � i=1 aixi∥ + (sup1≤i≤n|ai|)ǫ ≤ ∥ n � i=1 aixi∥ + ǫB∥ n � i=1 aixi∥ = (1 + ǫB)∥ n � i=1 aixi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' WEAK PHASE RETRIEVAL 7 Similarly, ∥ n � i=1 |ai|yi∥ ≥ (1 − ǫB)∥ n � i=1 aixi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So {xi}n i=1 is (1 + ǫB)-equivalent to {yi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For ǫi = ±1, ∥ n � i=1 ǫiaiyi∥ ≤ (1 + ǫB)∥ n � i=1 ǫiaixi∥ ≤ B(1 + ǫB)∥ n � i=1 aixi∥ ≤ B(1 + ǫB)2∥ n � i=1 aiyi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' and so {yi}n i=1 is a B(1 + ǫB) unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The family of m-element weak phase retrieval frames are not dense in the set of m-element frames in Rn for all m ≥ 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We may assume m = 2n−2 since for larger m we just repeat the (2n-2) vec- tors over and over until we get m vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {ei}n i=1 be the canonical orthonormal basis for Rn and let xi = ei for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' By [10], there is an orthonormal sequence {xi}2n−2 i=n+1 so that {xi}2n−2 i=1 is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let X = span{xi}n−1 i=1 and Y = span{xi}2n−2 i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then x = en ⊥ X and there is a ∥y∥ = 1 with y ⊥ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Note that ⟨x − y, en⟩ ̸= 0 ̸= ⟨x + y, en⟩, for otherwise, x = ±y ⊥ span{xi}i̸=n, contradicting the fact that the vectors are full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So there is a j = n and a δ > 0 so that |(x + y)(j)|, |(x − y)(j)| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We will show that there exists an 0 < ǫ so that whenever {yi}2n−2 i=1 are vectors in Rn satisfying �n i=1 ∥xi − yi∥ < ǫ, then {yi}n i=1 fails weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Fix 0 < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Assume {yi}2n−2 i=1 are vectors so that �2n−2 i=1 ∥xi−yi∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Choose unit vectors x′ ⊥ span{yi}i∈I, y′ ⊥ span{yi}i∈Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' By Proposition 2 and Lemma 1, we may choose ǫ small enough (and change signs if necessary) so that ∥x−x′∥, ∥y−y′∥ < δ 4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Hence, since the unconditional basis constant is B, |[(x + y) − (x′ + y′)](j)| ≤ |(x − x′)j| + |(y − y′)(j)| < B∥x − x′∥ + B∥y − y′∥ ≤ 2B δ 4B = δ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It follows that |(x′ + y′)(j)| ≥ |(x + y)(j)| − |[(x + y) − (x′ + y′)](j)| ≥ δ − 1 2δ = δ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Similarly, |(x′ − y′)(j)| > δ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So x′ + y′, x′ − y′ are not disjointly supported and so {yi}2n−2 i=1 fails weak phase retrieval by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Classifying Weak Phase Retrieval In this section we will give several surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' These results give a complete understanding of the difference between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Now we give a surprising and very strong classification of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' AKRAMI Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {xi}2n−2 i=1 be non-zero vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The following are equivalent: (1) The family {xi}2n−2 i=1 does weak phase retrieval in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (2) If x, y ∈ Rn and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='1) |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , 2n − 2, then one of the following holds: (a) x = ±y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (b) x and y are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (1) ⇒ (2): Given the assumption in the theorem, assume (a) fails and we will show that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since {xi}2n−2 i=1 does weak phase retrieval, replacing y by −y if necessary, Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='1 implies aj = bj whenever aj ̸= 0 ̸= bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let I = {1 ≤ i ≤ 2n − 2 : ⟨x, xi⟩ = ⟨y, yi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then x + y ⊥ xi for all i ∈ Ic and x − y ⊥ xi for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' By Theorem 2, x + y ∥x + y + x − y ∥x − y∥ and x + y ∥x + y∥ − x − y ∥x − y∥ are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Assume there is a 1 ≤ j ≤ n with aj = bj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then (x + y)(j) ∥x + y∥ + (x − y)(j) ∥x − y∥ = 2aj ∥x + y∥ and (x + y)(j) ∥x + y∥ − (x − y)(j) ∥x − y∥ = 2aj ∥x + y∥, Contradicting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (2) ⇒ (1): This is immediate since (a) and (b) give the conditions for weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ Phase retrieval is when (a) in the theorem holds for every x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So this the- orem shows clearly the difference between weak phase retrieval and phase retrieval: namely when (b) holds at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If {xi}2n−2 i=1 does weak phase retrieval in Rn, then there are disjointly supported non-zero vectors x, y ∈ Rn satisfying: |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since {xi}2n−2 i=1 must fail phase retrieval, (b) of Theorem 7 must hold at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {ei}n i=1 be the canonical orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If J ⊂ [n], we define PJ as the projection onto span{ei}i∈J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {xi}m i=1 be unit vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The following are equivalent: (1) Whenever I ⊂ [2n − 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic, there is no j ∈ [n] so that ⟨x, ej⟩ = 0 = ⟨y, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (2) For every J ⊂ [n] with |J| = n − 1, {Pjxi}2n−2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (3) For every J ⊂ [n] with |J| < n, {PJxi}2n−2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' WEAK PHASE RETRIEVAL 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (1) ⇒ (2): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So assume (2) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then choose J ⊂ [n] with |J| = n − 1, J = [n] \\ {j}, and {PJxi}2n−2 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' In particular, it fails complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' That is, there exists I ⊂ [2n− 2] and span {PJxi}i∈I ̸= PJRn and span {Pjxi}i∈Ic ̸= PJRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So there exists norm one vectors x, y in PJRn with PJx = x ⊥ PJxi for all i ∈ I and PJy = y ⊥ PJxi for all i ∈ Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Extend x, y to all of Rn by setting x(j) = y(j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Hence, x ⊥ xi for i ∈ I and y ⊥ xi for i ∈ Ic, proving (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (2) ⇒ (3): This follows from the fact that every projection of a set of vectors doing phase retrieval onto a subset of the basis also does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (3) ⇒ (2): This is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' (3) ⇒ (1): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So assume (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then there is a I ⊂ [2n− 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic and a j ∈ [n] so that ⟨x, ej⟩ = ⟨y, ej⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' It follows that x = PJx, y = PJy are non zero and x ⊥ Pjxi for all i ∈ I and y ⊥ Pjxi for i ∈ Ic, so {PJxi}2n−2 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' The assumptions in the theorem are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' That is, in general, {xi}m i=1 can do weak phase retrieval and {PJxi}m i=1 may fail phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For example, in R3 consider the row vectors {xi}4 i=1 of: \uf8ee \uf8ef\uf8ef\uf8f0 1 1 1 −1 1 1 1 −1 1 1 1 −1 \uf8f9 \uf8fa\uf8fa\uf8fb This set does weak phase retrieval, but if J = {2, 3} then x = (0, 1, −1) ⊥ PJxi for i = 1, 2 and y = (0, 1, 1) ⊥ xi for i = 3, 4 and {PJxi}4 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Assume {xi}2n−2 i=1 does weak phase retrieval in Rn and for every J ⊂ [n] {PJxi}2n−2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then if x, y ∈ Rn and |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' , 2n − 2, then there is a J ⊂ [n] so that x(j) = � aj ̸= 0 for j ∈ J 0 for j ∈ Jc y(j) = � 0 for j ∈ J bj ̸= 0 for j ∈ Jc Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let {ei}n i=1 be the unit vector basis of Rn and for I ⊂ [n], let PI be the projection onto XI = span{ei}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For every m ≥ 1, there are vectors {xi}m i=1 so that for every I ⊂ [1, n], {PIxi}m i=1 is full spark in XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We do this by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' For m=1, let x1 = (1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' This satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' So assume the theorem holds for {xi}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Choose I ⊂ [1, n] with |I| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Choose J ⊂ I with |J| = k − 1 and let XJ = span{xi}i∈J ∪ {xi}i∈Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Then XJ is a hyperplane in Rn for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since there only exist finitely many such J′s there is a vector xm+1 /∈ XJ for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' We will show that {xi}m+1 i=1 satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Let I ⊂ [1, n] and J ⊂ I with |J| = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' If PIxm+1 /∈ XJ, then {PIxi}i∈J is linearly independent by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' On the other hand, if m + 1 ∈ J then xm+1 /∈ XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' But, if PIxm+1 ∈ span{PIxi}i∈J\\m+1, since (I − PI)xm+1 ∈ span{ei}i∈Ic, it follows that xm+1 ∈ XJ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' □ 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' AKRAMI Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' In the above proposition, none of the xi can have a zero coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Since if it does, projecting the vectors onto that coordinate produces a zero vector and so is not full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content=' Akrami, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='akrami@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'} diff --git a/4dE4T4oBgHgl3EQf0w2N/content/2301.05285v1.pdf b/4dE4T4oBgHgl3EQf0w2N/content/2301.05285v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..87641e86ae05518a87aa8388517abc409632f894 --- /dev/null +++ b/4dE4T4oBgHgl3EQf0w2N/content/2301.05285v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4e3ab6d442bd90145c5eb19d9df1aabcad3b31f127cc327329a88bd27b1e33b0 +size 10347457 diff --git a/4tAyT4oBgHgl3EQfpPhP/content/2301.00521v1.pdf b/4tAyT4oBgHgl3EQfpPhP/content/2301.00521v1.pdf new file mode 100644 index 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APOLIN´ARIO, GEOFFREY BECK, LAURENT CHEVILLARD, ISABELLE GALLAGHER, +AND RICARDO GRANDE +Abstract. Turbulent cascades characterize the transfer of energy injected by a random force at +large scales towards the small scales. In hydrodynamic turbulence, when the Reynolds number +is large, the velocity field of the fluid becomes irregular and the rate of energy dissipation re- +mains bounded from below even if the fluid viscosity tends to zero. A mathematical description +of the turbulent cascade is a very active research topic since the pioneering work of Kolmogorov +in hydrodynamic turbulence and that of Zakharov in wave turbulence. In both cases, these turbu- +lent cascade mechanisms imply power-law behaviors of several statistical quantities such as power +spectral densities. For a long time, these cascades were believed to be associated with nonlinear +interactions, but recent works have shown that they can also take place in a dynamics governed by +a linear equation with a differential operator of degree 0. In this spirit, we construct a linear equa- +tion that mimics the phenomenology of energy cascades when the external force is a statistically +homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen +as a linear transport equation, which corresponds to an operator of degree 0 in physical space. Our +results give a complete characterization of the solution: it is smooth at any finite time, and, up to +smaller order corrections, it converges to a fractional Gaussian field at infinite time. +1. Introduction +1.1. Background and motivation. This work is mainly motivated by some important aspects +of the phenomenology of three-dimensional homogenous and isotropic fluid turbulence [34, 42, 20], +of which several aspects have been also observed and formalized for waves in various situations +when they are weakly interacting [43, 35]. As has been repeatedly observed in geophysical and +laboratory flows, and in numerical simulations of the incompressible Navier-Stokes equations, a +fluid that is stirred by a statistically stationary random force f(t, x), assumed to be smooth in +space, will eventually reach a statistically stationary state in which the velocity variance is finite. +To dissipate all the energy that is constantly injected into the system in such an efficient way, the +velocity field of that fluid will develop a complex multiscale structure ending up with high values +of velocity gradients such that viscosity can easily transform mechanical energy into heat. In other +words, the fluid has transferred the energy pumped at large scales by the forcing towards small +scales, at which viscous diffusion efficiently acts. This picture is known as the cascading process of +energy. +The purpose of this article is to model and reproduce this phenomenon of transfer of energy +as a cascading process through the scales. We propose a partial differential equation, which is +of course much simpler than the nonlinear Navier-Stokes equations, stochastically forced by an +additive random force f(t, x) that we take to be smooth in space and correlated over a typical large +lengthscale (known in the turbulence literature as the integral lengthscale), whose solution develops +roughness as time goes on. More precisely, our goal is to generate rough fractional Gaussian H¨older +continuous random fields of parameter H (see for instance the textbook [16]) from smooth forcing +through a dynamical evolution, which can be seen as a simple stochastic representation of the +phenomenology mainly developed by Kolmogorov [24]. +1 +arXiv:2301.00780v1 [math-ph] 2 Jan 2023 + +2 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +As mentioned earlier, a striking feature of three-dimensional turbulent motion is its ability to +efficiently dissipate the energy that is injected at large scales in a statistically stationary and homo- +geneous manner. To be more precise, let us consider a solution of the incompressible Navier-Stokes +equation, i.e. a divergence-free velocity field u(t, x) ∈ R3 with periodic boundary conditions. This +dynamics is stirred by a divergence-free vector forcing term f(t, x), that we take delta-correlated +in time and smooth in space, say Gaussian, of zero-average and of covariance +E [f(t, x) ⊗ f(s, y)] = δt−sCf(x − y), +where ⊗ stands for the matrix product, and the matrix Cf(x) is made of a linear combination of the +matrix x ⊗ x and the identity, with multiplicative coefficients depending only on |x|, i.e. a typical +covariance matrix of a statistically homogeneous and isotropic vector field [6, 36]. We furthermore +require that these scalar functions of |x| are smooth and compactly supported over a range of the +size order of the aforementioned large length scale, so as to mimic the energy injection at the so- +called integral length scale. As time goes on, it has been repeatedly observed that the velocity field +u reaches a statistically stationary state, which is furthermore statistically homogeneous, of finite +variance, and with the additional striking property that it becomes independent of the viscosity ν +as ν goes to zero, i.e. +(1.1) +lim +ν→0 lim +t→∞ E +� +|u(t, x)|2� +< +∞ +for all x. +The former asymptotic behavior of the velocity variance illustrates clearly how a turbulent fluid +can dissipate energy with high efficiency. For instance, in the same setup but considering the heat +equation instead of the Navier-Stokes equations, a statistically stationary regime would also be +reached at t → ∞. However, the variance of the solution is then inversely proportional to the +viscosity ν, see [14]. Instead, turbulent motion dissipates energy in a way that the velocity variance +is eventually independent of viscosity, which is a far more efficient way of dissipating energy. In +order to ensure the independence of said variance on viscosity, (1.1), the fluid develops a rough +behavior of H¨older-type at small scales, in such a way that the variance of the velocity increments +asymptotically behaves as follows: +(1.2) +lim +ν→0 lim +t→∞ E +� +|u(t, x + ℓ) − u(t, x)|2� +∝ +|ℓ|→0 |ℓ|2H +for all x, +where the power-law exponent is determined by Kolmogorov’s prediction H ≈ 1/3 [20]. Much +more could be said on a more precise characterization of the distribution of the increments than +only its variance, (1.2), such as its higher order moments that quantify its non Gaussian, skewed +and intermittent nature [20]. In this article, we will focus on a second-order modeling of these +fluctuations, (1.1) and (1.2); we leave finer descriptions for future research. +A first precise formalization of the cascade phenomenon could be built by imposing a particular +dynamical relation between the coefficients of a decomposition of the velocity field, such as a +continuous wavelet transform or a discrete (dyadic) decomposition on a tree [18]. This has been +explored in the literature [5, 4, 13] leading to precise statements on H¨older regularity and its +relationship with scaling behaviors of the coefficients. Although great progress has been made in +the understanding of such models and their formalization, which usually exploits a typical quadratic +interaction between neighboring coefficients, these approaches often avoid the important question +of the relation of these coefficients in space. This is necessary in order to design a model that leads +to statistically homogeneous velocity fields, as observed in nature and in numerical simulations. +Nonetheless, these models can be seen as a sophistication of the so-called shell models1. In this +spirit, we believe an important step was made in [32], where the authors investigate a simple linear +1See for instance [10, 9] which consist in exploring quadratic interactions between shells, that share some behaviors +with velocity Fourier modes and wavelet coefficients, along a single branch of a tree decomposition, lacking thus a +discussion of the spatial relationships between coefficients. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +3 +relation between shells, which is shown to be able to transfer energy from large to small scales. Let +us also mention [30] where some ideas to build a PDE from these shell models are proposed. +From a somewhat different side of fluid mechanics, more focused on the implications of global +rotation [40, 39] or stratification of the density field [29, 41] on a flow, it has been evidenced +a phenomenon of focusing of waves onto attractors, whose precise shape are determined by the +boundaries. Based on a linearization of the fluid equations, this phenomenon has been interpreted +as a cascading process through scales. These ideas have been then formalized and rigorously studied +from a mathematical viewpoint in a series of recent articles [17, 19], which underline the importance +of operators of degree 0 as a deterministic mechanism able to transfer energy through scales. +1.2. Main results. The rough and disordered nature of a turbulent velocity field u(t, x) has been +repeatedly observed in laboratory and numerical flows, and in geophysical situations [42, 20]. From +this signal, considering for instance a component of the velocity vector field as a function of space, +depending on the experimental possibilities and the large-scale geometry of the flows, one can +construct the energy spectrum |k| �→ E|�u(t, k)|2 where �u stands for the spatial Fourier transform. +According to the standard phenomenology of fluid turbulence, which has been multiply confirmed +by observations in very different situations, the energy-spectrum resembles a curve [42, 20] that +can be schematically decomposed as follows: +• (injection range) for small |k| of the order of the characteristic wavelength of energy injec- +tion, the energy-spectrum is mainly determined by the forcing and the associated large-scale +geometry of the flow, +• (inertial range) for intermediate |k|, the energy-spectrum develops a power-law behavior +whose exponent is found universal, i.e. independent of viscosity and of the nature of the +flow, and can be interpreted as the generation of small scales by the internal motion of the +fluid following a transfer of energy from small wave-numbers to large wave-numbers, +• (dissipative range) for large |k|, the energy-spectrum is governed by dissipation processes +which damp efficiently all the energy coming from the large scales, making the spatial +velocity profile a smooth function. +The intermediate range of scales, called the inertial range in the turbulence literature [42, 20], +is where this mechanism of transport of energy takes place. The universally observed power-law +exponent of the energy-spectrum can be written as −(2H + d), i.e. E|�u(t, k)|2 ∼ |k|−(2H+d), where +we have introduced for the sake of generality the space dimension d, and the parameter H that will +be eventually interpreted as a Hurst, or H¨older, exponent, in a statistically averaged sense. In real +situations, for d = 3, it is indeed observed that H ≈ 1/3, as predicted by dimensional arguments +mainly attributed to Kolmogorov [24, 27]. +The main goal of this paper is to propose a family of partial differential equations, such that, +when stirred by a statistically stationary and spatially homogenous, smooth in space forcing term, +its solution u(t, x) reaches at long times a statistically stationary state which displays the typical +spectral behavior detailed above. We will achieve this with the following transport equation in +Fourier space: +(1.3) +� +� +� +� +� +� +� +� +� +∂t�u(t, k) + divk +� +ck +|k| �u(t, k) +� ++ cH + 1 +2 +|k| +�u(t, k) = �f(t, k) +t > 0, k ∈ Rd, |k| > κ > 0, +�u(t, k) = 0 +t > 0, k ∈ Rd, |k| ≤ κ, +�u(0, k) = 0. +Here H ∈ R and κ > 0 are fixed, and the source f satisfies +E[f(t, x)f(s, y)] = δt−s Cf(x − y), + +4 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +where Cf is smooth and satisfies some additional assumptions detailed below. Our main result is +the following: +Theorem 1.1. Let H ∈ (0, 1) and let the forcing f be +(1.4) +f(t, x) = +� +Rdy +ϕ(x − y) dW(t, y), +where dW is a space-time Gaussian real white noise and ϕ ∈ S(Rd +x) is a radial function such +that �ϕ(k) = 0 for all |k| < κ. +(i) The transport equation in wavenumber space (1.3) with source (1.4) can be rigorously formu- +lated in physical space as an a.s. well-posed PDE. Moreover, at any t > 0, the solution u(t, x) +has finite variance and a.s. smooth paths with respect to x. +(ii) As t → ∞, u(t, x) converges to a zero-mean Gaussian field u∞(x) which has a.s. α-H¨older +continuous paths for any 0 < α < H. +(iii) The correlations are given by +E[u∞(x1)u∞(x2)] = C(d, H) KH(x1 − x2) − (KH ∗ JH)(x1 − x2), +where +KH := F−1 � +χ|k|>κ|k|−(2H+d)� +, +while C(d, H) is an explicit constant and the function JH ∈ S(Rd +x) depends explicitly on ϕ +in (1.4). +A more detailed version of this result is presented in Theorem 4.4 page 23. +Remark 1.2. The parameter κ can be chosen as the smallest non-vanishing wavenumber in the +support of the Fourier transform of the forcing. +Following the analogy with the Navier-Stokes +equations presented in the introduction, κ may be interpreted as a quantity linked to the inverse of +the integral lengthscale2, i.e the typical lengthscale of the correlations of the forcing. The fact that +our force acts at large but finite scales means that κ is small but non-zero. The kernel KH is a +function when H ∈ (0, 1) and κ > 0. However, when κ = 0 we have the following operator: +KH −→ +κ→0 (−∆)− +H+ d +2 +2 +. +Remark 1.3. Note that the limiting Gaussian field u∞ shares some statistical properties, such as +roughness, with statistical homogeneous fractional gaussian fields [16, 28] defined by +(−∆)− +H+ d +2 +2 +dW, +that are classically encountered in the turbulence literature [25, 12, 15]. Indeed, for H ∈ (0, 1) both +have a.s. α-H¨older continuous paths for any 0 < α < H and one can show that +(1.5) +u∞ += +(in law) C(d, H)F−1 � +χ|k|>κ +� +∗ (−∆)− +H+ d +2 +2 +dW − ureg. +Here, ureg is a smooth zero-mean Gaussian field with correlations +E[ureg(x1)ureg(x2)] = (KH ∗ JH)(x1 − x2), +2If Lf is the integral lengthscale, then there exists two real positive numbers a < b such that the support of �f is +contained in the annulus of inner radius +a +Lf and outer radius +b +Lf . Thus one may set κ = +a +Lf . + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +5 +which is a smooth function with respect to (x1 − x2) even if H ≤ 0. The case H ∈ [−d/2, 0] will be +discussed in Section 4. +Remark 1.4. When H ∈ [−d/2, 0], as t → ∞, u(t) still converges to a zero-mean Gaussian field +u∞, but this field is not necessarily H¨older continuous with respect to x anymore. In this case, +one may view u∞ as a distribution living in the dual of an appropriate test function space T , see +Theorem 4.5 for more details. The correlation structure of the limiting Gaussian measure is given +by: +(1.6) +E[ lim +t→∞⟨u(t), g1⟩⟨u(t), g2⟩] = +� +Rd +k +χ|k|>κ |k|−(2H+d) � +C(d, H) − � +JH(k) +� +�g1(k) �g2(k) dk +for any test functions g1, g2 ∈ T , where ⟨·, ·⟩ stands for the duality product in T . In Theorem 4.5, +we show that, for any test functions in T ∩S(Rx +d), (1.6) yields a rate of convergence proportional to +(ct)−(2H+d+2n) for n as large as desired. The expression (1.6) corresponds to the energy-spectrum +picture described above. Indeed, |k|−(2H+d) corresponds precisely to the inertial range previously +described, while +� +JH(k) captures the contribution from the forcing, which is a correction in the +injection range. In fact, if the source is spectrally supported in small wavenumbers, then � +JH(k) +vanishes in the inertial range. +Finally, let us highlight the difference between the properties of the solution at finite and infinite +time. At finite time, the solution is smooth with respect to x, whereas at infinite time the solution +is only H¨older continuous (or even rougher if H ≤ 0, as explained in Remark 1.4). This loss of +regularity at infinite time is what is expected in linear turbulence. Turbulence is usually associated +to a nonlinear equation. For example, in the case of wave turbulence, nonlinearities create wave +interactions which allow the transfer of energy to higher and higher wavenumbers. Such transfers +of energy typically result in a loss of regularity. However, nonlinearities might not be the only way +in which such loss of regularity can occur. Indeed, Y. Colin de Verdi`ere and L. Saint-Raymond have +shown that, in the context of internal waves, a loss of regularity can also take place in the case of a +linear equation with an operator of degree 0. Linear equations with operators of degree 0 are also +common whenever one introduces a dispersive perturbation in a hyperbolic system. In such cases, +these operators of degree 0 are used to model wave propagation under strong dispersive effects and +they are responsible for memory effects. For example, in the context of wave-energies, the second +author and D. Lannes show that the waves generated by a moving floating object are governed in +the linear regime by a non-local transport equation of degree 0, see [8]. In the context of electrical +circuits, there are cases in which 1D models of electromagnetic waves propagating along a coaxial +cable are governed by operators of degree 0, see for instance [7, Chapter 5]. +One issue of our model is that it only features a single H¨older exponent. The velocity field +of a concrete turbulent fluid consists of many H¨older exponents, i.e. the H¨older-regularity of the +velocity field u(t, x) around a point x ∈ Rd depends on the point itself. This is known as the +multifractal formalism [14, 16]. The term multifractal refers to the fact that the sets of points +with same regularity are often fractal. Moreover, our model does not capture finer descriptions +(beyond the variance) of the distribution of the increments of the velocity field. Such descriptions +should quantify its non-Gaussian and intermittent nature [15], and therefore our linear model does +not suffice. +It is known that one can construct a multifractal and intermittent field with the +theory of Gaussian multiplicative chaos [38], however our actual goal is to obtain a multifractal and +intermittent field dynamically, i.e. as the solution to a non-linear equation forced by a white-noise +in time that admits a rigorous mathematical treatment. We can also consider a forcing which is +not a white-noise in time whose temporal correlation function is given by an oscillating function + +6 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +in order to make a comparison with [17, 21, 1] and [11]. Finally, we could investigate other linear +models of cascade such as in the case of a compact operator plus a potential of degree 0 as in [31]. +These issues will be tackled in future papers. +1.3. Outline. The article is organized as follows. +In Section 2, we present a simple transport +equation that converges to a complex white noise (up to lower order terms). A small tweak to +this model allows us to construct a model that gives rise to a real white noise. In Section 3, we +explain how to generalize the latter model to higher dimensions and give a heuristic proof of the +main results in this paper. In Section 4, we provide a mathematically rigorous study of our model: +we introduce the right functional setting, we develop a global well-posedness theory and give a +complete description of the asymptotic behavior of the solution, as well as its properties. This +constitutes the proof of Theorem 1.1. Finally, in Section 5 we propose a numerical method and +conduct numerical simulations in dimensions 1, 2 and 3 to validate our model. +1.4. Acknowledgements. We thank Oliver B¨uhler for his interesting suggestion about replacing +the pure transport term ∂k in (2.1) by sign(k)∂k as a way of fixing the physically undesirable +behavior of the solution. All five authors were funded by the Simons Collaboration Grant on Wave +Turbulence, Simons Award ID: 651475 and 651675. +1.5. Notation. For an integrable function f : Rd → C, we denote by �f its Fourier transform, +namely +∀k ∈ Rd +k, +�f(k) := Ff(k) := +� +Rdx +e−2πix·k f(x) dx. +Whenever defined, the inverse Fourier transform is +∀x ∈ Rd +x, +f(x) = F−1 �f(x) = +� +Rd +k +e2πix·k �f(k) dk. +It is well know that the Fourier transform is an isometry from L2(Rd +x) to L2(Rd +k), from S(Rd +x) +to S(Rd +k), where S(Rd +x) denote the space of Schwartz functions (i.e. smooth functions whose +derivatives are rapidly decreasing), and from S′(Rd +x) to S′(Rd +k) where S′(Rd +x) denote the space of +tempered distribution (i.e. the dual space of S(Rd +x)). We will denote by ⟨·, ·⟩ the duality product +between S′ and S. +We will also need some spaces that quantify the regularity of functions more precisely. For a +fixed integer n, Sobolev spaces are defined by +Hn(Rd) := {u ∈ L2(Rd) | ∂j +xiu ∈ L2(Rd) with 1 ≤ i ≤ d and 0 ≤ j ≤ n}, +and their dual spaces are denoted by H−n(Rd). We will denote by ⟨·, ·⟩H−n,Hn the duality product +between Hn and H−n. For α ∈ (0, 1) the H¨older space C0,α(Rd) is defined by +C0,α(Rd) := {u continuous and bounded | ∃C > 0, ∀x, ℓ ∈ Rd, |ℓ| ≤ 1, |δℓu(x)| ≤ C|ℓ|α}, +where δℓ denotes the increment defined by +δℓu(x) := u(x + ℓ) − u(x) +for x, ℓ ∈ Rd. +We will denote by χA the characteristic function3 of the set A. +Let (Ω, σ(Ω), P) be a probability space. A Gaussian field u : Rd → L2(Ω) is a field such +that for all n ≥ 1 and for all (x1, x2, · · · , xn) ∈ (Rd)n, the random vector (u(x1), u(x2), · · · , u(xn)) +is a Gaussian random vector. +When d = 1, a 1D Gaussian field is usually called a Gaussian +process. A Gaussian random measure µ acting on S(Rd) is a random tempered distribution +3This means that χA(k) = 1 if k ∈ A and χA(k) = 0 if k /∈ A. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +7 +such that for every g ∈ S(Rd), the random variable ⟨µ, g⟩ is a centered Gaussian. A white noise +dW(x) is a Gaussian random measure acting on L2(Rd) that satisfies the following for any functions +f, g ∈ L2(Rd) +E +��� +Rd f(x)dW(x) +� �� +Rd g(x)dW(x) +�� += +� +Rd f(x)g(x)dx. +Since we always integrate a deterministic function against dW(x), the choice between Itˆo and +Stratonovich integrals is unimportant. +Sometimes we will change variables when integrating against a white noise measure: +� t +0 +f(t − s)dW(s) = +� t +0 +f(s)d� +W(s). +In such cases, we will use the d� +W to denote the new white noise measure after this change of +variables. +2. One-dimensional transport in wavenumber space +2.1. Building one-dimensional white noise: real vs complex. In order to mimic the trans- +port of energy from large scales to small scales, the authors in [3, 2] proposed a simple transport +equation in Fourier space. To present these ideas, we first consider a one-dimensional model for a +velocity field u(t, x), whose spatial Fourier transform aims to solve the linear evolution +(2.1) +� +∂t�u(t, k) + c ∂k�u(t, k) = �f(t, k), +(t, k) ∈ (0, ∞) × R, +�u(t, k)|t=0 = 0. +Here c > 0 is fixed and can be viewed as a transport rate in wavenumber space. On the right-hand +side of (2.1), we have included an additive term �f which is the Fourier transform of a spatial forcing +term. The support of �f is localized at small wavenumbers, which is consistent with the assumption +that the forcing term in physical space acts at large scales. +As we can see, the dynamical evolution proposed in (2.1) is a genuine transport equation, and +only the presence of a forcing makes it inhomogeneous. Adopting such a setup immediately imposes +the complex nature of the velocity field in physical space, as it can be seen when formally taking +the inverse Fourier transform of (2.1), and obtaining the following evolution in physical space: +(2.2) +� +∂tu(t, x) − 2πicx u(t, x) = f(t, x), +(t, x) ∈ (0, ∞) × R, +u(t, x)|t=0 = 0. +Note that the operator in (2.2) corresponds to multiplication by the space variable 2πicx. In [3], it +is shown that when the forcing f is a white noise in time and statistically homogeneous in space, +then the solution to (2.2), u : (0, ∞)t × Rx −→ C, converges4 to a complex white noise in space +as t → ∞. In other words, the evolution that has been proposed, expressed in Fourier space as a +transport equation (2.1) and in physical space as an equation (2.2) involving an operator of degree +0 (multiplication by −2πicx), is able to transfer energy through scales. Moreover, the solution is +statistically homogeneous at any time and it develops the regularity of a white noise as time goes +on (technically it’s a sudden drop in regularity at t = ∞). To complete the program suggested by +the phenomenology of turbulence, the additional linear action of a fractional operator allows, in a +similar setup, to generate a solution with asymptotic H¨older-type regularity of parameter H ∈ (0, 1) +instead of the one of the white noise, as explained in [3]. +If some energy is introduced by the forcing at a negative wavelength k < 0, the transport +equation (2.1) will move it to smaller negative wavenumbers, going through k = 0, and then to +4Up to lower order terms, see Theorem 2.4 for a full asymptotic expansion. + +8 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +infinitely large positive wavenumbers k → ∞. In order to avoid this pathological behavior, one +could replace ∂k by ∂|k| = sign(k)∂k which leads to a transport in the direction of |k| instead of k. +Note however that sign(k)∂k is not properly defined at k = 0, and so one needs to be careful in +order to propose a well-posed mathematical problem. With this in mind, the heart of this article +will be the theoretical and numerical study of the following formal evolution: +(2.3) +� +� +� +� +� +∂t�u(t, k) + c ∂|k|�u(t, k) = �f(t, k), +(t, k) ∈ (0, ∞) × R, +�u(t, k)|t=0 = 0, +�u(t, k)||k|=0 = 0. +Note that it is necessary to introduce a transmission condition between negative and positive k. +In (2.3), we have decided to add the boundary condition �u(t, k)||k|=0 = 0 to decouple negative from +positive wavenumbers, so that no energy crosses k = 0. In particular, this means that the integral +over space of u is zero for all times. This new dynamics proposed in (2.3) can be written in physical +space after formally applying the inverse Fourier transform: +(2.4) +� +� +� +� +� +� +� +∂tu + 2πc xH(u) = f, +(t, x) ∈ (0, ∞) × R, +u|t=0 = 0, +� +Rx +u dx = 0, +where H denotes the Hilbert transform defined in the usual way: +(2.5) +Hf(x) := 1 +πp.v. +� +Ry +f(y) +x − y dy = − 1 +π lim +κ→0+ +� ∞ +κ +f(x + y) − f(x − y) +y +dy. +Notice that we have used the fact the integral of u(t, x) over space vanishes to get the expression +of (2.4). +Notice also that, despite the fact that the spectral evolutions (2.1) and (2.3) (whose +equivalent expressions in physical space are provided respectively in (2.2) and (2.4)), look very +similar, the solution to the new dynamics (2.4) is now real-valued. Equivalently, the dynamics +in Fourier space (2.3) conserves the Hermitian symmetry of an appropriate initial condition, here +assumed to be zero. +Moreover, the solution u : (0, ∞)t × Rx → R of (2.4) can be shown to +asymptotically converge to a real white noise in space. +As we will explain in the sequel, the +additional linear action of a fractional operator will allow the generation, from smooth forcing, of +a real fractional Gaussian field. +Finally, it is tempting to generalize (2.3) to higher dimensions by replacing ∂|k| by +k +|k| · ∇k. As +we will develop in Section 4, this eventually generates a statistically homogeneous and isotropic +solution that will converge to a real d-dimensional Gaussian random measure which is rougher than +a white noise (in space) whenever d > 1. As we will explain in the sequel, the additional linear +action of a fractional operator will us to generate a real d-dimensional Gaussian random measure +with the desired H¨older regularity. +Interestingly, it is not obvious to generalize (2.1) to space +dimension d ≥ 1 which would generate a similar statistically homogeneous and isotropic solution +in physical space. We provide at the end of the section some additional discussions on this matter. +However, for the time being we focus on developing a good understanding in the one-dimensional +setting. In the case of (2.4), we have the following result: +Theorem 2.1. Let the forcing f in (2.4) be +(2.6) +f(t, x) = +� +Ry +ϕ(x − y) dW(t, y), +where dW is a space-time Gaussian real white noise, and ϕ ∈ S(R) is a non-negative, non- +identically null, even function with null average. Then: + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +9 +(i) Equation (2.4) admits a global (in time) solution u(t, x), which is a Gaussian process with +a.s. smooth paths in x, and with α-H¨older continuous paths in t for any 0 < α < 1/2. +(ii) As t → ∞, the solution u(t) converges in S′(R) to a random Gaussian measure u∞ acting on +S(R) with zero-mean, i.e. for any g ∈ S(R), E[⟨u∞, g⟩] = 0. +(iii) We have the following asymptotic behavior: +E[⟨u∞, g1⟩⟨u∞, g2⟩] = lim +t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩] += C +� +Rx +g1(x) g2(x) dx − +� +Rx×Ry +I(x − y) g1(x) g2(y) dx dy. +(2.7) +for any g1, g2 ∈ S(R). Here C > 0 is a constant and I is an explicit continuous, even function +that depends on ϕ in (2.6). +A more detailed version of this result is presented in Theorem 4.5, see also Section 4.4. Note +that the first term on the right-hand side of (2.7) corresponds to a delta function, while the second +term given by I is a smooth lower order term. +Remark 2.2. The forcing introduced in (2.6) is indeed a Gaussian white noise in time and statis- +tically homogeneous in space, i.e. +(2.8) +E[f(s, x)f(t, y)] = Cf(x − y) δs−t , +where the spatial correlation function Cf = ϕ∗ϕ is a convolution. Given that ϕ ∈ S(R), Cf ∈ S(R). +The constant C in (2.7) is precisely +C = Cf(0) +2c += 1 +2c +� +Rx +|ϕ(x)|2 dx > 0. +Remark 2.3. The solution to the stochastic PDE (2.4) is an explicit Gaussian Itˆo process. A +precise formula will be given in Section 4. Even if this solution is continuous in time and space, +one can lose regularity at t = +∞, which is why one needs to consider u∞ on the left-hand side of +(2.7) as a distribution. +The proof of this theorem is posponed to the next section where a more general case, i.e. +multidimesional white noise, will be tackled. Before we develop the techniques needed to prove +Theorem 2.1, it is important to understand the asymptotic behavior of solutions to (2.2) in the +complex setting, which is less technical and informative. +In this setting, we have the following result: +Theorem 2.4. Let the forcing f in (2.2) be +(2.9) +f(t, x) = +� +Ry +ϕ(x − y) dW(t, y), +where dW is a space-time Gaussian complex white noise, and ϕ ∈ S(R) is a complex, non-identically +null, even function. Then: +(i) Equation (2.2) admits a global (in time) solution u(t, x), which is a Gaussian process with +a.s. continuous paths in time and space. +(ii) As t → ∞, the solution u(t) converges (in S′(Rd)) to a random Gaussian measure u∞ acting +on S(Rd). + +10 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +(iii) We have the following asymptotic behavior (in the sense of distributions). +For any g1, g2 ∈ S(R), +E[⟨u∞, g1⟩⟨u∞, g2⟩] = lim +t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩] += 1 +2c Cf(0) +� +Rz +g1(z)g2(z)dz ++ +1 +2πic p.v. +� +Rz +Cf(z) +z +�� +Ry +g1(z + y)g2(y)dy +� +dz . +(2.10) +The function Cf = ϕ ∗ ϕ is the spatial correlation function given by +(2.11) +E[f(s, x)f(t, y)] = δs−t Cf(x − y), +and p.v. Cf(z) +z +is the principal value of the distribution Cf(z)/z. That is, for any test function +g ∈ S(R), +(2.12) +� +p.v.Cf(z) +z +, g +� +:= p.v. +� +R +Cf(z)g(z) +z +dz = +� ∞ +0 +Cf(z) g(z) − g(−z) +z +dz. +As we mentioned in Remark 2.2, the forcing in (2.9) is a complex Gaussian white noise in time +and statistically homogeneous in space. Moreover, the solution admits an explicit formula: +(2.13) +u(t, x) = +� t +0 +� +Ry +e2πicx(t−s) ϕ(x − y) dW(s, y). +The same comments as in Remark 2.3 apply in this case. +Remark 2.5. The asymptotic expansion (2.10) remains valid when testing against functions with +a finite number of derivatives. Indeed, u∞ in (2.10) can also be interpreted as a Gaussian random +measure in H−n(R) for any integer n ≥ 2. More precisely, we will show that for any test functions +g1, g2 ∈ Hn(R) one gets +E[⟨u(t), g1⟩H−n,Hn⟨u(t), g2⟩H−n,Hn] +∼ +t→∞ +Cf(0) +2c +� +R +g1(z)g2(z) dz ++ +1 +2πic p.v. +� +Rz +Cf(z) +z +�� +Ry +g1(z + y)g2(y)dy +� +dz ++ d(n) ∥g1∥Hn ∥g2∥Hn +� 1 +ct +�n−1 +. +(2.14) +where d(n) depends only on n. This characterizes the rate of convergence. +Remark 2.6. One interpretation of (2.7) (resp. (2.10)) is that the correlation function (resp. the +real part of it) asymptotically behaves like a white noise. However, this theorem gives a lower-order +correction, in the sense that the regularity of the correction is higher than that of the white noise. +In (2.7), such a regular correction is given by a Schwartz function, whereas in (2.10), the regular +correction is a purely imaginary principal value which has no singularity at zero: by (2.12), the +principal value is “controlled” by C′ +f(0) near zero. It is important to note that in both cases the +regular correction is fast-decaying. +Remark 2.7. Note that we recover the result in proposition 2.1 in [3], i.e. +(2.15) +lim +t→∞ E[u(t, x)u(t, y)] = 1 +2c Cf(0)δx−y + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +11 +as long as one only tests against even functions with respect to the variable x − y, as is easily seen +from the right-hand side of (2.12). +One way to recover a result similar to that in [3] that holds for all test functions is to define the +function v(t, x) = e−πictxu(t, x). This function now satisfies +E[v(t, x)v(t, y)] = t sinc (ct(x − y)) Cf(x − y) +where sinc(x) := sin(πx) +πx +denotes the normalized sinc function. It immediately follows that +(2.16) +lim +t→∞ E[v(t, x)v(t, y)] = Cf(0) +c +δx−y. +However, it is unclear whether this transformation of u is an interesting object from the physical +viewpoint. Assuming one can take the Fourier transform, (2.16) can be rewritten in wavenumber +space as +(2.17) +lim +t→∞ E[�v(t, k)�v(t, k′)] = lim +t→∞ E +� +�u (t, k + πct) �u (t, k′ + πct) +� += δk−k′ Cf(0). +The transformation given by v is therefore equivalent to computing the correlation between the k+πct +and k′ + πct Fourier modes as t → ∞. +2.2. Proof of Theorem 2.4. First of all, note that equation (2.2) admits the explicit solu- +tion (2.13) thanks to the Duhamel formula. +Step 1: The solution u is a well defined Gaussian field whose limit at t → ∞ is a Gaussian +random measure. +Clearly, u(t, x) is a well defined Itˆo process with zero average and variance +(2.18) +E[|u(t, x)|2] = +� t +0 +� +Ry +|ϕ(x − y)|2 dyds = t ∥ϕ∥2 +L2 . +For any test function g (we will soon see that actually g ∈ H1(R) suffices), one gets +⟨u(t), g⟩ = +� t +0 +� +Ry +�� +Rx +e2πicx(t−s) ϕ(x − y) g(x) dx +� +dW(s, y). +Since Brownian motion has independent, stationary increments we can rewrite the above equation +as +⟨u(t), g⟩ = +� t +0 +� +Ry +�� +Rx +e−2πicxs ϕ(x − y) g(x) dx +� +d� +W(s, y) +where d� +W(s, y) is another Gaussian white noise. As t → ∞, we find that +(2.19) +lim +t→∞⟨u(t), g⟩ = +� ∞ +0 +G(s, y) d� +W(s, y) +where +G(s, y) = +� +Rx +e−2πicxs ϕ(x − y) g(x) dx. +This limit is justified only if G ∈ L2(R+ +s × Ry), which we set out to prove next. Firstly, the Young +convolution inequality immediately yields +∥G∥L2([0,1]s×Ry) ≤ ∥ϕ∥L1 ∥g∥L2 . +Next, to handle the case |s| ≥ 1, we assume that g ∈ H1(R) and we integrate by parts: +2πics G(s, y) = − +� +Rx +e−2πicxs ∂x +� +ϕ(x − y) g(x) +� +dx. + +12 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +By the Young convolution inequality, +∥2πicsG(s, y)∥L2y ≤ +��g′�� +L2 ∥ϕ∥L1 + ∥g∥L2 +��ϕ′�� +L1 . +Thus one easily finds that: +∥G(s, y)∥L2(R+ +s ×Ry) ≲ 1 +c ∥g∥H1 ∥ϕ∥W 1,1 , +which justifies (2.19). +Step 2: Exchanging expectation and limit as t → ∞. +One could directly use the right-hand side of (2.19) in order to compute the correlations of the +limiting Gaussian measure. However, this doesn’t allow us to quantify the speed of convergence +as t → ∞, so we take a slightly different approach. +We would like to show that +E[ lim +t→∞ ⟨u(t), g1⟩⟨u(t), g2⟩] = lim +t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩], +which happens to be of independent interest. We justify this exchange of expectation and limit using +the Dominated Convergence theorem. It suffices to show that there exists a random variable X +with finite expectation such that +|⟨u(t), g1⟩⟨u(t), g2⟩| ≤ X +∀t ∈ [0, ∞). +In fact one can choose +X := sup +t≥0 +|⟨u(t), g1⟩|2 + sup +t≥0 +|⟨u(t), g2⟩|2. +In order to show that EX < ∞, we use the Monotone Convergence theorem and Doob’s submartin- +gale inequality (see for instance Theorem 3.8 in [22]): +E[ sup +t≥0 +|⟨u(t), g⟩|2] = lim +N→∞ E[ sup +0≤t≤N +|⟨u(t), g⟩|2] ≤ lim +N→∞ 4 E[|⟨u(N), g⟩|2] = 4 ∥G(s, y)∥2 +L2(R+ +s ×Ry) , +which is finite by Step 1. +Step 3: Calculation of the correlations as t → ∞. +We start by computing the correlations for a finite time t > 0. +E[u(t, x1)u(t, x2)] = +� t +0 +e2πic(x1−x2)sCf(x1 − x2) ds = e2πict(x1−x2) − 1 +2πic(x1 − x2) Cf(x1 − x2). +This is a well-defined function for each finite t, but we must treat it as a distribution if we want to +take the limit t → ∞. To do so, we test it against some g1, g2 ∈ S(R): +E[⟨u(t), g1⟩⟨u(t), g2⟩] = +� +Rx1×Rx2 +E[u(t, x1)u(t, x2)]g1(x1)g2(x2) dx1dx2 += +� +Rz +e2πictz − 1 +2πicz +ψ(z) dz +(2.20) +with ψ(z) = Cf(z) +� +Ry g1(z + y)g2(y)dy, z = x1 − x2 and y = x2. Our goal is to study the last +integral as t → ∞. +We start with a simple identity that gives us a way to integrate the function (e2πictz −1)/(2πicz). +Note that this function is not absolutely integrable in R. +However, its integral does converge + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +13 +conditionally, i.e. the final result might depend on how we integrate it. More precisely, we recall +that for all t > 0 +(2.21) +� +R +eitz − 1 +iz +dz := lim +R→∞ +� R +−R +eitz − 1 +iz +dz = π. +As a result of (2.21), we have that +lim +R→∞ +� R +−R +e−2πictz − 1 +2πicz +ψ(z)dz − ψ(0) +2c += lim +R→∞ +� R +−R +e−2πictz − 1 +2πicz +[ψ(z) − ψ(0)] dz. +In view of +ψ(0) = Cf(0) +� +Ry +g1(y)g2(y)dy, +it suffices to prove the following in order to obtain (2.10): +lim +t→∞ lim +R→∞ +� R +−R +e−2πictz − 1 +2πicz +[ψ(z) − ψ(0)] dzdy = +1 +2πc +� ∞ +0 +ψ(z) − ψ(−z) +iz +dz += +1 +2πic p.v. +� +Rz +ψ(z) +z +dz. +(2.22) +To prove (2.22), we rewrite the left-hand side as follows: +(2.23) +lim +R→∞ +� R +−R +e−2πictz − 1 +2πc +ψ(z) − ψ(0) +iz +dz = +1 +2πic +� +lim +R→∞ +� R +−R +e−2πictz ψ(z) − ψ(0) +z +dz +− lim +R→∞ +� R +−R +ψ(z) − ψ(0) +z +dz +� +Note that the last term gives the desired limit after using the fact that +� R +−R +ψ(z) − ψ(0) +z +dz = +� R +0 +ψ(z) − ψ(−z) +z +dz +and taking R → ∞. +The final step is to show that the first term on the right-hand side of (2.23) tends to zero +as t → ∞. Note that +F(z) := ψ(z) − ψ(0) +z +is not integrable in Rz. However the following lemma shows that its derivatives have better prop- +erties. Its proof is postponed to the end of the proof of Theorem 2.4. Recall that ψ was defined in +terms of g1, g2 right after (2.20). +Lemma 2.8. Let n ≥ 1. If g1 ∈ Hn+1(R) and g2 ∈ L2(R) then F ∈ W n,1(Rz) and +lim +|z|→∞(∂m +z F)(z) = 0, for any 0 ≤ m ≤ n, +(2.24) +∥∂m +z F∥L1(Rz) ≲ ∥g1∥Hm+1 ∥g2∥L2 for any 1 ≤ m ≤ n. +(2.25) +We integrate by parts the first term on the right-hand side of (2.23) +� R +−R +e−ictz ψ(z) − ψ(0) +z +dz = − +� R +−R +1 +ict ∂ze−ictz F(z)dz += − 1 +ict e−ictz F(z) +��� +R +z=−R + 1 +ict +� R +−R +e−ictz∂zF(z) dz. + +14 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +Using Lemma 2.8, we are able to take the limit R → ∞: +lim +R→∞ +� R +−R +e−ictz ψ(z) − ψ(0) +z +dz = 1 +ict +� +Rz +e−ictz∂zF(z) dz. +Next, we continue to integrate by parts using (2.24) (which gets rid of the boundary terms) +lim +R→∞ +� R +−R +e−ictz ψ(z) − ψ(0) +z +dz = +� 1 +ict +�n � +Rz +e−ictz∂n +z F(z) dz, +thus we have +���� lim +R→∞ +� R +−R +e−ictz ψ(z) − ψ(0) +z +dz +���� ≤ +� 1 +ct +�n +∥∂n +z F∥L1(R2) . +We use (2.25) to finish the proof of Theorem 2.4. Now, we need to prove the technical Lemma 2.8. +Proof of Lemma 2.8. First we can easily show that +∂n +z F(z) = +� 1 +0 +sn∂n+1 +z +ψ(zs)ds +(2.26) += +� z +0 +sn +zn+1 ∂n+1 +z +ψ(s)ds. +(2.27) +Then we show that for all 0 ≤ m ≤ n + 1 and all p ∈ [1, ∞] +(2.28) +∥∂m +z ψ∥Lp(R) ≲ ||Cf||W m,p(Rz) ∥g1∥Hm ∥g2∥L2 . +Indeed, the Leibniz rule yields +∂m +z ψ(z) = +m +� +j=0 +� +m +j +� +∂j +zCf(z) +� +Ry +∂m−j +z +g1(y + z)g2(y)dy. +By Cauchy-Schwarz inequality, +|∂m +z ψ(z)| ≤ +� +� +m +� +j=0 +� +m +j +� +|∂j +zCf(z)| +� +� ∥g1∥Hm ∥g2∥L2 +thus one gets (2.28). +Then we show that for all 0 ≤ m ≤ n, +lim +|z|→∞ ∂m +z F(z) = 0. +Indeed, by the Cauchy-Schwarz inequality, (2.27) and (2.28), +|∂m +z F(z)| ≤ +��∂m+1 +z +ψ +�� +L2(Rz) +�� |z| +0 +s2m +|z|2(m+1) ds +� 1 +2 += +��∂m+1 +z +ψ +�� +L2(Rz) +|z| +1 +2 +−→ +|z|→∞ 0. +Finally, for all 1 ≤ m ≤ n, (2.26) implies +∥∂m +z F(z)∥L1(R2) ≤ +� 1 +0 +sm ��∂m+1 +z +ψ(zs) +�� +L1(R2) ds ≤ +��∂m+1 +z +ψ +�� +L1(R2) +�� 1 +0 +sm−1ds +� +and thus we conclude with (2.28). +□ + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +15 +Natural generalizations of the dynamics in (2.1)-(2.2) to higher dimensions could be obtained +in two ways. Firstly, the product c ∂k�u in the transport equation (2.1) could be generalized to +a scalar product of a given unit vector e ∈ Rd with the gradient ∇k�u(t, k). Unfortunately, this +puts too much weight on the constant vector e and results in an obvious statistical anisotropy in +physical space. For applications to turbulence, we must require that statistical laws are not only +invariant by translation, but also under rotation (i.e. statistical isotropy), as commonly observed +in laboratory and numerical experiments, and as expected from a physical point of view. Another +option would be to replace the multiplication by ix in the physical space formulation (2.2) by +the multiplication by i|x|, where |x| is the modulus of x ∈ Rd. Once again, this would introduce +anisotropy in the system, and more importantly, it would break statistical homogeneity even in +dimension d = 1. One may check these claims directly using the exact solution (2.2) to compute +the covariance function at a given time t and any two positions x, y (see [3] for details). Indeed, +this covariance function eventually depends on the difference |x| − |y|, and not on x − y as would +be desirable. Beyond these issues, none of these propositions would ensure that a given real-valued +initial condition u(0, x) ∈ R gives rise to a real-valued solution u(t, x) ∈ R at all future times. In +other words, in order to construct a real-valued solution in physical space, one needs to propose +a dynamical picture able to preserve the Hermitian symmetry of the Fourier transform �u(t, k), as +does the dynamics in (2.3). +3. Higher dimensional real fractional gaussian fields: heuristic +In the previous section we gave a rigorous proof of the construction of a dynamical complex +white noise. This proof was carried out in physical space. For the dynamical real white noise of +(2.3), on the other hand, it is more convenient to think of its wavenumber formulation. However, +even in the case of a dynamical complex white noise, the solution u(t, x) is not in Lp(Rx) for any +1 ≤ p < ∞ (see (2.18)), hence its Fourier transform is not defined pointwise. In conclusion, it +is difficult to make sense of equation (2.1) in wavenumber space. However, we will not concern +ourselves with such difficulties in this section, and we will work as if the solution to (2.1) were +well-defined pointwise: we refer to Section 4 for a rigorous analysis. As we will later see, working +in wavenumber space is very convenient to formally show that (2.3) builds a dynamical real white +noise, as well as to extend this construction to d-dimensional fractional Gaussian fields. +3.1. A transport equation in wavenumber space. We propose the following initial value +problem as a generalization of (2.3) +(3.1) +� +� +� +� +� +� +� +� +� +∂t�u(t, k) + divk +� +ck +|k| �u(t, k) +� ++ cH + 1 +2 +|k| +�u(t, k) = �f(t, k) +t > 0, k ∈ Rd, |k| > κ > 0, +�u(t, k) = 0 +t > 0, k ∈ Rd, |k| ≤ κ, +�u(0, k) = 0. +where H is a real constant (which will be eventually connected with the H¨older exponent of the +solution), and divk stands for the usual divergence operator in wavenumber space. As one can +easily verify with a few vector calculus identities, +(3.2) +divk +� ck +|k| �u(k) +� ++ c H + 1 +2 +|k| +�u(k) = c ∂|k| �u(k) + c H + d − 1 +2 +|k| +�u(k) +with +∂|k| := k +|k| · ∇k. +When d = 1 and H = −1/2, one recovers (2.3) from the above. The initial value problem (3.1) +can be regarded as a conservation law in wavenumber space with a source term �f(t, k), a damping +term (H + 1/2) �u(t, k)/|k| and Dirichlet boundary conditions at the sphere |k| = κ. +Equation (3.1) has been thoroughly studied when �f(t, k) is regular enough (see [26], [33]), but +in our case �f(t, k) is too rough for such classical results to be applicable. In particular, we will + +16 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +assume that the forcing term �f satisfies +(3.3) +E[ �f(t, k1) �f(s, k2)] = � +Cf(k1) δt−s δk1−k2, +where � +Cf(k) is radial and null in the ball |k| < κ. Condition (3.3) formally follows from considering +the Fourier transform of a white noise in time satisfying +(3.4) +E[f(t, x)f(s, y)] = δt−s Cf(x − y). +Remark 3.1. As part of equation (3.1) one has the following technical condition +(3.5) +�u(t, k) = 0 +t > 0, +k ∈ Rd, +|k| ≤ κ, +for κ > 0. Indeed, if |k| = 0, then divk +� +k +|k| · +� +and +H+ 1 +2 +|k| +are not well-defined. This condition (3.5) +implies in particular that F−1�u has null spatial average, whenever defined. It might be possible +to make sense of the problem (3.1) for κ = 0 by adequately changing condition (3.5), imposing +�f(t, k = 0) = 0 and an appropriate behaviour near k = 0, but this is outside the scope of this paper. +3.2. Asymptotic behavior: power-law. In this section, we show that the two-point correlation +of the solution to (3.1) displays a power-law behavior. In our first result (Theorem 3.2), we obtain +this asymptotic behavior as t → ∞ under fairly mild assumptions on the forcing �f. Under stronger +assumptions on �f, we derive a second result (Proposition 3.5) showing this power law behavior in +finite time. Among other things, such power laws are important because their exponent determines +the H¨older regularity of the solution in physical space (should it be possible to take the inverse +Fourier transform). The main idea of the heuristic proof of our desired results is to perform a change +of variables to rewrite (3.1) as a 1D transport equation with respect to |k| and parametrized by the +“angular variable” +k +|k|. Such an equation admits an explicit solution that we will exploit. We will +further discuss such consequences in Section 4. +Theorem 3.2 (Heuristic version). Let the forcing �f satisfy (3.3) in such a way that � +Cf(k) = ψ(|k|) +is radial, non-negative, non identically null, with s2H+d ψ(s) ∈ L1(R+ +s ). Furthermore, we assume +that ψ is null when |k| < κ. Then, (3.1) admits a solution that satisfies the following asymptotic +behavior +lim +t→∞ E[�u(t, k)�u(t, k′)] = |k|−(2H+d) (C(d, H) − Ψd,H(|k|)) δk−k′ +where +(3.6) +Ψd,H(|k|) := 1 +c +� ∞ +|k| +s2H+d ψ(s) ds. +is a positive non-increasing absolutely continuous function and +(3.7) +C(d, H) := Ψd,H(0) > 0. +As we have already pointed out in Remark 2.6, the function Ψd,H can be seen as lower-order +correction in comparison with the Dirac distribution. +Moreover, if we assume that ψ is fast- +decaying, then Ψd,H will also be fast-decaying. +Heuristic proof. In order to find a formal solution to (3.1), we let +�v(t, k) := |k|H+d− 1 +2 �u(t, k), +�g(t, k) := |k|H+d− 1 +2 �f(t, k). + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +17 +Next we rewrite (3.1) in terms of �v, namely +(3.8) +� +� +� +� +� +∂t�v(t, k) + c ∂|k|�v(t, k) = �g(t, k) +t > 0, |k| > κ, +�v(t, k) = 0 +t > 0, |k| ≤ κ, +�v(0, k) = 0, +where we use the fact that ∂|k| = +k +|k| · ∇k. +Equation (3.8) can be regarded as a 1D transport equation with respect to |k| and parametrized +by the “angular variable” k/|k|. Thus it is easy to give an explicit solution: +(3.9) +�v(t, k) = +� t +� +t− |k|−κ +c +� ++ +�g +� +s, (|k| − ct + cs) k +|k| +� +ds. +Next we compute the correlations of �v(t) using those of �f. We have +E[�v(t, k1)�v(t, k2)] = +� t +� +t− |k1|−κ +c +� ++ +� t +� +t− |k2|−κ +c +� ++ +δcs1−cs2δ |k1|−ct+s1 +|k1| +k1− |k2|−t+cs2 +|k2| +k2 +� +Cg +�|k1| − ct + cs1 +|k1| +k1 +� +ds1ds2. +We assume that we can write +δcs1−cs2δ |k1|−ct+cs1 +|k1| +k1− |k2|−ct+cs2 +|k2| +k2 = δcs1−cs2δ |k1|−ct+cs1 +|k1| +k1− |k2|−ct+cs1 +|k2| +k2 +even if this not mathematically rigorous. Moreover, by the change of variables from cartesian to +polar coordinates +δ |k1|−ct+cs +|k1| +k1− |k2|−ct+cs +|k2| +k2 = +�|k1| − ct + cs +|k1| +�−(d−1) +δk1−k2. +This is possible since the Jacobian +� +|k1|−ct+cs +|k1| +�−(d−1) +has no singularities in the region of integration +thanks to κ > 0. As a result, one obtains +E[�v(t, k)�v(t, k′)] = +� +� +� t +� +t− |k|−κ +c +� ++ +�|k| − ct + cs +|k| +�−(d−1) +� +Cg +�|k| − ct + cs +|k| +k +� +ds +� +� δk−k′. +This immediately implies +E[�u(t, k)�u(t, k′)] = |k|−(2H+d) +� +� +� t +� +t− |k|−κ +c +� ++ +(|k| − ct + cs)2H+d � +Cf +�|k| − ct + cs +|k| +k +� +ds +� +� δk−k′. +The change of variables s �→ |k1| − ct + cs yields +E[�u(t, k)�u(t, k′)] = |k|−(2H+d) +� +χ|k|>ct+κ +� |k| +|k|−ct +s2H+d � +Cf +� s +|k| k +� ds +c +� +δk−k′ ++ |k|−(2H+d) +� +χ|k|≤ct+κ +� |k| +κ +s2H+d � +Cf +� s +|k| k +� ds +c +� +δk−k′ +where χA denotes the characteristic function of the set A. Remember that ψ(|k|) := � +Cf(k) since Cf +is radial. Using (3.6), we may rewrite +E[�u(t, k)�u(t, k′)] = |k|−(2H+d) � +χ|k|>ct+κ [Ψd,H(|k| − ct) − Ψd,H(|k|)] ++ χ|k|≤ct+κ [Ψd,H(κ) − Ψd,H(|k|)] +� +δk−k2 . +(3.10) + +18 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. 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+tdzOo5tCqKuRevokLcaZiha6ayRM9TSZ0E+BlnEq+RnEik0IYPGxH6YPAgCLswEdJTRx4GAuKaGBHCTnKznAHqY3Ii5GHSWyPvl3a1WPWo72MGSq1TVlcejkpdeyTxic/Tlhm05U9UpEl+1vskYopzakvxXH6hMrcEPsX7qx5/91k+eV7FAB6eqUocqDxQje2DHuSLVO1mfPlG7oAgBcRK3yc4J20o5vg1daULVIXkDprK/K0/Jyr0d+0b4mFNLnTIMKEdAtum7uZ2ukuYo/3NqZkHlKJc/zhUuC9niWTxRSexiDwc0NSco4gIlCmyh3s84F70l60V+3t21VLxJodTC3t8wsBNKjbt2 + r⇤ +Figure 1. The functions |k| �→ F(t1, |k|) and |k| �→ F(t2, |k|) for t1 < t2 and ψ +supported in [κ, r∗]. +We conclude by taking the limit t → ∞ and noting that Ψd,H(κ) = Ψd,H(0). +□ +Remark 3.3. In the previous proof, we have shown that (see (3.10)) for any fixed t > 0 +E[�u(t, k)�u(t, k′)] = |k|−(2H+d) F(t, |k|) δk−k′ +where F is a non-negative absolutely continuous function in t and |k| that admits the following +expression +(3.11) +F(t, |k|) = +� +Ψd,H(κ) − Ψd,H(|k|), +|k| < ct + κ, +Ψd,H(|k| − ct) − Ψd,H(|k|), +|k| > ct + κ. +In particular, for any time t1 ≤ t2, F(t1, |k|) = F(t2, |k|) for all |k| ≤ ct1 + κ. As t grows the +window of |k| where F(t, |k|) is stationary (and displays a power-law behavior) grows too (see figure +1). As a result, it is not necessary to wait until t = ∞ in order to observe the power-law. This is +helpful when performing numerical simulations. +Remark 3.4. The limits t → ∞ and |k| → ∞ in (3.11) don’t commute. Indeed, one gets +lim +t→∞ E[�u(t, k)�u(t, k′)] +∼ +|k|→∞ |k|−(2H+d) C(d, H) δk−k′, +whereas for any fixed time t > 0 +lim +|k|→∞ E[�u(t, k)�u(t, k′)] = 0. +In simulations, it is usual to look for the power-law for large |k|. But, as pointed out in the +previous remark, if we don’t wait long enough then we will see nothing. The next proposition shows +that if the two-point correlation of the source is spectrally located in the ball |k| ≤ r∗, then it is +enough to wait until t ≥ r∗ to observe the desired power-law. +Proposition 3.5. Let ψ supported in [κ, r∗] for r∗ > κ > 0. For any t > r∗ fixed, the function F +given in Remark 3.3 satisfies +F(t, |k|) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +|k| ∈ (0, κ) +stationary in t and non-decreasing in |k| +|k| ∈ (κ, r∗), +constant +|k| ∈ (r∗, ct + κ), +non-decreasing in t and non-increasing in |k| +|k| ∈ (ct + κ, ct + r∗), +0 +|k| ∈ (ct + r∗, ∞). + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +19 +Remark 3.6. Since ψ vanishes in the ball |k| < κ, +Ψ(0) = Ψ(κ) = Ψ(|k| − ct) +for +ct ≤ |k| ≤ ct + κ, +and thus the expression +F(t, |k|) = +� +Ψd,H(0) − Ψd,H(|k|), +|k| < ct, +Ψd,H(|k| − ct) − Ψd,H(|k|), +|k| > ct. +is independent of κ. On this note, see Remark 3.1. +In order that the solution of (3.12) be closer to the experimental energy-spectrum picture (see +beginning of section 1.2), one can add some viscosity in the equation. More precisely, one can +consider f − ν∆u instead of a sole forcing term f. +Theorem 3.7 (Heuristic version). Let ν ≥ 0 and let the forcing �f satisfy (3.3) such that � +Cf(k) = +ψ(|k|) is radial, non-negative, non identically null, with s2H+de +8π2ν +3c s3 ψ(s) ∈ L1(R+ +s ) and ψ null in +the ball |k| < κ. Then +(3.12) +� +� +� +� +� +� +� +� +� +∂t�uν(t, k) + divk +� +ck +|k| �uν(t, k) +� ++ +� +cH + 1 +2 +|k| ++ ν|2πk|2 +� +�uν(t, k) = �f(t, k) +t > 0, k ∈ Rd, |k| > κ > 0, +�uν(t, k) = 0 +t > 0, k ∈ Rd, |k| ≤ κ, +�uν(0, k) = 0 +admits a solution that satisfies for any fixed t > 0 +E[�uν(t, k)�uν(t, k′)] = |k|−(2H+d)e− 8π2ν +3c |k|3 Fν(t, |k|) δk−k′ +where Fν is a non-negative absolutely continuous function in t and |k| that admits the following +expression +(3.13) +Fν(t, |k|) = +� +Ψd,H,ν(κ) − Ψd,H,ν(|k|), +|k| < ct + κ, +Ψd,H,ν(|k| − ct) − Ψd,H,ν(|k|), +|k| > ct + κ. +and where +(3.14) +Ψd,H,ν(|k|) := 1 +c +� ∞ +|k| +s2H+de +8π2ν +3c s3 ψ(s) ds. +is a positive non-increasing absolutely continuous function. +Proof. In order to find a formal solution to (3.12), we let +�v(t, k) := |k|H+d− 1 +2 e +4π2ν +3c |k|3�u(t, k), +�g(t, k) := |k|H+d− 1 +2 e +4π2ν +3c |k|3 �f(t, k) +and rewriting (3.12) for �v, we find that �v solves (3.8). Thus the proof is similar to the one of +Theorem 3.2. +□ +Note that in the presence of viscosity, the limits ν → 0 and t → ∞ commute. + +20 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +3.3. Main issues with the heuristic proof. The calculations in this section provide invaluable +intuition regarding the construction of an equation whose solution displays a power-law behavior as +well as the right parameters involved. Nevertheless, there are several steps in our calculations that +are difficult to justify from a rigorous mathematical viewpoint (if possible at all). More precisely: +(1) Fourier-transform. As we already mentioned, the Fourier transform of a rough non-decrea- +sing source (and therefore the solution of equation (3.1)) is unlikely to be defined pointwise. +(2) Transport equation in wavenumber space. We have never showed that (3.1) is well-posed. +Even if it were in some weak sense, it is unclear whether one can define the trace of the +solution at the boundary |k| = κ or to define the domain of the operator divk +� +k +|k|· +� +. +(3) Solution of a stochastic transport equation. +In the simplest case, (3.1) becomes (3.8) +(for H = 1 +2 − d) whose solution we gave in (3.9), namely +(3.15) +�u(t, k) = +� t +(ct−(|k|−κ))+ +�f +� +s, (|k| − ct + cs) k +|k| +� +ds. +In the case where f is a white noise in time of the form (2.6) (in Rd instead of R), the integral +on the right-hand side of (3.15) is not a Riemann integral. Worse than that, one would +need to define the object �f(t, k) and evaluate it along the characteristics of the transport +equation. To sum up, one would need to make sense of the “stochastic integral”: +� t +(t−(|k|−κ))+ +�ϕ +� +(|k| − ct + cs) k +|k| +� +� +dW +� +s, (|k| − ct + cs) k +|k| +� +. +Moreover, without a rigorous functional setup it is hard to study the important problem of +uniqueness of solutions. In fact, it is unclear whether the solution (3.15) is unique. +(4) Two-point correlations. Even if we can make sense of (3.1) weakly, e.g. as a tempered +distribution, the two-point correlation is not necessary well defined. Indeed, it involves a +product of distributions whose singular supports might overlap. +(5) Mixing coordinates in Dirac distributions. In the proof, we use the following identity +δs1−s2δ |k1|−ct+cs1 +|k1| +k1− |k2|−ct+cs2 +|k2| +k2 = δs1−s2δ |k1|−ct+cs1 +|k1| +k1− |k2|−ct+cs1 +|k2| +k2. +In other words, we first apply δs1−s2 to δ |k1|−ct+cs1 +|k1| +k1− |k2|−ct+cs2 +|k2| +k2. Can this be justified? +This is in particular possible if the inner delta were a function, which is of course not the +case. Later, in the proof, we perform a change of coordinates to obtain +δs1−s2δ |k1|−ct+cs1 +|k1| +k1− |k2|−ct+cs2 +|k2| +k2 = +�|k1| − ct + cs1 +|k1| +�−(d−1) +δs1−s2δk1−k2. +Does the same final result hold if we exchanged the order in which the deltas on the left-hand +side are applied? +Ultimately, we would also like to write equation (3.1) in physical space and to interpret the +power-law in physical space. One first idea is to perform an inverse Fourier transform of (3.1) but +the difficulties mentioned above make it complicated. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +21 +4. Higher dimensional real fractional gaussian fields: results +As mentioned in the previous subsection, it is difficult to justify the Fourier transform of a rough +source such as the one that interests us. Similarly, the solution to such a problem (3.1) may not +admit a Fourier transform either. For that reason, our goal is to try to formulate a weaker notion +of (3.1) which is entirely given in physical space, thereby avoiding such problems. To do so, we +first introduce the Hilbert space +(4.1) +X := {u ∈ L2(Rd +x) | ˆu(k) = 0 for all |k| < κ } +together with the inner product +(u, v) := +� +Rd +k\B(0,κ) +�u(k) �v(k)dk. +Next we introduce the unbounded operator +(4.2) +A : u �→ F−1 +� +c divk +� k +|k| �u +� ++ c H + 1 +2 +|k| +�u +� +with domain +D(A) := {u ∈ X |Au ∈ X and �u||k|=κ = 0 } += {u ∈ X |∂|k|�u ∈ L2(Rd +k \ B(0, κ)) and �u||k|=κ = 0 }. +(4.3) +For any function u ∈ X such that ∂|k|�u ∈ L2(Rd +k \ B(0, κ)), the trace in Fourier space �u||k|=κ is +well-defined thanks to the Sobolev embedding theorem. +This allows us to impose �u||k|=κ = 0. +Consequently, the transport equation in wavenumber space (3.1) can be written in physical space +after formally applying the inverse Fourier transform: +(4.4) +� +∂tu + Au = f +for t > 0, +u|t=0 = 0. +At this stage, note that for any regular enough forcing f ∈ L1 +loc((0, +∞)t, X), the problem (4.4) +is rigorously defined. Its mild solution is given by: +(4.5) +u(t) = +� t +0 +e−(t−s)Af(s)ds +where e−tA : X → X is defined by +(4.6) +e−tAu0 := F−1 +� +χ|k|>ct+κ +�|k| − ct +|k| +�H+d− 1 +2 +� +u0 +�|k| − ct +|k| +k +�� +. +Indeed, it is easy to check that (4.5) solves (4.4). Moreover, it is also easy to check that if u0 ∈ +X ∩ S(Rd +x) then e−tAu0 ∈ X ∩ S(Rd +x). +Unfortunately, our source f is a white noise in time (and colored in space) and therefore the +above considerations do not apply. Rigorously speaking, f is a Gaussian random measure acting +on L2((0, +∞)t), i.e. +(4.7) +⟨f, g⟩(x) := +� +(0,+∞)t×Rdy +τyϕ(x) g(t)dW(t, y), +for all test functions g ∈ L2((0, +∞)t) where dW is a space-time Gaussian real white noise, τy is a +translation by y (i.e. τyg(x) = g(x − y)) and ϕ ∈ X ∩ S(Rd +x) is a real, non-identically null, radial +function. + +22 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +In order to give a meaning to solutions of (4.4) with rough source, we need a weak formulation. +To propose one, one first needs to introduce the adjoint of A. +Proposition 4.1. The adjoint of A in (4.2) is defined by +(4.8) +A∗ : u �→ F−1 +� +−c divk +� k +|k| �u +� ++ cH + d − 1 +2 +|k| +�u +� +and its domain is +D(A∗) := {u ∈ X |∂|k|�u ∈ L2(Rd +k \ B(0, κ)) } ⊋ D(A). +Proof. Let u ∈ D(A) and v ∈ X. Using (3.2), one gets +c−1(Au, v)L2 = +� +Rd +k\B(0,κ) +∂|k|�u(k) �v(k) dk + +� +Rd +k\B(0,κ) +H + d − 1 +2 +|k| +�u(k) �v(k) dk. +We perform a change of variables from cartesian to polar coordinates in wavenumber space: +� +Rd +k +∂|k|�u(k) �v(k) dk = +� +R+ +� +Sθ +∂|k|�u(k) �v(k) |k|d−1dσθd|k|. +We now assume that ∂|k|�v ∈ L2(Rd +k \ B(0, κ)) so that we can integrate by parts in variable |k| and +immediately return to cartesian coordinates: +� +Rd +k +∂|k|�u �v dk = − +� +Rd +k +�u ∂|k|�v dk − +� +Rd +k +(d − 1) +|k| +�u �vdk − +� +Sθ +κd−1 �u||k|=κ �v||k|=κdσθ. +Since �u||k|=κ = 0, the previous equality yields +c−1(Au, v)L2 = − +� +Rd +k +�u ∂|k|�v + +� +Rd +k +H + 1 +2 +|k| +�u �v dk. +Using (3.2) once again, one readily gets +c−1(Au, v)L2 = +� +Rd +k\B(0,κ) +�u(k) +� +−divk +� k +|k|�v(k) +� ++ H + d − 1 +2 +|k| +�v(k) +� +dk +for all u ∈ D(A) and v ∈ X such that ∂|k|�v ∈ L2(Rd +k \ B(0, κ)). +Note that it is not necessary to impose �v||k|=κ = 0 and thus D(A∗) is strictly larger than D(A). +□ +Remark 4.2. In the definition of D(A) the condition �u||k|=κ = 0 can be interpreted as a boundary +condition in wavenumber space. This boundary condition is lost in D(A∗), which is strictly larger +than D(A). It might be possible to take κ = 0 by adequately exchanging �u||k|=κ = 0 by �u(t, k = 0) = +0, and by imposing an appropriate behaviour near k = 0, but this is outside the scope of this paper. +We are ready to give the weak formulation of (4.4). +Definition 4.3. We say that a stochastic process u is a weak solution to (4.4) with f given by (4.7), +if u ∈ L1 +loc([0, ∞)t × Rd) almost surely and +(4.9) +� +Rdx +u(t, x)g(t, x)dx + +� t +0 +� +Rdx +u(s, x)(−∂s + A∗)g(s, x)dsdx + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +23 += +� t +0 +� +Rdy +�� +Rdx +ϕ(x − y)g(s, x)dx +� +dW(s, y) +for all g ∈ C∞((0, ∞), D(A∗) ∩ S(Rd +x)), and such that +(4.10) +� t +0 +� +Rdx +u(s, x)g(s, x)dsdx = 0 +for all g ∈ C∞((0, ∞), S(Rd +x)) with �g(t, k) = 0 for all |k| > κ. +Note that for a test function g ∈ X ∩ S(Rd +x), g ∈ D(A∗) and A∗g ∈ D(A∗) ∩ S(Rd +x) and thus the +terms on the left-hand side of (4.9) are well-defined. +We are ready to give the main result of this paper for positive H ∈ (0, 1). +Theorem 4.4. Let H ∈ (0, 1), and let the forcing f in (4.4) be +(4.11) +f(t, x) = +� +Rdy +τyϕ(x) dW(t, y), +where dW is a space-time Gaussian real white noise, τy is a translation by y (i.e. τyg(x) = g(x−y)) +and ϕ ∈ S(Rd +x) is a real, non-identically null, radial function such that �ϕ(k) = 0 for all |k| < κ. +Then: +(i) Equation (4.4) admits a unique global (in time) weak solution u(t, x) in C((0, ∞), (X∩S(Rd +x))′) +given by +(4.12) +u(t, x) = +� t +0 +� +Rdy +e−(t−s)A[τyϕ](x) dW(s, y) +where e−tA : X ∩ S(Rd +x) → X ∩ S(Rd +x) is defined in (4.6). +(ii) For each t > 0 fixed, u(t, x) is a Gaussian process with a.s. smooth paths in x, and with +α-H¨older continuous paths in t for any 0 < α < 1/2. +(iii) As t → ∞, the solution u(t, x) converges to a zero-mean Gaussian field u∞(x) with α-H¨older +continuous paths in x for any 0 < α < H. More precisely, +lim +t→∞ sup +x∈Rd E|u(t, x) − u∞(x)|2 = 0. +(iv) The limiting process u∞(x) is characterized by the correlations: +E[u∞(x1)u∞(x2)] = lim +t→∞ E[u(t, x1)u(t, x2)] += C(d, H) KH(x1 − x2) − (KH ∗ JH)(x1 − x2), +(4.13) +where +KH := F−1 � +χ|k|>κ|k|−(2H+d)� +and +JH := F−1 � +χ|k|>κ Ψd,H(|k|) +� +and where C(d, H) and Ψd,H are given in Theorem 3.2. +In the case of non-positive H, one needs to be more careful since the solution converges to a +very rough object. In this direction, we have the following result: +Theorem 4.5. Let H ∈ [−d/2, 0], and let the forcing f as in (4.11). Then points (i) and (ii) in +Theorem 4.4 still hold. Moreover, we have that: + +24 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +(iii) As t → ∞, the solution u(t) converges (in (X ∩ S(Rd +x))′) to a random Gaussian measure +acting on X ∩ S(Rd +x) with zero-mean, i.e. for any g ∈ X ∩ S(Rd), E[⟨u∞, g⟩] = 0. +(iv) We have the following asymptotic behavior: +lim +t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩] = E[ lim +t→∞⟨u(t), g1⟩⟨u(t), g2⟩] += C(d, H) +� +Rd +k +χ|k|>κ |k|−(2H+d) �g1(k) �g2(k) dk +− +� +Rd +k +χ|k|>κ |k|−(2H+d) Ψd,H(|k|) �g1(k) �g2(k) dk. +(4.14) +for any g1, g2 ∈ X ∩ S(Rd), where Ψd,H are given in Theorem 3.2. +Before we prove these theorems, let us make a final comment about the operator A. +This +operator can be regarded as an operator of degree 0 plus a bounded operator of degree −1 as +shown in the following proposition. +Proposition 4.6. For all u ∈ D(A) and v ∈ D(A∗), the following equalities hold: +(4.15) +Au(x) = +� +Rd +k/Bκ +a(x, k)eik·x �u(k) dk +and +A∗v(x) = +� +Rd +k/Bκ +a∗(x, k)eik·x �v(k)dk. +where +a(x, k) := −i k +|k| · x + H + 1 +2 +|k| +and +a∗(x, k) := i k +|k| · x + H + d − 1 +2 +|k| +· +For any multi-index (α, β) with |α| ≥ 1, one has +|x||k|−|β| ≲ |∂β +k a(x, k)| + |∂β +k a∗(x, k)| ≲ |x||k|−|β| + |k|−1−|β| +and +|∂α +x ∂β +k a(x, k)| + |∂α +x ∂β +k a∗(x, k)| ≲ |k|−|β|, +for any (x, k) ∈ R2d such that |k| ≥ κ. The implicit constant doesn’t depend on κ, x or k. +Proof. Let u ∈ D(A). Using the vector calculus identities (3.2), one gets +Au = +� +Rd +k/Bκ +� +k +|k| · ∇k�u + H + d − 1 +2 +|k| +�u +� +eik·xdk. +Then, by integration by part +Au = − +� +Rd +k/Bκ +divk +� k +|k|eik·x +� +�u + +� +Rd +k/Bκ +H + d − 1 +2 +|k| +�ueik·xdk. +As one can easily verify with a few vector calculus identities +(4.16) +divk +� k +|k|eik·x +� += ik · x + (d − 1) +|k| +eik·x +such that +Au = − +� +Rd +k/Bκ +ik · x + (d − 1) +|k| +�ueik·xdk + +� +Rd +k/Bκ +H + d − 1 +2 +|k| +�ueik·xdk +which gives the expected expression in (4.15). Similar computations hold for A∗. +□ + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +25 +Several recent works [17, 11, 21] show that linear equations involving operators of degree 0 +may be useful in showcasing behavior related to turbulence, such as loss of regularity at long +time. +In [17], Yves Colin de Verdi`ere and Laure Saint-Raymond in [17] consider an equation +∂tu+iAu = f where f is an L2 deterministic monochromatic forcing and A is an homogeneous self- +adjoint operator of degree 0 that satisfies some technical (but general) assumptions. The authors +show that, as t → ∞, the solution resembles more and more the generalized eigenfunctions of the +operator A, which do not have L2 finite energy. The fact that A is of order 0 is essential to ensure +that its generalized eigenfunctions live in a space akin to H− 1 +2 −, which is less regular than L2. R. +Carles and C. Cheverry in [11] have also introduced an operator of degree 0 in the context of nuclear +magnetic resonance. More precisely, they have shown that for some highly oscillatory source, at +long times (i-e diffractive time), the solution produces constructive and destructive interferences +which are interpreted as turbulent effects. +It is therefore interesting to compare our model with this existing literature. A first difference +is that these other works are developed in deterministic setting. But even if one were to consider +our model with a monochromatic deterministic source, it is hard to use similar techniques as those +in [17]. Firstly, because our operator satisfies D(A) ⊊ D(A∗) and it cannot thus be self-adjoint +or skew-adjoint. As a result, one cannot use Mourre’s commutator theory as in [17] to prove a +limiting amplitude principle which allows a nice spectral representation of the solution at infinite +time. Secondly, the principal symbol of our operator A (see (4.15)) grows with respect to x. This +is not a standard pseudo-differential operator5 of degree 0. A more profound study of this special +operator (4.2) is therefore interesting, and it is postponed for a future work. Nevertheless one can +mention some conserved quantities in equation (4.4) when the source is null and initial datum is +considered: +• conservation of volume for all H, i-e +∂t +� +Rdx +u(t, x) dx = 0, +• conservation of L2-norm for H = −d/2, i-e +∂t +� +Rdx +|�u(t, x)|2 dx = 0, +• conservation of trace at origin H = −1/2, i-e +∂tu|x=0 = 0. +4.1. Well-posedness. +4.1.1. Uniqueness. Suppose that we have two solutions which live a.s. in L1 +loc([0, ∞)t × Rd) satis- +fying (4.9). When tested against test functions whose Fourier transform is supported in B(0, κ), +both solutions are null and thus they agree. When tested against g ∈ C∞((0, ∞), D(A∗) ∩ S(Rd +x)), +their difference v a.s. satisfies +(4.17) +� +Rdx +v(t, x)g(t, x)dx + +� t +0 +� +Rdx +v(s, x)(−∂s + A∗)g(s, x)dsdx = 0 +for all t > 0. +5A standard operator of degree m is an operator whose symbol a ∈ C∞(R2d) statisfies +|∂α +x ∂β +k a(x, k)| ≲ (1 + |k|)m−|β| +for all multi-index α and β. + +26 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +If g0 ∈ D(A∗) ∩ S(Rd +x)) then +gt(s, ·) = F−1 +� +|k|−H− 1 +2 �g0 +�|k| − c(s − t) +|k| +k +�� +∈ C∞((0, ∞), D(A∗) ∩ S(Rd +x)) +satisfies +(−∂s + A∗)gt = 0 +and +gt|s=t = g0. +Using this choice of test function, (4.17) yields that a.s. for all t > 0 +� +Rdx +v(t, x)g0(x)dx = 0. +As a result, almost surely, for all t, v(t, ·) = 0 almost everywhere. More precisely, for any two +solutions u1, u2 of our problem, we have: +P [∀t > 0, u1(t, x) = u2(t, x) almost everywhere in x] = 1. +Remark 4.7. Note that one could obtain a stronger version of uniqueness by slightly chang- +ing the definition of weak solution. For instance, one could add the condition that E|u(t, x)|2 ∈ +L1 +loc((0, ∞)t × Rd +x). Under this new definition one could show that any two solutions u1 and u2 +must satisfy +E|u1(t, x) − u2(t, x)| = 0 +(t, x)-almost everywhere. This follows directly from the above argument to prove uniqueness together +with the fact that the variance is a.e. finite. +4.1.2. Existence of a weak solution. Let us start by giving an expression for e−tA[τyϕ](x), which +will enable us to prove that (4.12) does provide a weak solution to (4.4). +Proposition 4.8. For all ϕ ∈ X ∩ S(Rd +x), the function G(t, x, y) := e−tA[τyϕ](x) given by +G(t, x, y) = +� +Rd +k +χ|k|>ct+κ +�|k| − ct +|k| +�H+d− 1 +2 +�ϕ +�|k| − ct +|k| +k +� +e−i2π +� |k|−ct +|k| +� +k·y+i2πk·x dk += +� +Rd +k +� +|k| +|k| + ct +�H+ 1 +2 +�ϕ(k) ei2π +� |k|+ct +|k| +� +k·x−i2πk·y dk. +(4.18) +is a smooth function in all variables. Moreover: +• For each multi-index β and each fixed y ∈ Rd, ∂β +y G(·, ·, y) ∈ C1([0, ∞)t, X ∩ S(Rd +x)). Ad- +ditionally, any Schwartz seminorm of ∂β +y G(·, ·, y) (with respect to x) is locally uniform in +y. +• For each x ∈ Rd, ∂β +xG(·, x, ·) ∈ C1([0, ∞)t, S(Rd +y)). Additionally, any Schwartz seminorm +of ∂β +xG(·, x, ·) (with respect to y) is locally uniform in x. +Finally, its Fourier transform in y is given by +(F2G)(t, x, k) = +� +|k| +|k| + ct +�H+ 1 +2 +�ϕ(k) e−i2π +� +|k|+ct +|k| +� +k·x. +Proof. The expression (4.18) comes from the definition of e−tA in (4.6), and the change of vari- +ables k �→ (|k| + ct)/|k|k. The second expression directly yields the formula for (F2G)(t, x, k). +The fact that G is smooth in all three variables follows easily from the integral expression and +the fact that we can interchange differentiation and the integral. The latter step will be justified +next. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +27 +Let us show that for any multi-index β and any fixed x, ∂β +xG(·, x, ·) ∈ C1([0, ∞)t, S(Rd +y)) (the +case of fixed y follows a similar proof). Note that any derivative ∂n +y ∂β +xG(t, x, y) results in a factor +of (−i2πk)n, which can be absorbed by �ϕ(k), and thus smoothness is no problem. Let us focus on +decay. For |y| ≤ 1, note that +|∂β +xG(t, x, y)| ≤ (2π)|β| +� +Rd +k +�|k| + ct +|k| +�|β| +|�ϕ(k)| dk. +Recall that the support of ϕ guarantees that there are not problems of integration around k = 0. +For |y| ≥ 1 and for any j ∈ {1, . . . , d}, +���(−iyj)n∂β +xG(t, x, y) +��� = +��� +� +Rd +k +� +|k| +|k| + ct +�H+ 1 +2 −|β| +�ϕ(k) ei2π +� |k|+ct +|k| +� +k·x∂n +kje−i2πk·y dk +��� +≤ (2π)|β| +� +Rd +k +���∂n +kj +�� +|k| +|k| + ct +�H+ 1 +2 −|β| +�ϕ(k) ei2π +� |k|+ct +|k| +� +k·x +� ��� dk +The integrand of the last expression can be shown to be absolutely integrable on the support of �ϕ. +Note, however, that one needs to pay some powers of x (which is fixed) to control this term. The +implicit constant will therefore depend on such powers, but it is locally uniform in x (i.e. the same +constant for all x in a compact set). +This concludes the proof that for each fixed y ∈ Rd, G(·, ·, y) ∈ C([0, ∞)t, X ∩ S(Rd +x)). In order +to improve this space to C1([0, ∞)t, X ∩ S(Rd +x)), it suffices to differentiate G in time and repeat +the above procedure. +An analogous argument based on integration by parts shows that for each fixed y ∈ Rd and each +multi-index β, ∂β +y G(·, ·, y) ∈ C1([0, ∞)t, X ∩ S(Rd +x)). +□ +Using these nice properties of G, one obtains the following corollary. +Corollary 4.9. The function u(t, x) defined in (4.12) is a well-defined Gaussian process. For any +0 < α < 1/2, it has a.s. α-Holder continuous paths with respect to t and a.s. smooth paths with +respect to x. +In particular, for any multi-index β, any t1, t2 > 0 and any x1, x2 ∈ Rd there exists a (locally +uniform) constant C > 0 such that +(4.19) +E|∂β +xu(t1, x1) − ∂β +xu(t2, x2)|2 ≤ C +� +|t1 − t2| + |x1 − x2|2� +. +Proof. Let G be as in (4.18), then u can be written as +u(t, x) = +� t +0 +� +Rdy +G(t − s, x, y) dW(s, y) = +� t +0 +� +Rdy +G(s, x, y) d� +W(s, y), +where d� +W is another Gaussian white noise. By Proposition 4.8, for any fixed (t, x), G(·, x, ·) ∈ +C1([0, ∞)s, X ∩ S(Rd +x)) ⊂ L2([0, t]s × Rd +y) and therefore u is a well-defined Itˆo process with zero +mean and variance +E|u(t, x)|2 = +� t +0 +� +Rdy +|G(s, x, y)|2 dy ds. +Next let us study the regularity. Firstly, note that for any multi-index β +∂β +xu(t, x) = +� t +0 +� +Rdy +∂β +xG(s, x, y) d� +W(s, y). + +28 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +A simple calculation shows that for t1 < t2, +E|∂β +xu(t1, x) − ∂β +xu(t2, x)|2 = +� t2 +t1 +� +Rdy +|∂β +xG(s, x, y)|2 dy ds +≤ |t1 − t2| +���∂β +xG(·, x, ·) +��� +2 +L∞([0,t2],L2(Rdy)) , +(4.20) +using the fact that ∂β +xG(·, x, ·) ∈ C([0, ∞)s, X ∩ S(Rd +x)). +Similarly, for any x1, x2, we have that +E|∂β +xu(t, x1) − ∂β +xu(t, x2)|2 = +� t +0 +� +Rdy +|G(s, x1, y) − G(s, x2, y)|2 dy ds += +� t +0 +� +Rdy +��� +� 1 +0 +(x2 − x1) · ∇xG(s, (1 − λ)x1 + λx2, y) dλ +��� +2 +dy ds +≤ |x2 − x1|2 +� t +0 +� +Rdy +� 1 +0 +|∇xG(s, (1 − λ)x1 + λx2, y)|2 dλ dy ds +≤ |x2 − x1|2 +� 1 +0 +∥∇xG(s, (1 − λ)x1 + λx2, y)∥2 +L2([0,t]×Rdy) dλ +(4.21) +The latter factor is finite since ∂β +xG(·, x, ·) ∈ C1([0, ∞)t, S(Rd +y)) locally uniformly in x. +One may use (4.20) and (4.21) to obtain (4.19). Given that ∂β +xu(t, x) is a Gaussian process, +(4.19) immediately implies that for any m ∈ N, +E|∂β +xu(t1, x1) − ∂β +xu(t2, x2)|2m ≤ Cm +� +|t1 − t2|m + |x1 − x2|2m� +. +The Kolmogorov continuity theorem guarantees that ∂β +xu(t, x) is therefore a.s. α-H¨older continuous +in time for any 0 < α < (m − d)/(2m). +□ +We are ready to show that the function u defined in (4.12) is indeed a weak solution to (4.4) in +the sense given by Definition 4.3. +Proof that u is a weak solution. Step 1. We start by checking (4.10). Choose any g ∈ C∞((0, ∞), S(Rd +x)) +with �g(t, k) = 0 for all |k| > κ. Then +(4.22) +� t +0 +� +Rdx +�� s +0 +� +Rdy +G(s′, x, y) d� +W(s′, y) +� +g(s, x) dsdx = +� t +0 +� s +0 +� +Rdy +�� +Rdx +G(s′, x, y) g(s, x) dx +� +d� +W(s′, y) ds, +where d� +W is a different Gaussian white noise from dW in (4.11). In order to justify this, it suffices +to show that G(s′, x, y) g(s, x) is absolutely integrable with respect to the Lebesgue measure in +x ∈ Rd and s ∈ [0, t], and with respect to d� +W(s′, y)) for s′ ∈ [0, t] and y ∈ Rd. +As proved in Proposition 4.8, one may show that |G(s′, x, y)| ≤ C(s′, x) ⟨y⟩−d−1 where C(s′, x) +depends on s′ and x in a polynomial way. To compensate for this growth in x, recall that for any +N, |g(s, x)| ≲N ⟨x⟩−N. As a result, |G(s′, x, y) g(s, x)| ≤ C(t, N) ⟨y⟩−d−1⟨x⟩−N. This guarantees +that we may integrate in whichever order we prefer. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +29 +In order to check (4.10), it suffices to show that for all s′ ∈ [0, s], s ∈ [0, t] and y ∈ Rd, +(4.23) +� +Rdx +G(s′, x, y) g(s, x) dx = 0. +This follows from the Plancherel theorem, which allows us to rewrite the integral in (4.23) as +� +Rd +k +�G(s′, k, y) �g(s, k) dk = +� +Rd +k +χ|k|>cs′+κ +�|k| − cs′ +|k| +�H+d− 1 +2 +�ϕ +�|k| − cs′ +|k| +k +� +e−i2π +� |k|−cs′ +|k| +� +k·y �g(s, k) dk. +Note that the support of �g(s, ·) and that of �ϕ do not overlap. Therefore this integral is zero. +Step 2. Next we check (4.9). Fix some g ∈ C∞((0, ∞), D(A∗) ∩ S(Rd +x)). After exchanging the +order of integration (which may be justified using Proposition 4.8 as in Step 1), (4.9) is equivalent +to: +(4.24) +� +Rdx +G(t − s′, x, y) g(t, x) dx + +� t +s′ +� +Rdx +G(s − s′, x, y) (−∂s + A∗)g(s, x) dx ds += +� +Rdx +ϕ(x − y)g(s′, x)dx +for all s′ ∈ [0, t] and all y ∈ Rd. Next we may use the pre-dual of −∂s + A∗ to rewrite the second +term on the left-hand side. After using the fact that G(0, x, y) = ϕ(x−y), we find that (4.24) holds +if and only if: +(4.25) +� t +s′ +� +Rdx +(∂s + A)G(s − s′, x, y) g(s, x) dx ds = 0 +∀ s′ ∈ [0, t], y ∈ Rd. +Therefore, it suffices to prove that (∂t + A)G(t, x, y) = 0. Taking the Fourier transform in x, which +is allowed by Proposition 4.8, it suffices to check that: +(4.26) +∂t �G(t, k, y) + c divk +� k +|k| +�G(t, k, y) +� ++ cH + d − 1 +2 +|k| +�G(t, k, y) = 0 +One may easily check that (4.26) does indeed hold by direct calculation using the fact that: +�G(t, k, y) = χ|k|>ct+κ +�|k| − ct +|k| +�H+d− 1 +2 +�ϕ +�|k| − ct +|k| +k +� +e−i2π +� |k|−ct +|k| +� +k·y. +□ +4.2. Correlations for H ∈ (0, 1). Our first result is an explicit formula for the correlations of our +solution at any finite time. Note that this result holds regardless of the value of H, because t > 0 +is finite. However, we will only be able to take the limit (as a function) when H ∈ (0, 1). +Proposition 4.10. For any finite time, the correlation function satisfies +E[u(t, x1)u(t, x2)] = +� +Rd +k +|k|−(2H+d)F(t, |k|)ei2πk(x1−x2) dk +where F is the function given in Remark 3.3. +Proof. For any finite time t > 0, one has +E[u(t, x1)u(t, x2)] = +� t +0 +� +Rd e−(t−s)A[τyϕ](x1)e−(t−s)A[τyϕ](x2) dy ds. + +30 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +The change of variables s �→ t − s yields +E[u(t, x1)u(t, x2)] = +� t +0 +� +Rd e−sA[τyϕ](x1)e−sA[τyϕ](x2) dy ds. +Using Proposition 4.8 together with the Plancherel theorem (in the y variable), we obtain +E[u(t, x1)u(t, x2)] = +� t +0 +� +Rd +� +|k| +|k| + cs +�2H+1 +|�ϕ(k)|2 e−i2π +� |k|+cs +|k| +� +k·(x1−x2)dk ds. +Next we perform the change of variables k �→ +|k| +|k|+cs k, which yields +E[u(t, x1)u(t, x2)] = +� t +0 +� +Rd χ|k|>cs+κ +�|k| − cs +|k| +�2H+d ����ϕ +�|k| − cs +|k| +k +� ��� +2 +e−i2πk·(x1−x2) dk ds += +� +Rd e−i2πk·(x1−x2) |k|−2H−d +�� t +0 +χ|k|>cs+κ (|k| − cs)2H+d ����ϕ +�|k| − cs +|k| +k +� ��� +2 +ds +� +dk. +This concludes the proof. +□ +As a corollary, we take the limit as t → ∞ and show points (ii) and (iv) of Theorem 4.4. +Corollary 4.11. Suppose that H ∈ (0, 1). One has +lim +t→∞ sup +x∈Rd E|u∞(x) − u(t, x)|2 = 0. +where u∞ is the following zero-mean Gaussian field. +(4.27) +u∞(x) = +� ∞ +0 +� +Rdy +G(s, x, y) d� +W(s, y), +and where d� +W is a different Gaussian white noise from dW in (4.11). This Gaussian field have +the following correlations +E[u∞(x1)u∞(x2)] = lim +t→∞ E[u(t, x1)u(t, x2)] +and one gets +E[u(t, x1)u(t, x2)] = C(d, H) KH(x1 − x2) − (KH ∗ JH)(x1 − x2) + O +� +(ct)−2H� +where +(4.28) +KH := F−1 � +χ|k|>κ |k|−(2H+d)� +and +JH := F−1 � +χ|k|>κ Ψd,H(|k|) +� +and where C(d, H) and Ψd,H are given in Theorem 3.2. +Proof. First we show that +u∞(x) = +� ∞ +0 +� +Rdy +G(s, x, y) d� +W(s, y) +is a well-defined Gaussian field. This is due to the fact that for fixed x ∈ Rd, the function G(·, x, ·) ∈ +L2([0, ∞]s × Rd +y). Indeed, following the arguments in Proposition 4.10 one easily finds that +� ∞ +0 +� +Rdy +|G(s, x, y)|2 dy ds = +� ∞ +0 +� +Rd +� +|k| +|k| + cs +�2H+1 +|�ϕ(k)|2 dk ds. +The convergence of u(t) to u∞ is now standard. Note that: +u∞(x) − u(t, x) = +� ∞ +t +� +Rdy +G(s, x, y) d� +W(s, y), + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +31 +and therefore +E|u∞(x) − u(t, x)|2 = +� ∞ +t +� +Rdy +|G(s, x, y)|2 dy ds += +� ∞ +t +� +Rd +� +|k| +|k| + cs +�2H+1 +|�ϕ(k)|2 dk ds += +1 +2cH +� +Rd(|k| + ct)−2H|k|2H+1 |�ϕ(k)|2 dk +It readily follows that +lim +t→∞ sup +x∈Rd E|u∞(x) − u(t, x)|2 = 0. +The same argument allows us to take the limit as t → ∞ in Proposition 4.10 in order to find +the correlations of u∞. The fact that we can exchange the limit and the expectation admits the +same argument as in Step 2 in the proof of Theorem 2.4. Indeed, note that the variance E|u∞(x)|2 +is finite and independent of x ∈ Rd. +The rate of convergence of the correlations as t → ∞ depend directly on the function F in +Proposition 4.10. A careful analysis shows that the rate of convergence is given by a multiple of +(ct)−2H. +□ +Now that we know that, whenever H ∈ (0, 1), u∞ is a well-defined Gaussian field indexed by +x ∈ Rd, we may wonder about its continuity and its regularity. In this direction, we have the +following result: +Corollary 4.12. Suppose that H ∈ (0, 1). The limit u∞ = limt→∞ u(t) is a Gaussian field in +x ∈ Rd with a.s. α-H¨older continuous paths for any 0 < α < H. More precisely, for all x, ℓ ∈ Rd, +the variance of the increment satisfies +(4.29) +E|δℓu∞(x)|2 ≤ +CK +H(1 − H)|ℓ|2H + CJ |ℓ|2, +where CK and CJ only depend of H and d without blowing up when H tends to 0 or 1. +Proof. By (4.13), we have that +(4.30) +E|δℓu∞(x)|2 = 2C(d, H)[KH(0) − KH(ℓ)] − 2[(KH ∗ JH)(0) − (KH ∗ JH)(ℓ)]. +Step 1. The function (KH ∗JH) in (4.28) is smooth, since Ψd,H(|k|) decays as rapidly as desired +in k. Moreover, we may write: +(4.31) +1 − e2πik·ℓ = 1 − cos (2πk · ℓ) − i sin (2πk · ℓ) . +Then note that +� +Rd +k +sin (2πk · ℓ) χ|k|>κ |k|2−(2H+d) |Ψd,H(|k|)| dk = 0 +thanks to the fact that χ|k|>κ |k|−(2H+d) Ψd,H(|k|) is rotationally invariant together with the change +of variables k �→ −k. + +32 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +As a result, +|(KH ∗ JH)(0) − (KH ∗ JH)(ℓ)| = +� +Rd +k +[1 − cos (2πk · ℓ)] χ|k|>κ |k|−(2H+d) |Ψd,H(|k|)| dk +≤ +� +Rd +k +(2π|k|ℓ)2 +2 +χ|k|>κ |k|−(2H+d) |Ψd,H(|k|)| dk +≤ 2π2 |ℓ|2 +� +Rd +k +|k|2−(2H+d) |Ψd,H(|k|)| dk. +The latter integral is finite thanks to the rapid decay of Ψd,H(|k|). This takes care of the second +term on the right-hand side of (4.30). +Step 2. The leading term, for the purpose of studying the H¨older regularity, is the first term +on the right-hand side of (4.30). We separate the integral into two parts: +KH(0) − KH(ℓ) = +� +Rd +k +χ|k|>κ +� +1 − e2πik·ℓ� +|k|−(2H+d) dk += +� +Rd +k +χ|k|κ +� +1 − e2πik·ℓ� +|k|−(2H+d) dk ++ +� +Rd +k +χ|k|>rχ|k|>κ +� +1 − e2πik·ℓ� +|k|−(2H+d) dk. +(4.32) +We choose r = (2π|ℓ|)−1; one can carefully check that this choice is optimal. Let us further assume +that |ℓ| < κ, the alternative scenario is easier and can be studied separately. +The second term on the right-hand side of (4.32) yields: +��� +� +Rd +k +χ|k|>rχ|k|>κ +� +1 − e2πik·ℓ� +|k|−(2H+d) dk +��� ≤ +��� +� +Rd +k +χ|k|>r |k|−(2H+d) dk +���. +Then, changing to spherical coordinates in the variable k, +��� +� +Rd +k +χ|k|>r |k|−(2H+d) dk +��� ≲ +��� +� +|k|>r +|k|−(2H+1) d|k| +��� ≲ 1 +H r−2H ≲ 1 +H |ℓ|2H. +The first term on the right-hand side of (4.32) requires more care. Using (4.31) as before, note that +� +|k|κ |k|−(2H+d) dk = 0, +as is easily seen from the change of variables k �→ −k. Therefore it suffices to bound +��� +� +|k|κ |k|−(2H+d) dk +��� ≤ +� +|k|κ |k|−(2H+d) dk +≲ 2π2|ℓ|2 +� r +κ +|k|1−2H d|k| +≲ +1 +2 − 2H |ℓ|2H. +This finishes the proof of (4.29). Using this bound on the variance, a standard argument based on +the Gaussianity of u∞ and the Kolmogorov continuity theorem show that u∞ has a.s. α-H¨older +continuous paths for any 0 < α < H. +□ + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +33 +4.3. Correlations for H ∈ [−d/2, 0]. For values of H in this range, we will still show that u∞ = +limt→∞ u(t) exists. However, u∞ cannot be interpreted as a function, in fact its variance is not +finite. Instead, u∞ is a random distribution acting on a space of test functions X ∩ S(Rd). Our +first result is the analogue of Proposition 4.10 in the context of distributions. +The rate of convergence depends on some integrability and differentiability properties of the test +functions chosen. If one tests against Schwartz functions in X ∩ S(Rd), convergence will happen +faster than t−n for any n ∈ N (as we will show below). However, one may want to test agains less +regular or less decaying functions, in which case this rate of convergence can be quantified. +Before doing so, one needs to identify a good space of test functions for which “testing” is still +well-defined. Given that u(t, x) does not generally decay in the x-variable, one needs to impose +a certain decay on the test functions. For n ∈ N, let us define the space6 Hn,n(Rd), which is the +closure of S(Rd) with respect to the norm +∥g∥Hn,n(Rd) = +n +� +j=0 +��⟨x⟩jg +�� +Hn−j(Rdx) . +It is not hard to show that Hn,n(Rd) is a Banach space algebra living inside L2(Rd +x), and thus +the Fourier transform is well-defined for such functions. Moreover, g ∈ Hn,n(Rd +x) if and only if +�g ∈ Hn,n(Rd +k) (and both norms are comparable). +Proposition 4.13. Let H ∈ [−d/2, 0]. For any t > 0 and any test functions g1, g2 ∈ Hn,n(Rd) +with 3(d + 1)/2 < n, the correlations satisfy +E[⟨u(t), g1⟩⟨u(t), g2⟩] = C(d, H) +� +Rd +k +χ|k|>κ |k|−(2H+d) �g1(k) �g2(k) dk +− +� +Rd +k +χ|k|>κ |k|−(2H+d) Ψd,H(|k|) �g1(k) �g2(k) dk ++ +� +Rd +k +χ|k|>ct+κ |k|−(2H+d) [Ψd,H(|k| − ct) − Ψd,H(0)] �g1(k) �g2(k) dk += C(d, H) +� +Rd +k +χ|k|>κ |k|−(2H+d) �g1(k) �g2(k) dk +− +� +Rd +k +χ|k|>κ |k|−(2H+d) Ψd,H(|k|) �g1(k) �g2(k) dk ++ C(d, H) +� 1 +ct +�−(2H+d)−2n +∥g1∥Hn,n ∥g2∥Hn,n +(4.33) +where C(d, H) and Ψd,H are given in Theorem 3.2, and F is given by (3.3). +Remark 4.14. If one chooses test functions g1, g2 ∈ X ∩ S(Rd), then the above result yields a rate +of convergence proportional to (ct)−(2H+d)−2n for n as large as desired. +Proof. Step 1. Firstly, note that ⟨u(t), g1⟩ is well-defined for each t > 0. This follows after a similar +argument to the one that justifies (4.22), i.e. the fact that G(s, x, y)g1(x) is absolutely integrable +with respect to the Lebesgue measure in x and with respect dW(s, y). Indeed, we showed that +6Let us mention that a better space might be possible, in the sense that one might require less decay in physical +space. However, this requires a careful study of the right functional space for u(t, x) and G(s, x, y). We leave this +study for future research. + +34 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +|G(s, x, y)| ≲t ⟨x⟩d+1⟨y⟩−d−1, so one needs only show that ⟨x⟩d+1|g1(x)| ∈ L1(Rd +x). But this follows +from the choice of n together with the Cauchy-Schwarz inequality: +� +Rd⟨x⟩d+1|g1(x)| dx ≲ ∥g1∥Hn,n +�� +Rd⟨x⟩−d−1 dx +�1/2 +. +By Proposition 4.10, we have: +E[⟨u(t), g1⟩⟨u(t), g2⟩] = +� +Rdx1×Rdx2 +�� +Rd +k +|k|−2H−dF(t, |k|)ei2πk(x1−x2) dk +� +g1(x1)g2(x2) dx1dx2 += +� +Rd +k +|k|−2H−dF(t, |k|) �g1(k) �g2(k)dk +Finally, we use the expression for F(t, |k|) obtained in Remark 3.3. This yields (4.33). +Step 2. In order to obtain an asymptotic expansion in terms of t, note that it suffices to study +the last term on the right-hand side of (4.33) which, using the definition of Ψd,H, can be rewritten +as follows: +− +� +Rd +k +χ|k|>ct+κ |k|−(2H+d) +�� |k|−ct +κ +s2H+dψ(s) ds +� +�g1(k) �g2(k) dk. +Note that generally, we cannot expect a better bound than +���χ|k|>ct+κ +�� |k|−ct +κ +s2H+dψ(s) ds +� ��� ≤ C(d, H) ≲ 1. +Certainly, better bounds could be obtained should ψ vanish on a large ball around zero, but this is +not the physical setting we are interested in. +Therefore, by the Cauchy-Schwarz inequality, +��� +� +Rd +k +χ|k|>ct+κ |k|−(2H+d) +�� |k|−ct +κ +s2H+dψ(s) ds +� +�g1(k) �g2(k) dk +��� +≤ C(d, H) +� +Rd +k +χ|k|>ct+κ |k|−(2H+d) | �g1(k)| | �g2(k)| dk +≲ C(d, H) (ct)−(2H+d)−2n ∥g1∥Hn,n ∥g2∥Hn,n . +□ +Thanks to the formula for the correlations obtained in Proposition 4.13, we may now study the +asymptotic behavior of u(t) as t → ∞. That is the content of the next result. +Corollary 4.15. Let H ∈ [−d/2, 0] and n > 3(d + 1)/2. The solution u(t) in (4.12) converges, +as t → ∞, to a random Gaussian measure acting on X ∩ Hn,n(Rd). Moreover, the correlation +structure of the limiting Gaussian measure is given by: +lim +t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩] = E[ lim +t→∞⟨u(t), g1⟩⟨u(t), g2⟩] += C(d, H) +� +Rd +k +χ|k|>κ |k|−(2H+d) �g1(k) �g2(k) dk +− +� +Rd +k +χ|k|>κ |k|−(2H+d) Ψd,H(|k|) �g1(k) �g2(k) dk. +(4.34) +for any g1, g2 ∈ X ∩ Hn,n(Rd). + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +35 +Proof. The convergence of u(t) as t → ∞ follows from (4.34) for g1 = g2, so let us focus on proving +this formula. The second equality in (4.34) follows by taking t → ∞ in (4.33). Finally, the fact +that we may commute the limit in t and the expectation admits the same argument as Step 2 in +the proof of Theorem 2.4, so we omit it. +□ +4.4. The case H = −d/2. In this case, note that the limiting Gaussian field u∞ obtained in +Corollary 4.15 seems to differ from the usual white noise measure in two ways: +(i) the space of test functions X ∩ S(Rd), which only admits functions whose Fourier transform +vanishes in |k| < κ; and +(ii) the top order term in (4.34), which involves a cut-off to wavenumbers |k| > κ. +Point (ii) can be fixed by defining a continuous zero-mean Gaussian field uI with correlations +E[uI(x1)uI(x2)] = J−d/2(x1 − x2) + F−1 � +χ|k|≤κ +� +(x1 − x2) += +� +Rd +k +χ|k|>κ Ψd,H(|k|) e2πik·(x1−x2) dk + F−1 � +χ|k|≤κ +� +(x1 − x2) +=: I(x1 − x2), +(4.35) +where J was defined in (4.28). Note that I(x) is a continuous, radial, square-integrable function, +and therefore they are much more regular than a delta function. For instance, in the one-dimensional +case d = 1, the second term on the right-hand side of (4.35) corresponds to the sinc function. +Using (4.35), we may rewrite the correlations of our solution u∞ in (4.34) as follows: +E[⟨u∞, g1⟩⟨u∞, g2⟩] = E[⟨dW, g1⟩⟨dW, g2⟩] − +� +R2d I(x1 − x2) g1(x1) g2(x2) dx1dx2 += +� +Rdx +g1(x)g2(x) dx − +� +R2d I(x1 − x2) g1(x1) g2(x2) dx1dx2 +(4.36) +for any g1, g2 ∈ X ∩ S(Rd +x). This solves point (ii). +Let us next focus on point (i), regarding the space of test functions. There is a natural way to +regard u(t) as a distribution in S(Rd)′ (as opposed to (X ∩ S(Rd))′). Fix a test function h ∈ S(Rd) +such that +�h(k) = 0 +if |k| ≥ κ, +�h(k) is radial and takes values in [0, 1] for |k| < κ. +(4.37) +For any test function g ∈ S(Rd), one may decompose +g = L1g + L2g := F−1[ �g �h ] + F−1[ �g (1 − �h) ]. +Then one may make a small modification to the notion of solution in Definition 4.3. Indeed, one +could define a weak solution to to (4.4) to be a stochastic process u such that u ∈ L1 +loc([0, ∞)t ×Rd) +almost surely and such that for all g ∈ C∞((0, ∞), S(Rd +x)) +� +Rdx +u(t, x)L2g(t, x)dx + +� t +0 +� +Rdx +u(s, x)(−∂s + A∗)L2g(s, x)dsdx += +� t +0 +� +Rdy +�� +Rdx +ϕ(x − y) L2g(s, x)dx +� +dW(s, y) +and +� t +0 +� +Rdx +u(s, x) L1g(s, x)dsdx = 0. + +36 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +Note that the solution found in (4.12) is still a solution with this new definition. Moreover, the +solution does not depend on the choice of function h as long as it satisfies (4.37). +This allows us to view u(t) and u∞ as distributions in S(Rd)′, and thus to extend (4.36) to test +functions in S(Rd). Then the top order of (4.36) does indeed correspond to a white noise in the +classical sense. +5. Numerical Simulations +Our goal in this section is to perform numerical simulations to illustrate the theory presented +in the previous sections. We recall the continuous problem (3.12): +(5.1) +� +� +� +� +� +∂t�u(t, k) + L +� +�u +� +(t, k) + D +� +�u +� +(t, k) = �f(t, k) +for t > 0, k ∈ Rd, |k| > κ +�u(t, k) = 0 +for t > 0, k ∈ Rd, |k| ≤ κ, +�u(0, k) = 0 +for k ∈ Rd, +where +(5.2) +L +� +�u +� +(t, k) := c divk +� k +|k| �u(t, k) +� +and +D +� +�u +� +(t, k) := +� +c H + 1 +2 +|k| ++ (2π)2ν|k|2 +� +�u(t, k) +Here, H is eventually the Hurst exponent of the solution (at infinite time), and ν > 0 denotes +viscosity. The introduction of viscosity is necessary in order to reach a statistically stationary state +at a finite time, state in which a statistical analysis is possible. In the inviscid problem proposed +in (3.1), on the other hand, scales as small as 1/ct are populated at time t, leading to numerical +instabilities and a blow up when simulated in a finite periodic box with a finite resolution. +Our numerical method combines ideas from time predictor-corrector schemes [23] and pseudo- +spectral methods [36]. In particular, this implies using the Discrete Fourier Transform (DFT), +which forces certain choices regarding the discretization. +We will denote by �u[t, k] the (discrete) vector whose continuous counterpart is �u(t, k). We denote +by ∆t > 0 the time stepping and by ∆x > 0 the mesh size. The mesh size is also the smallest +accessible length scale. +5.1. Numerical method. +5.1.1. Discretization. We discretize our physical space by considering a discrete periodic box x ∈ +(Z/NZ)d of unit length Ltot = 1, using N = 2n collocation points in each direction, with n ∈ N +where we adopt the convention that 0 is not a natural number. Therefore, the mesh size is +∆x = Ltot/N. +The wave vector k = (ki)1≤i≤d is discretized as +ki = [0, 1, ..., N/2, −N/2 + 1, −N/2 + 2, ..., −1]∆k, +where the spectral resolution is given by +∆k = 1/Ltot. +This discretization is standard when using the Discrete Fourier Transform (DFT). The choice of +starting from ki = 0 as the first element of the array is dictated by the convention that is used to +define the DFT and its inverse. +In order to discretize the divergence term L +� +�u +� +in (5.2), we introduce the following discretization +of derivatives in each component kj of the wavenumber: +(5.3) +∂kj�g[t, k] = DFT +� +−2πi˜xjDFT−1 [�g[t, k]] +� +, + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +37 +where the component ˜xj = [0, 1, ..., N/2 − 1, 0, −N/2 + 1, −N/2 + 2, ..., −1]∆x, considered as a +vector and DFT−1 [�g[k]] as a scalar, corresponds to the component j of the position xj where the +“Nyquist” mode has been set to 0 in order to respect the parity of the differentiation. +5.1.2. Discretization of the forcing. In order to produce the forcing term �f[t, k], we generate an +instance of Nd independent, zero-average and unit-variance Gaussian random variables at each time +step, which we store in a vector g[t, x]. Then we weigh them by the appropriate factor (∆x)d/2, we +take the DFT and multiply by the indicator function χ3≤|k|Ltot≤5, which ensures in particular that +no energy is injected at the mode k = 0, and mostly at large scales. Mathematically, one gets +(5.4) +�f[t, k] = χ3≤|k|Ltot≤5 DFT +� +(∆x)d/2 g[t, x] +� +. +Finally, we choose κ = 1/Ltot which is smaller than the smallest non vanishing wavenumber of the +source �f[t, k] (see Remark 1.2). +5.1.3. Algorithm. +• We pre-compute the forcing �f[t, k] as in (5.4). +• Initialization step: �u[0, k] = 0. +• Induction step: given �u[t, k], we compute �u[t + ∆t, k] via the following procedure: +(1) (Prediction step - spatial part) Compute L +� +�u +� +[t, k] according to (5.3), and compute the +numerical damping D +� +�u +� +[t, k]. +(2) (Prediction step - temporal part) For each k such that |k| ≥ κ, compute the predictor �u∗[t, k] +according to +�u∗[t, k] − �u[t, k] +∆t ++ L +� +�u +� +[t, k] + D +� +�u +� +[t, k] = �f[t, k](∆t)− 1 +2 . +(3) (Correction step - spatial part) Compute L +� +�u∗� +[t, k] with (5.3), and the numerical damping +D +� +�u∗� +[t, k]. +(4) (Correction step - temporal part) For each k such that |k| ≥ κ, compute the corrector +�u[t + ∆t, k] according to +�u[t + ∆t, k] − �u[t, k] +∆t ++ L +� +�u∗� +[t, k] + D +� +�u∗� +[t, k] + L +� +�u +� +[t, k] + D +� +�u +� +[t, k] +2 += �f[t, k](∆t)− 1 +2 . +5.2. Discussion of the numerical method. +5.2.1. Discretization of the operators L. For any choice made for the estimation of the deriva- +tives entering in L +� +�u +� +(see (5.2)), the way the wavevector is discretized in a Cartesian fashion is +not well adapted to this numerical problem which has natural spherical symmetry. Besides the +forcing �f, which is only statistically isotropic, the deterministic part of the evolution is spheri- +cally symmetric7. Nonetheless, the way it is presently discretized in a Cartesian form allows an +easy and standard implementation of the DFT (using the Fast Fourier Transform algorithm) such +that the solution u[t, x] in physical space, over the discrete set of positions x = (xi)1≤i≤d with +xi = [0, 1, ..., N/2, −N/2 + 1, −N/2 + 2, ..., −1]∆x, is obtained with an inverse DFT of �u[t, k]. +As we have already mentioned, we cannot make sense of �u(t, k) as a pointwise function of k. +To this regard, the formulation of the problem in Fourier space needs to be justified and a proper +meaning has to be given to the divergence of a rough field, see Sections 3 and 4 for full details. +7Indeed, all quantities involve the vector amplitude |k|. Moreover, L +� +�u +� +can be written as a radial derivative with +respect to |k| (see (3.2)). + +38 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +For the purpose of numerical simulations we have thus proposed (5.3) as a numerical realization +of the derivatives, with the use of back and forth DFTs and an appropriate multiplication by a +space-dependent factor. +5.2.2. Discretization of the operators ∂t. Concerning time integration for a given time stepping ∆t, +we use an explicit predictor-corrector method with a single independent realization of the forcing, +consisting in predicting the solution �u[t+∆t, k] using an explicit Euler discretization scheme taking +previous time step �u[t, k] as an initial condition, and correcting by another Euler scheme which +weighs the initial condition and the prediction equally. +5.2.3. Choice of discretization parameters. Let us give the last relevant parameters that we use +for our simulations, keeping in mind that the spatial resolution ∆x = Ltot/N is determined by +the physical length of the side of the periodic box (henceforth, we choose Ltot = 1 without loss of +generality), and the number of collocation points N in each direction. As we will see, the viscosity +ν is chosen accordingly. +Firstly, the time step ∆t has to be chosen. At this stage, a CFL criterion for the dynamics +proposed in our algorithm is not clear. Thus for the time being, we assume that a reasonable +stability criterion may come from the time derivative and the viscous term. In that spirit, it seems +reasonable to consider the numerical stability of the heat equation, which is imposed by the viscous +term and which requires8 that ν∆t < (∆x)2/2 (see for instance Ref. [37]). +Forthcoming simulations will be done for very high values of the number of collocation points +N in order to consider small values for the viscosity ν. Even in the most comfortable situation +where ν∆t is of order ∆x, the numerical cost will eventually become prohibitive. For this reason, +we choose a time step independent of resolution and dimension d, and given by ∆t = 5 × 10−3. +Additional simulations (data not shown) for d = 1 and d = 2 using ∆t = (∆x)2/2 and for moderate +numbers of collocation points N have given similar numerical results as the ones obtained with +∆t = 5 × 10−3. It is unclear at this stage why this chosen value of the time step does not lead to +numerical instabilities, although we could invoke the fact that viscosities will be chosen very small. +5.3. Simulations. +5.3.1. Determination of viscosity and averaging procedure. All predictions that have been made in +the theoretical sections concern the statistical behavior of the solution u(t, x) of the continuous +problem recalled in (3.12), with or without viscosity. From a numerical point of view, we need +to define a time T∗ at which the system has reached a statistically stationary regime, in which +mathematical expectations will be approximated by an empirical average in time. Because of the +cascading process of energy will eventually populate modes at higher and higher wavenumber k as +time goes on, we introduce viscosity in order to damp all the energy once it has reached the highest +accessible wave-number +kmax := (N/2)∆k. +In an inviscid (ν = 0) regime, energy injected at low wavenumbers eventually reaches kmax at a +time scale of the order of +T∗ := kmax/c. +We choose ν smaller than ck−3 +max to ensure a strong decrease of the spectral density at high wavenum- +bers (see Theorem 3.7). In practice, exploratory simulations will be carried out with various values +of ν, all of them satisfying this above criterion. This criterion is essential to ensure that no waves +are reflected at the boundary of the artificial wavenumber periodic box. We will consider that T∗ +is the time when the statistically stationary regime is reached. +8A more sophisticated scheme could be written including explicitly the exact solution of the underlying heat +equation in the time stepping; this is known as an exponential scheme and would allow for ν∆t of order ∆x. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +39 +We let then the simulation run after T∗, and we will average statistical estimators that we will +define later over a set made of 100 instances every (103∆t) to ensure statistical independence9. +More explicitly, forthcoming time averages ⟨g[t]⟩t of a given function of time g[t] will be taken as +(5.5) +⟨g[t]⟩t := +1 +100 +100 +� +n=1 +g[T∗ + n × (103∆t)]. +Accordingly, we stop the integration in time at +T ∗ = T∗ + 100 × (103∆t). +With these choices, simulations are longer and longer as the number of collocation points N +increases, but it allows for smaller and smaller viscosities, which is necessary to develop an extended +inertial range. +5.3.2. Definition and estimation of key statistical quantities. First of all, the L2-norm σ2 +u[t] of the +solution u[t, x] at a given instant t is defined by +(5.6) +σ2 +u[t] = +� +k +|�u[t, k]|2∆k, +where the sum is taken over all possible values of the discrete wave vector k. Initially, we have +σ2 +u[0] = 0, it will then quickly grow and end up fluctuating around a certain average value way before +T∗ (data not shown). This aforementioned mean value ⟨σ2 +u[t]⟩t, where the time average procedure +is defined in (5.5), could be considered as the variance of the solution, and we have checked that +it is independent of viscosity as expected if ν is chosen small enough. From the physical point of +view, this is consistent with the observation that the variance of the solution of the Navier-Stokes +equations get independent of viscosity at large Reynolds numbers (1.1). +To characterize more precisely the statistical behavior of the solution when the L2-norm starts +fluctuating around a mean value (5.6), we define the energy spectral density estimated as a peri- +odogram, i.e. the norm square of the Fourier mode, that is +(5.7) +�Cu(t, k) = |�u[t, k]|2, +and its averaged version +(5.8) +�Cu(k) = ⟨|�u[t, k]|2⟩t +where the time-average is defined in (5.5). +Another quantity of great importance is the so-called second-order structure function, i.e. the +variance of the increment over a scale ℓ ∈ Rd, and given by +(5.9) +S2(ℓ) = ⟨(u[t, x + ℓ] − u[t, x])2⟩t,x, +where the time average is defined in (5.5) and the additional spatial-average is taken over x ∈ Rd +for a given function of space g[x]: +⟨g[x]⟩x := +1 +Ld +tot +� +x +g[x](∆x)d. +where the sum is taken over all possible values of the discrete wave vector x. +We recall that +concerning solutions of the Navier-Stokes equations, it is observed that S2(ℓ) behaves as |ℓ|2/3 at +infinite Reynolds numbers, corresponding to a regularity of H¨older type with H = 1/3 (c.f. (1.2)). + +40 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +Figure 2. Local and statistical behaviors of the solution u[t, x] to the evolution +provided in paragraph 5.1.3 for space dimension d = 1, for a given viscosity ν +in the statistically stationary range. All simulations have been done with c = 1 +and H = 1/3 (a): spatial profiles of u[·, x] at a given time t in the statistically +stationary regime for ν = 10−8 obtained using N = 210 collocation points. (b): +estimations of the energy spectral density based on the averaged periodograms (see +Section 5.3.2 and Equation (5.8)) of the solution for various values of viscosity +ν = 10−8, 10−9, 10−10, 10−11 and 10−12 (from left to right), using respectively +N = 210, 211, 212, 213 and 214 collocation points. We superimpose with a dashed +line the asymptotic prediction |k|−5/3. (c): same plot as in (a), but for a lower +value of viscosity ν = 10−12. (d): Similar plot as for (b) but for the second order +structure function S2(ℓ) (5.9), i.e. +the variance of the increments, following an +averaging procedure detailed in the text. We superimpose the expected asymptotic +power-law behavior ℓ2/3. +5.3.3. Results and comments in dimension d = 1. We display in Figure 2 the results of our simula- +tions for space dimension d = 1. We have used c = 1 and H = 1/3, and the time step ∆t = 5×10−3, +whose value is motivated in section 5.2.3. As explained in Section 5.3.1, we run the simulation until +T∗ = kmax/c at which all accessible length scales and wave lengths have been populated. After this +transient, we then average various estimators such as the energy spectral density (5.8) and the sec- +ond order structure function (5.9) every 2 units of time. We indeed observe (data not shown) much +before T∗ that the L2-norm of the solution (5.6) fluctuates around a mean value. Typical snapshots +of the solution u[·, x] are displayed in Figs. 2(a) and (c) at a time pertaining to the statistically +stationary regime. A moderate viscosity ν = 10−8 has been used in (a), whereas a much smaller +9The number of instances and the time between each samples are empirically chosen. + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +41 +one ν = 10−12 is used in (c). We can see that as ν gets smaller, the velocity profile develops smaller +scales and for the smallest viscosity that has been considered, it looks rougher. The averaged peri- +odograms (5.8) obtained for all considered viscosities are represented in Figure 2(b). All estimated +power spectral densities coincide at low wave lengths since the same forcing term has been used in +all simulations. Then, at higher wave lengths, the spectra develop a universal power-law, indepen- +dent of the characteristics of the forcing, with an exponent −5/3 which coincides with the expected +−(2H + d) exponent (see Eq. (3.10)) when choosing H = 1/3. Then, at a characteristic wave +length determined by the value of viscosity, spectra undergo a strong decrease, which is reminiscent +of viscous damping. Similar behaviors are observed on the second order structure function S2(ℓ) +(5.9). At large length scales ℓ of the order of the scales where energy is injected into the system, +determined by the spectral support of the forcing, S2(ℓ) is independent of viscosity. At these scales, +larger than the correlation length of the spatial profile of u[t, x], S2(ℓ) is approximatively equal to +two times the variance of the solution. The fact that all curves superimposed at these scales show +that this variance is indeed independent of viscosity. In the so-called inertial range of scales, as +it is recalled in Section 1.2, S2(ℓ) develops a power-law behavior, whose exponent 2H governs the +H¨older regularity of the asymptotic solution. At smaller length scales, viscosity smooths out any +irregular variations such that S2(ℓ) can be Taylor expanded and becomes proportional to ℓ2. +5.3.4. Results and comments in dimension d = 2. We now display in Figure 3 the results of our +simulations for space dimension d = 2. We have used again c = 1 and H = 1/3, and the time step +∆t = 5 × 10−3. Similarly to the d = 1 case, we propagate in time the discrete evolution provided +in paragraph 5.1.3 using the predictor-corrector method until time T∗ at which all accessible scales +have been populated. Before then, viscosity has damped all the energy at small scales such that the +Fourier mode at kmax is exponentially small. Hence, we will consider that starting from time T∗, +the statistically stationary regime has also been reached. We display in Figure 3(a) the amplitude +of the Fourier modes �u[t, k] at a time t lying in the statistically stationary regime, as a function of +the two components kx and ky of the wave vector k, in a logarithmic representation. We can see +that a lot of energy is concentrated along the two lines (kx, 0) and (0, ky), whereas elsewhere in the +plane, up to fluctuations, energy is distributed in a rotation invariant (i.e. isotropic) way. Recall +that in a continuous framework, the statistical properties of Fourier modes �u(t, k) at any time t +are expected to depend only on the amplitude |k|. This shows that our numerical representation +�u[t, k] is intrinsically anisotropic. We interpret this spurious anisotropy as the consequence of the +finiteness of our simulation domain, with the implied finiteness of the resolution ∆k = 1/Ltot, +but also the fact that the Fourier modes �u[t, k] are distributed on a Cartesian grid, whereas the +continuous solution �u(t, k) is expected in average to be spherically symmetric. Another limitation +of our numerical approach is related to the rough nature of the expected solution. It is clear from +the inspection of Figure 3(a) that Fourier modes are correlated and smoother than what is expected +from the Fourier transform of a statistically isotropic random field. This is very possibly related to +the estimation of the divergence operator entering in the evolution (3.12) with finite differences, as +it is implicitly done using back and forth DFTs (see Equation (5.3)). A specifically devoted article +aimed at exploring the numerical representation of the continuous formulation provided in (3.12) +would be needed, designing for instance some finite volume algorithms able to deal with the rough +nature of the underlying fields. It will be the subject of future publications. +Accordingly, the representation of the solution in physical space, that we display in Figure 3(c), +exhibits two types of anisotropies. The first type of anisotropy can be observed along the two +directions x and y of the Cartesian frame as straight lines. +This anisotropy is consistent with +what is observed on the Fourier transform displayed in Figure 3(a) along kx and ky. From the +inspection of Figure 3(c), another type of anisotropy can be evidenced around the origin, i.e. +around x = y = 0. Recall that for a statistically homogenous field, probability laws are expected +to be invariant by translation, and thus the origin of the domain should not play a particular + +42 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +-24 +-16 +-8 +0 +8 +16 +-78 +-46 +-13 +19 +51 +84 +Figure 3. Local and statistical behaviors of the solution u[t, x] to the evolution +provided in paragraph 5.1.3 for space dimension d = 2, for a given viscosity ν in +the statistically stationary range. All simulations have been done with c = 1 and +H = 1/3 (a): Representation of the logarithm of the absolute value of �u[·, k] at a +given time in the statistically stationary regime for ν = 10−9 obtained using N = 211 +collocation points. (b): estimations of the angle-averaged energy spectral density +based on the averaged periodograms (see Section 5.3.2 and (5.8)) of the solution for +various values of viscosity ν = 10−5, 10−6, 10−7, 10−8 and 10−9 (from left to right), +using respectively N = 27, 28, 29, 210 and 211 collocation points. We superimpose +with a dashed line the asymptotic prediction |k|−2H−d, with H = 1/3 and d = 2. +(c): similar plot as in (a) but for the corresponding spatial profiles of u[·, x]. (d) +Similar plot as for (b) but for the second order structure function S2(ℓ) (5.9), i.e. +the variance of the increments, following an averaging procedure detailed in the text. +We superimpose the expected asymptotic power-law behavior ℓ2H, with H = 1/3. +role. In our numerical solution, this is clearly not the case, and we believe that this anisotropy is +related to the aforementioned correlated nature of the Fourier modes displayed in Figure 3(a). Once +again, more work is needed to design proper numerical schemes able to get rid of these spurious +anisotropies. +Nonetheless, despite these anisotropies, that are not present in the solution of the continuous +framework, our numerical solution behaves in a statistical sense in the expected way. For instance, +we display in Figure 3(b) the estimation of the power spectral densities, obtained while averaging in +time the square of the amplitude of Fourier modes. We underline that these spectral densities are +furthermore averaged over the angles that the wave vector k makes with the axes of the Cartesian + +A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS +43 +Figure 4. Local and statistical behaviors of the solution u[t, x] to the evolution +provided in paragraph 5.1.3 for space dimension d = 3, for a given viscosity ν +in the statistically stationary range. All simulations have been done with c = 1 +and H = 1/3 (a): Representation of some slices of u[·, x] at a given time in the +statistically stationary regime for ν = 10−7 obtained using N = 29 collocation +points in each directions. (b): estimations of the angle-averaged energy spectral +density based on the averaged periodograms (see Section 5.3.2 and (5.8)) of the +solution for various values of viscosity ν = 10−3, 10−4, 10−5, 10−6 and 10−7 (from +left to right), using respectively N = 25, 26, 27, 28 and 29 collocation points. We +superimpose with a dashed line the asymptotic prediction |k|−2H−d, with H = 1/3 +and d = 3. +frame, such that displayed spectral densities depend only on the wave length amplitude |k|. We +indeed observe that, as viscosity gets smaller and smaller, spectral densities develop a power-law +behavior in the inertial range of scales, with the expected exponent −(2H + d) which is derived +in (3.10). At higher wavenumbers, Fourier modes are exponentially damped by viscosity. Similar +conclusions could be drawn from inspection of the scale dependence of the second order structure +function S2(ℓ) that is shown in Figure 3(d). Let us mention here that spatial averages have been +obtained over the full spatial domain, averaging furthermore over two structure functions obtained +while considering spatial lags ℓ = |ℓ|ex and ℓ = |ℓ|ey, where ex and ey are the two orthonormal +unit vectors of the Cartesian frame. At the largest scales, above the characteristic ones of the +forcing, we can see that S2(ℓ) reaches a plateau, which gets independent of viscosity as ν → 0. +Similarly to the d = 1 case, above the correlation length of u[t, x], S2(ℓ) coincides with two times +the variance of u[t, x], which says in other words that the variance gets itself independent of viscosity +if ν is chosen small enough. At lower scales, i.e. in the inertial range, S2(ℓ) develops a power-law +behavior of exponent 2H, as it is expected from the predictions made in the continuous framework +(Corollary 4.12), over a range of scales which grows as viscosity gets smaller and smaller. Finally, +at even smaller scales, viscous effects dominate and smooth out the spatial profiles, such that S2(ℓ) +gets proportional to ℓ2. + +44 +G. B. APOLIN´ARIO, G. BECK, L. CHEVILLARD, I. GALLAGHER, AND R. GRANDE +5.3.5. Results and comments in dimension d = 3. Let us now finish this Section devoted to numer- +ical simulations by presenting in Figure 4 the results for space dimension d = 3. Let us mention +that in a three-dimensional setup, simulations are much more demanding from a computational +perspective, and the cost of performing back and forth FFTs gets much higher, because derivatives +along the 3 directions have to be considered, and also because the number of discretization points, +N3, increase tremendously as N increases. For these reasons, and because we are propagating the +integration in time from a vanishing initial condition towards the statistically stationary regime +before taking averages, we could not go above N = 29 = 512 collocation points along each direction. +Consequently, we have not been able to perform simulations for viscosities smaller than ν = 10−7. +Nonetheless, we observe (data not shown) that fluctuations as quantified by the L2-norm (5.6) get +independent of viscosity in a good approximation starting from ν = 10−6, as it is expected from +the behavior of the solution in a continuous framework. As mentioned, and similarly to the d = 1 +and d = 2 cases, we go through the transient while integrating the solution until time T∗, and only +then we start taking averages. +We display in Figure 4(a) a rendering of our three-dimensional simulations in physical space +in the Cartesian frame. For the sake of clarity, and because visualizations gets more complicated, +we only show three slices along the planes (x, y, 0), (x, 0, z) and (0, y, z). Bright and dark colors +correspond respectively to large positive and large negative fluctuations of the solution, similarly +to what has been observed for the d = 2 case, which is displayed in Figure 3(c). Once again, we +observe the two types of anisotropies that we evidenced in the d = 2 case, one along the three +directions of the Cartesian frame, and one around the origin. +Nonetheless, as it is displayed in Figure 4(b), the simulations behave as expected in a statistical +manner, as it can be observed on the power spectral densities (5.8). Once again, these densities are +not only averaged in time once the statistically stationary regime has been reached, but they are +also averaged over the two angles that the wave vector k does with the Cartesian axes, such that +�Cu(k) is a function of the norm |k| only. We can see that at large scales, i.e. at low wavenumbers +|k|, fluctuations are independent of viscosity, even for the highest value ν = 10−3 which is used +for the lowest number of collocation points N = 25. It is nonetheless crucial to consider smaller +values of viscosity, necessitating thus higher values of N, up to N = 29, in order to develop an +extended inertial range. It is clear for the smallest value of viscosity (ν = 10−7 for N = 29) that +the spectral density has developed a power-law behavior in the inertial range, with the expected +exponent −(2H + d), in a consistent manner with our theoretical predictions (3.10). +We thus see that we are able to give an appropriate numerical representation of the continuous +framework using the DFT to define the divergence operator entering in the spectral evolution pro- +vided in (3.12). In particular, we are able to reproduce in a discrete setup the statistical behaviors +of the spectral densities (Eq. 3.10) and second-order structure functions (Corollary 4.12), and their +related power-law behaviors. Nonetheless, more work is needed to get rid of the anisotropies that +are clearly observed for the d = 2 and d = 3 cases. To do so, a promising direction could be +given while designing a finite volume scheme able to respect the inherent spherical symmetry of +the deterministic part of the evolution (3.12). This is required in order to propose a realistic model +of fully developed fluid turbulence that could be used in an efficient way in various applications, +in which spatial fluctuations of the velocity field have crucial consequences on the evolutions of +dynamical quantities of interest. We keep these important developments for future investigations. +References +[1] J. Almonacid and N. 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Valdettaro, Inertial waves in a rotating spherical shell: attractors and +asymptotic spectrum, Journal of Fluid Mechanics, 435 (2001), pp. 103–144. +[40] M. Rieutord and L. Valdettaro, Inertial waves in a rotating spherical shell, Journal of Fluid Mechanics, +341 (1997), pp. 77–99. +[41] H. Scolan, E. Ermanyuk, and T. Dauxois, Nonlinear fate of internal wave attractors, Physical review letters, +110 (2013), p. 234501. +[42] H. Tennekes and J. L. Lumley, A first course in turbulence, MIT press, 1972. +[43] V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov spectra of turbulence I: Wave turbulence, +Springer Science & Business Media, 2012. +Theoretical Physics I, University of Bayreuth, Universit¨atsstr. 30, 95447 Bayreuth, Germany +Email address: gapolinario@uni-bayreuth.de +Univ Rennes, IRMAR UMR 6625 & Centre Inria de l’Universit´e de Rennes (MINGuS) & ENS Rennes, +France +Email address: geoffrey.a.beck@inria.fr +Univ Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, 46 all´ee d’Italie, +69342 Lyon, France +Email address: laurent.chevillard@ens-lyon.fr +D´epartement de math´ematiques et applications, ´Ecole normale sup´erieure, CNRS, PSL University, +and Universit´e Paris Cit´e, 75005 Paris, France +Email address: isabelle.gallagher@ens.fr +D´epartement de math´ematiques et applications, ´Ecole normale sup´erieure, CNRS, PSL University, +75005 Paris, France +Email address: ricardo.grande@ens.fr + diff --git a/A9AyT4oBgHgl3EQf3_rL/content/tmp_files/load_file.txt b/A9AyT4oBgHgl3EQf3_rL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39337f2dec0a0c1d7ff245e728ee4c318d15807e --- /dev/null +++ b/A9AyT4oBgHgl3EQf3_rL/content/tmp_files/load_file.txt @@ -0,0 +1,1913 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf,len=1912 +page_content='A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS GABRIEL B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, GEOFFREY BECK, LAURENT CHEVILLARD, ISABELLE GALLAGHER, AND RICARDO GRANDE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In hydrodynamic turbulence, when the Reynolds number is large, the velocity field of the fluid becomes irregular and the rate of energy dissipation re- mains bounded from below even if the fluid viscosity tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A mathematical description of the turbulent cascade is a very active research topic since the pioneering work of Kolmogorov in hydrodynamic turbulence and that of Zakharov in wave turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In both cases, these turbu- lent cascade mechanisms imply power-law behaviors of several statistical quantities such as power spectral densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For a long time, these cascades were believed to be associated with nonlinear interactions, but recent works have shown that they can also take place in a dynamics governed by a linear equation with a differential operator of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In this spirit, we construct a linear equa- tion that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In the Fourier variable, this equation can be seen as a linear transport equation, which corresponds to an operator of degree 0 in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Background and motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This work is mainly motivated by some important aspects of the phenomenology of three-dimensional homogenous and isotropic fluid turbulence [34, 42, 20], of which several aspects have been also observed and formalized for waves in various situations when they are weakly interacting [43, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As has been repeatedly observed in geophysical and laboratory flows, and in numerical simulations of the incompressible Navier-Stokes equations, a fluid that is stirred by a statistically stationary random force f(t, x), assumed to be smooth in space, will eventually reach a statistically stationary state in which the velocity variance is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' To dissipate all the energy that is constantly injected into the system in such an efficient way, the velocity field of that fluid will develop a complex multiscale structure ending up with high values of velocity gradients such that viscosity can easily transform mechanical energy into heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In other words, the fluid has transferred the energy pumped at large scales by the forcing towards small scales, at which viscous diffusion efficiently acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This picture is known as the cascading process of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The purpose of this article is to model and reproduce this phenomenon of transfer of energy as a cascading process through the scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We propose a partial differential equation, which is of course much simpler than the nonlinear Navier-Stokes equations, stochastically forced by an additive random force f(t, x) that we take to be smooth in space and correlated over a typical large lengthscale (known in the turbulence literature as the integral lengthscale), whose solution develops roughness as time goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' More precisely, our goal is to generate rough fractional Gaussian H¨older continuous random fields of parameter H (see for instance the textbook [16]) from smooth forcing through a dynamical evolution, which can be seen as a simple stochastic representation of the phenomenology mainly developed by Kolmogorov [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='00780v1 [math-ph] 2 Jan 2023 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE As mentioned earlier, a striking feature of three-dimensional turbulent motion is its ability to efficiently dissipate the energy that is injected at large scales in a statistically stationary and homo- geneous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' To be more precise, let us consider a solution of the incompressible Navier-Stokes equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' a divergence-free velocity field u(t, x) ∈ R3 with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This dynamics is stirred by a divergence-free vector forcing term f(t, x), that we take delta-correlated in time and smooth in space, say Gaussian, of zero-average and of covariance E [f(t, x) ⊗ f(s, y)] = δt−sCf(x − y), where ⊗ stands for the matrix product, and the matrix Cf(x) is made of a linear combination of the matrix x ⊗ x and the identity, with multiplicative coefficients depending only on |x|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' a typical covariance matrix of a statistically homogeneous and isotropic vector field [6, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We furthermore require that these scalar functions of |x| are smooth and compactly supported over a range of the size order of the aforementioned large length scale, so as to mimic the energy injection at the so- called integral length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As time goes on, it has been repeatedly observed that the velocity field u reaches a statistically stationary state, which is furthermore statistically homogeneous, of finite variance, and with the additional striking property that it becomes independent of the viscosity ν as ν goes to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) lim ν→0 lim t→∞ E � |u(t, x)|2� < +∞ for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The former asymptotic behavior of the velocity variance illustrates clearly how a turbulent fluid can dissipate energy with high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For instance, in the same setup but considering the heat equation instead of the Navier-Stokes equations, a statistically stationary regime would also be reached at t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, the variance of the solution is then inversely proportional to the viscosity ν, see [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Instead, turbulent motion dissipates energy in a way that the velocity variance is eventually independent of viscosity, which is a far more efficient way of dissipating energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In order to ensure the independence of said variance on viscosity, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1), the fluid develops a rough behavior of H¨older-type at small scales, in such a way that the variance of the velocity increments asymptotically behaves as follows: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) lim ν→0 lim t→∞ E � |u(t, x + ℓ) − u(t, x)|2� ∝ |ℓ|→0 |ℓ|2H for all x, where the power-law exponent is determined by Kolmogorov’s prediction H ≈ 1/3 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Much more could be said on a more precise characterization of the distribution of the increments than only its variance, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2), such as its higher order moments that quantify its non Gaussian, skewed and intermittent nature [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In this article, we will focus on a second-order modeling of these fluctuations, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' we leave finer descriptions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A first precise formalization of the cascade phenomenon could be built by imposing a particular dynamical relation between the coefficients of a decomposition of the velocity field, such as a continuous wavelet transform or a discrete (dyadic) decomposition on a tree [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This has been explored in the literature [5, 4, 13] leading to precise statements on H¨older regularity and its relationship with scaling behaviors of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Although great progress has been made in the understanding of such models and their formalization, which usually exploits a typical quadratic interaction between neighboring coefficients, these approaches often avoid the important question of the relation of these coefficients in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This is necessary in order to design a model that leads to statistically homogeneous velocity fields, as observed in nature and in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Nonetheless, these models can be seen as a sophistication of the so-called shell models1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In this spirit, we believe an important step was made in [32], where the authors investigate a simple linear 1See for instance [10, 9] which consist in exploring quadratic interactions between shells, that share some behaviors with velocity Fourier modes and wavelet coefficients, along a single branch of a tree decomposition, lacking thus a discussion of the spatial relationships between coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 3 relation between shells, which is shown to be able to transfer energy from large to small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let us also mention [30] where some ideas to build a PDE from these shell models are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' From a somewhat different side of fluid mechanics, more focused on the implications of global rotation [40, 39] or stratification of the density field [29, 41] on a flow, it has been evidenced a phenomenon of focusing of waves onto attractors, whose precise shape are determined by the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Based on a linearization of the fluid equations, this phenomenon has been interpreted as a cascading process through scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' These ideas have been then formalized and rigorously studied from a mathematical viewpoint in a series of recent articles [17, 19], which underline the importance of operators of degree 0 as a deterministic mechanism able to transfer energy through scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The rough and disordered nature of a turbulent velocity field u(t, x) has been repeatedly observed in laboratory and numerical flows, and in geophysical situations [42, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' From this signal, considering for instance a component of the velocity vector field as a function of space, depending on the experimental possibilities and the large-scale geometry of the flows, one can construct the energy spectrum |k| �→ E|�u(t, k)|2 where �u stands for the spatial Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' According to the standard phenomenology of fluid turbulence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' which has been multiply confirmed by observations in very different situations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the energy-spectrum resembles a curve [42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 20] that can be schematically decomposed as follows: (injection range) for small |k| of the order of the characteristic wavelength of energy injec- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the energy-spectrum is mainly determined by the forcing and the associated large-scale geometry of the flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (inertial range) for intermediate |k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the energy-spectrum develops a power-law behavior whose exponent is found universal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' independent of viscosity and of the nature of the flow, and can be interpreted as the generation of small scales by the internal motion of the fluid following a transfer of energy from small wave-numbers to large wave-numbers, (dissipative range) for large |k|, the energy-spectrum is governed by dissipation processes which damp efficiently all the energy coming from the large scales, making the spatial velocity profile a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The intermediate range of scales, called the inertial range in the turbulence literature [42, 20], is where this mechanism of transport of energy takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The universally observed power-law exponent of the energy-spectrum can be written as −(2H + d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' E|�u(t, k)|2 ∼ |k|−(2H+d), where we have introduced for the sake of generality the space dimension d, and the parameter H that will be eventually interpreted as a Hurst, or H¨older, exponent, in a statistically averaged sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In real situations, for d = 3, it is indeed observed that H ≈ 1/3, as predicted by dimensional arguments mainly attributed to Kolmogorov [24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The main goal of this paper is to propose a family of partial differential equations, such that, when stirred by a statistically stationary and spatially homogenous, smooth in space forcing term, its solution u(t, x) reaches at long times a statistically stationary state which displays the typical spectral behavior detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We will achieve this with the following transport equation in Fourier space: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) � � � � � � � � � ∂t�u(t, k) + divk � ck |k| �u(t, k) � + cH + 1 2 |k| �u(t, k) = �f(t, k) t > 0, k ∈ Rd, |k| > κ > 0, �u(t, k) = 0 t > 0, k ∈ Rd, |k| ≤ κ, �u(0, k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Here H ∈ R and κ > 0 are fixed, and the source f satisfies E[f(t, x)f(s, y)] = δt−s Cf(x − y), 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE where Cf is smooth and satisfies some additional assumptions detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Our main result is the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let H ∈ (0, 1) and let the forcing f be (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) f(t, x) = � Rdy ϕ(x − y) dW(t, y), where dW is a space-time Gaussian real white noise and ϕ ∈ S(Rd x) is a radial function such that �ϕ(k) = 0 for all |k| < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (i) The transport equation in wavenumber space (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) with source (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) can be rigorously formu- lated in physical space as an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' well-posed PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, at any t > 0, the solution u(t, x) has finite variance and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' smooth paths with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (ii) As t → ∞, u(t, x) converges to a zero-mean Gaussian field u∞(x) which has a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' α-H¨older continuous paths for any 0 < α < H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (iii) The correlations are given by E[u∞(x1)u∞(x2)] = C(d, H) KH(x1 − x2) − (KH ∗ JH)(x1 − x2), where KH := F−1 � χ|k|>κ|k|−(2H+d)� , while C(d, H) is an explicit constant and the function JH ∈ S(Rd x) depends explicitly on ϕ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A more detailed version of this result is presented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4 page 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The parameter κ can be chosen as the smallest non-vanishing wavenumber in the support of the Fourier transform of the forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Following the analogy with the Navier-Stokes equations presented in the introduction, κ may be interpreted as a quantity linked to the inverse of the integral lengthscale2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e the typical lengthscale of the correlations of the forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The fact that our force acts at large but finite scales means that κ is small but non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The kernel KH is a function when H ∈ (0, 1) and κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, when κ = 0 we have the following operator: KH −→ κ→0 (−∆)− H+ d 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that the limiting Gaussian field u∞ shares some statistical properties, such as roughness, with statistical homogeneous fractional gaussian fields [16, 28] defined by (−∆)− H+ d 2 2 dW, that are classically encountered in the turbulence literature [25, 12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, for H ∈ (0, 1) both have a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' α-H¨older continuous paths for any 0 < α < H and one can show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5) u∞ = (in law) C(d, H)F−1 � χ|k|>κ � ∗ (−∆)− H+ d 2 2 dW − ureg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Here, ureg is a smooth zero-mean Gaussian field with correlations E[ureg(x1)ureg(x2)] = (KH ∗ JH)(x1 − x2), 2If Lf is the integral lengthscale, then there exists two real positive numbers a < b such that the support of �f is contained in the annulus of inner radius a Lf and outer radius b Lf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Thus one may set κ = a Lf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 5 which is a smooth function with respect to (x1 − x2) even if H ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The case H ∈ [−d/2, 0] will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' When H ∈ [−d/2, 0], as t → ∞, u(t) still converges to a zero-mean Gaussian field u∞, but this field is not necessarily H¨older continuous with respect to x anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In this case, one may view u∞ as a distribution living in the dual of an appropriate test function space T , see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The correlation structure of the limiting Gaussian measure is given by: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6) E[ lim t→∞⟨u(t), g1⟩⟨u(t), g2⟩] = � Rd k χ|k|>κ |k|−(2H+d) � C(d, H) − � JH(k) � �g1(k) �g2(k) dk for any test functions g1, g2 ∈ T , where ⟨·, ·⟩ stands for the duality product in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5, we show that, for any test functions in T ∩S(Rx d), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6) yields a rate of convergence proportional to (ct)−(2H+d+2n) for n as large as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6) corresponds to the energy-spectrum picture described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, |k|−(2H+d) corresponds precisely to the inertial range previously described, while � JH(k) captures the contribution from the forcing, which is a correction in the injection range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In fact, if the source is spectrally supported in small wavenumbers, then � JH(k) vanishes in the inertial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Finally, let us highlight the difference between the properties of the solution at finite and infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' At finite time, the solution is smooth with respect to x, whereas at infinite time the solution is only H¨older continuous (or even rougher if H ≤ 0, as explained in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This loss of regularity at infinite time is what is expected in linear turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Turbulence is usually associated to a nonlinear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For example, in the case of wave turbulence, nonlinearities create wave interactions which allow the transfer of energy to higher and higher wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Such transfers of energy typically result in a loss of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, nonlinearities might not be the only way in which such loss of regularity can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Colin de Verdi`ere and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Saint-Raymond have shown that, in the context of internal waves, a loss of regularity can also take place in the case of a linear equation with an operator of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Linear equations with operators of degree 0 are also common whenever one introduces a dispersive perturbation in a hyperbolic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In such cases, these operators of degree 0 are used to model wave propagation under strong dispersive effects and they are responsible for memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For example, in the context of wave-energies, the second author and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Lannes show that the waves generated by a moving floating object are governed in the linear regime by a non-local transport equation of degree 0, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In the context of electrical circuits, there are cases in which 1D models of electromagnetic waves propagating along a coaxial cable are governed by operators of degree 0, see for instance [7, Chapter 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' One issue of our model is that it only features a single H¨older exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The velocity field of a concrete turbulent fluid consists of many H¨older exponents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the H¨older-regularity of the velocity field u(t, x) around a point x ∈ Rd depends on the point itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This is known as the multifractal formalism [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The term multifractal refers to the fact that the sets of points with same regularity are often fractal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, our model does not capture finer descriptions (beyond the variance) of the distribution of the increments of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Such descriptions should quantify its non-Gaussian and intermittent nature [15], and therefore our linear model does not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' It is known that one can construct a multifractal and intermittent field with the theory of Gaussian multiplicative chaos [38], however our actual goal is to obtain a multifractal and intermittent field dynamically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' as the solution to a non-linear equation forced by a white-noise in time that admits a rigorous mathematical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We can also consider a forcing which is not a white-noise in time whose temporal correlation function is given by an oscillating function 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE in order to make a comparison with [17, 21, 1] and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Finally, we could investigate other linear models of cascade such as in the case of a compact operator plus a potential of degree 0 as in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' These issues will be tackled in future papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In Section 2, we present a simple transport equation that converges to a complex white noise (up to lower order terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A small tweak to this model allows us to construct a model that gives rise to a real white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In Section 3, we explain how to generalize the latter model to higher dimensions and give a heuristic proof of the main results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In Section 4, we provide a mathematically rigorous study of our model: we introduce the right functional setting, we develop a global well-posedness theory and give a complete description of the asymptotic behavior of the solution, as well as its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This constitutes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Finally, in Section 5 we propose a numerical method and conduct numerical simulations in dimensions 1, 2 and 3 to validate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We thank Oliver B¨uhler for his interesting suggestion about replacing the pure transport term ∂k in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) by sign(k)∂k as a way of fixing the physically undesirable behavior of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' All five authors were funded by the Simons Collaboration Grant on Wave Turbulence, Simons Award ID: 651475 and 651675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For an integrable function f : Rd → C, we denote by �f its Fourier transform, namely ∀k ∈ Rd k, �f(k) := Ff(k) := � Rdx e−2πix·k f(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Whenever defined, the inverse Fourier transform is ∀x ∈ Rd x, f(x) = F−1 �f(x) = � Rd k e2πix·k �f(k) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' It is well know that the Fourier transform is an isometry from L2(Rd x) to L2(Rd k), from S(Rd x) to S(Rd k), where S(Rd x) denote the space of Schwartz functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' smooth functions whose derivatives are rapidly decreasing), and from S′(Rd x) to S′(Rd k) where S′(Rd x) denote the space of tempered distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the dual space of S(Rd x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We will denote by ⟨·, ·⟩ the duality product between S′ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We will also need some spaces that quantify the regularity of functions more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For a fixed integer n, Sobolev spaces are defined by Hn(Rd) := {u ∈ L2(Rd) | ∂j xiu ∈ L2(Rd) with 1 ≤ i ≤ d and 0 ≤ j ≤ n}, and their dual spaces are denoted by H−n(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We will denote by ⟨·, ·⟩H−n,Hn the duality product between Hn and H−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For α ∈ (0, 1) the H¨older space C0,α(Rd) is defined by C0,α(Rd) := {u continuous and bounded | ∃C > 0, ∀x, ℓ ∈ Rd, |ℓ| ≤ 1, |δℓu(x)| ≤ C|ℓ|α}, where δℓ denotes the increment defined by δℓu(x) := u(x + ℓ) − u(x) for x, ℓ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We will denote by χA the characteristic function3 of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let (Ω, σ(Ω), P) be a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A Gaussian field u : Rd → L2(Ω) is a field such that for all n ≥ 1 and for all (x1, x2, · · · , xn) ∈ (Rd)n, the random vector (u(x1), u(x2), · · · , u(xn)) is a Gaussian random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' When d = 1, a 1D Gaussian field is usually called a Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A Gaussian random measure µ acting on S(Rd) is a random tempered distribution 3This means that χA(k) = 1 if k ∈ A and χA(k) = 0 if k /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 7 such that for every g ∈ S(Rd), the random variable ⟨µ, g⟩ is a centered Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A white noise dW(x) is a Gaussian random measure acting on L2(Rd) that satisfies the following for any functions f, g ∈ L2(Rd) E ��� Rd f(x)dW(x) � �� Rd g(x)dW(x) �� = � Rd f(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Since we always integrate a deterministic function against dW(x), the choice between Itˆo and Stratonovich integrals is unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Sometimes we will change variables when integrating against a white noise measure: � t 0 f(t − s)dW(s) = � t 0 f(s)d� W(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In such cases, we will use the d� W to denote the new white noise measure after this change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' One-dimensional transport in wavenumber space 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Building one-dimensional white noise: real vs complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In order to mimic the trans- port of energy from large scales to small scales, the authors in [3, 2] proposed a simple transport equation in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' To present these ideas, we first consider a one-dimensional model for a velocity field u(t, x), whose spatial Fourier transform aims to solve the linear evolution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) � ∂t�u(t, k) + c ∂k�u(t, k) = �f(t, k), (t, k) ∈ (0, ∞) × R, �u(t, k)|t=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Here c > 0 is fixed and can be viewed as a transport rate in wavenumber space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' On the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1), we have included an additive term �f which is the Fourier transform of a spatial forcing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The support of �f is localized at small wavenumbers, which is consistent with the assumption that the forcing term in physical space acts at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we can see, the dynamical evolution proposed in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) is a genuine transport equation, and only the presence of a forcing makes it inhomogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Adopting such a setup immediately imposes the complex nature of the velocity field in physical space, as it can be seen when formally taking the inverse Fourier transform of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1), and obtaining the following evolution in physical space: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) � ∂tu(t, x) − 2πicx u(t, x) = f(t, x), (t, x) ∈ (0, ∞) × R, u(t, x)|t=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that the operator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) corresponds to multiplication by the space variable 2πicx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In [3], it is shown that when the forcing f is a white noise in time and statistically homogeneous in space, then the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2), u : (0, ∞)t × Rx −→ C, converges4 to a complex white noise in space as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In other words, the evolution that has been proposed, expressed in Fourier space as a transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) and in physical space as an equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) involving an operator of degree 0 (multiplication by −2πicx), is able to transfer energy through scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, the solution is statistically homogeneous at any time and it develops the regularity of a white noise as time goes on (technically it’s a sudden drop in regularity at t = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' To complete the program suggested by the phenomenology of turbulence, the additional linear action of a fractional operator allows, in a similar setup, to generate a solution with asymptotic H¨older-type regularity of parameter H ∈ (0, 1) instead of the one of the white noise, as explained in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' If some energy is introduced by the forcing at a negative wavelength k < 0, the transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) will move it to smaller negative wavenumbers, going through k = 0, and then to 4Up to lower order terms, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4 for a full asymptotic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE infinitely large positive wavenumbers k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In order to avoid this pathological behavior, one could replace ∂k by ∂|k| = sign(k)∂k which leads to a transport in the direction of |k| instead of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note however that sign(k)∂k is not properly defined at k = 0, and so one needs to be careful in order to propose a well-posed mathematical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' With this in mind, the heart of this article will be the theoretical and numerical study of the following formal evolution: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) � � � � � ∂t�u(t, k) + c ∂|k|�u(t, k) = �f(t, k), (t, k) ∈ (0, ∞) × R, �u(t, k)|t=0 = 0, �u(t, k)||k|=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that it is necessary to introduce a transmission condition between negative and positive k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3), we have decided to add the boundary condition �u(t, k)||k|=0 = 0 to decouple negative from positive wavenumbers, so that no energy crosses k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In particular, this means that the integral over space of u is zero for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This new dynamics proposed in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) can be written in physical space after formally applying the inverse Fourier transform: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) � � � � � � � ∂tu + 2πc xH(u) = f, (t, x) ∈ (0, ∞) × R, u|t=0 = 0, � Rx u dx = 0, where H denotes the Hilbert transform defined in the usual way: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5) Hf(x) := 1 πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' � Ry f(y) x − y dy = − 1 π lim κ→0+ � ∞ κ f(x + y) − f(x − y) y dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Notice that we have used the fact the integral of u(t, x) over space vanishes to get the expression of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Notice also that, despite the fact that the spectral evolutions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) (whose equivalent expressions in physical space are provided respectively in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4)), look very similar, the solution to the new dynamics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) is now real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Equivalently, the dynamics in Fourier space (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) conserves the Hermitian symmetry of an appropriate initial condition, here assumed to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, the solution u : (0, ∞)t × Rx → R of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) can be shown to asymptotically converge to a real white noise in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we will explain in the sequel, the additional linear action of a fractional operator will allow the generation, from smooth forcing, of a real fractional Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Finally, it is tempting to generalize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) to higher dimensions by replacing ∂|k| by k |k| · ∇k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we will develop in Section 4, this eventually generates a statistically homogeneous and isotropic solution that will converge to a real d-dimensional Gaussian random measure which is rougher than a white noise (in space) whenever d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we will explain in the sequel, the additional linear action of a fractional operator will us to generate a real d-dimensional Gaussian random measure with the desired H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Interestingly, it is not obvious to generalize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) to space dimension d ≥ 1 which would generate a similar statistically homogeneous and isotropic solution in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We provide at the end of the section some additional discussions on this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, for the time being we focus on developing a good understanding in the one-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In the case of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4), we have the following result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let the forcing f in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) be (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6) f(t, x) = � Ry ϕ(x − y) dW(t, y), where dW is a space-time Gaussian real white noise, and ϕ ∈ S(R) is a non-negative, non- identically null, even function with null average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Then: A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 9 (i) Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) admits a global (in time) solution u(t, x), which is a Gaussian process with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' smooth paths in x, and with α-H¨older continuous paths in t for any 0 < α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (ii) As t → ∞, the solution u(t) converges in S′(R) to a random Gaussian measure u∞ acting on S(R) with zero-mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' for any g ∈ S(R), E[⟨u∞, g⟩] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (iii) We have the following asymptotic behavior: E[⟨u∞, g1⟩⟨u∞, g2⟩] = lim t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩] = C � Rx g1(x) g2(x) dx − � Rx×Ry I(x − y) g1(x) g2(y) dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7) for any g1, g2 ∈ S(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Here C > 0 is a constant and I is an explicit continuous, even function that depends on ϕ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A more detailed version of this result is presented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5, see also Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that the first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7) corresponds to a delta function, while the second term given by I is a smooth lower order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The forcing introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6) is indeed a Gaussian white noise in time and statis- tically homogeneous in space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8) E[f(s, x)f(t, y)] = Cf(x − y) δs−t , where the spatial correlation function Cf = ϕ∗ϕ is a convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Given that ϕ ∈ S(R), Cf ∈ S(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The constant C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7) is precisely C = Cf(0) 2c = 1 2c � Rx |ϕ(x)|2 dx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The solution to the stochastic PDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) is an explicit Gaussian Itˆo process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A precise formula will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Even if this solution is continuous in time and space, one can lose regularity at t = +∞, which is why one needs to consider u∞ on the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7) as a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The proof of this theorem is posponed to the next section where a more general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' multidimesional white noise, will be tackled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Before we develop the techniques needed to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1, it is important to understand the asymptotic behavior of solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) in the complex setting, which is less technical and informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In this setting, we have the following result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let the forcing f in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) be (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='9) f(t, x) = � Ry ϕ(x − y) dW(t, y), where dW is a space-time Gaussian complex white noise, and ϕ ∈ S(R) is a complex, non-identically null, even function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Then: (i) Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) admits a global (in time) solution u(t, x), which is a Gaussian process with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' continuous paths in time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (ii) As t → ∞, the solution u(t) converges (in S′(Rd)) to a random Gaussian measure u∞ acting on S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE (iii) We have the following asymptotic behavior (in the sense of distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For any g1, g2 ∈ S(R), E[⟨u∞, g1⟩⟨u∞, g2⟩] = lim t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩] = 1 2c Cf(0) � Rz g1(z)g2(z)dz + 1 2πic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' � Rz Cf(z) z �� Ry g1(z + y)g2(y)dy � dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10) The function Cf = ϕ ∗ ϕ is the spatial correlation function given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='11) E[f(s, x)f(t, y)] = δs−t Cf(x − y), and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Cf(z) z is the principal value of the distribution Cf(z)/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' That is, for any test function g ∈ S(R), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='12) � p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='Cf(z) z , g � := p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' � R Cf(z)g(z) z dz = � ∞ 0 Cf(z) g(z) − g(−z) z dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2, the forcing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='9) is a complex Gaussian white noise in time and statistically homogeneous in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, the solution admits an explicit formula: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='13) u(t, x) = � t 0 � Ry e2πicx(t−s) ϕ(x − y) dW(s, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The same comments as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3 apply in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10) remains valid when testing against functions with a finite number of derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, u∞ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10) can also be interpreted as a Gaussian random measure in H−n(R) for any integer n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' More precisely, we will show that for any test functions g1, g2 ∈ Hn(R) one gets E[⟨u(t), g1⟩H−n,Hn⟨u(t), g2⟩H−n,Hn] ∼ t→∞ Cf(0) 2c � R g1(z)g2(z) dz + 1 2πic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' � Rz Cf(z) z �� Ry g1(z + y)g2(y)dy � dz + d(n) ∥g1∥Hn ∥g2∥Hn � 1 ct �n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='14) where d(n) depends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This characterizes the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' One interpretation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10)) is that the correlation function (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the real part of it) asymptotically behaves like a white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, this theorem gives a lower-order correction, in the sense that the regularity of the correction is higher than that of the white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7), such a regular correction is given by a Schwartz function, whereas in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10), the regular correction is a purely imaginary principal value which has no singularity at zero: by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='12), the principal value is “controlled” by C′ f(0) near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' It is important to note that in both cases the regular correction is fast-decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that we recover the result in proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1 in [3], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='15) lim t→∞ E[u(t, x)u(t, y)] = 1 2c Cf(0)δx−y A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 11 as long as one only tests against even functions with respect to the variable x − y, as is easily seen from the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' One way to recover a result similar to that in [3] that holds for all test functions is to define the function v(t, x) = e−πictxu(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This function now satisfies E[v(t, x)v(t, y)] = t sinc (ct(x − y)) Cf(x − y) where sinc(x) := sin(πx) πx denotes the normalized sinc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' It immediately follows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='16) lim t→∞ E[v(t, x)v(t, y)] = Cf(0) c δx−y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, it is unclear whether this transformation of u is an interesting object from the physical viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Assuming one can take the Fourier transform, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='16) can be rewritten in wavenumber space as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='17) lim t→∞ E[�v(t, k)�v(t, k′)] = lim t→∞ E � �u (t, k + πct) �u (t, k′ + πct) � = δk−k′ Cf(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The transformation given by v is therefore equivalent to computing the correlation between the k+πct and k′ + πct Fourier modes as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' First of all, note that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) admits the explicit solu- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='13) thanks to the Duhamel formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Step 1: The solution u is a well defined Gaussian field whose limit at t → ∞ is a Gaussian random measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Clearly, u(t, x) is a well defined Itˆo process with zero average and variance (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='18) E[|u(t, x)|2] = � t 0 � Ry |ϕ(x − y)|2 dyds = t ∥ϕ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For any test function g (we will soon see that actually g ∈ H1(R) suffices), one gets ⟨u(t), g⟩ = � t 0 � Ry �� Rx e2πicx(t−s) ϕ(x − y) g(x) dx � dW(s, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Since Brownian motion has independent, stationary increments we can rewrite the above equation as ⟨u(t), g⟩ = � t 0 � Ry �� Rx e−2πicxs ϕ(x − y) g(x) dx � d� W(s, y) where d� W(s, y) is another Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As t → ∞, we find that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='19) lim t→∞⟨u(t), g⟩ = � ∞ 0 G(s, y) d� W(s, y) where G(s, y) = � Rx e−2πicxs ϕ(x − y) g(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This limit is justified only if G ∈ L2(R+ s × Ry), which we set out to prove next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Firstly, the Young convolution inequality immediately yields ∥G∥L2([0,1]s×Ry) ≤ ∥ϕ∥L1 ∥g∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Next, to handle the case |s| ≥ 1, we assume that g ∈ H1(R) and we integrate by parts: 2πics G(s, y) = − � Rx e−2πicxs ∂x � ϕ(x − y) g(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE By the Young convolution inequality, ∥2πicsG(s, y)∥L2y ≤ ��g′�� L2 ∥ϕ∥L1 + ∥g∥L2 ��ϕ′�� L1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Thus one easily finds that: ∥G(s, y)∥L2(R+ s ×Ry) ≲ 1 c ∥g∥H1 ∥ϕ∥W 1,1 , which justifies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Step 2: Exchanging expectation and limit as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' One could directly use the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='19) in order to compute the correlations of the limiting Gaussian measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, this doesn’t allow us to quantify the speed of convergence as t → ∞, so we take a slightly different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We would like to show that E[ lim t→∞ ⟨u(t), g1⟩⟨u(t), g2⟩] = lim t→∞ E[⟨u(t), g1⟩⟨u(t), g2⟩], which happens to be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We justify this exchange of expectation and limit using the Dominated Convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' It suffices to show that there exists a random variable X with finite expectation such that |⟨u(t), g1⟩⟨u(t), g2⟩| ≤ X ∀t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In fact one can choose X := sup t≥0 |⟨u(t), g1⟩|2 + sup t≥0 |⟨u(t), g2⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In order to show that EX < ∞, we use the Monotone Convergence theorem and Doob’s submartin- gale inequality (see for instance Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8 in [22]): E[ sup t≥0 |⟨u(t), g⟩|2] = lim N→∞ E[ sup 0≤t≤N |⟨u(t), g⟩|2] ≤ lim N→∞ 4 E[|⟨u(N), g⟩|2] = 4 ∥G(s, y)∥2 L2(R+ s ×Ry) , which is finite by Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Step 3: Calculation of the correlations as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We start by computing the correlations for a finite time t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' E[u(t, x1)u(t, x2)] = � t 0 e2πic(x1−x2)sCf(x1 − x2) ds = e2πict(x1−x2) − 1 2πic(x1 − x2) Cf(x1 − x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This is a well-defined function for each finite t, but we must treat it as a distribution if we want to take the limit t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' To do so, we test it against some g1, g2 ∈ S(R): E[⟨u(t), g1⟩⟨u(t), g2⟩] = � Rx1×Rx2 E[u(t, x1)u(t, x2)]g1(x1)g2(x2) dx1dx2 = � Rz e2πictz − 1 2πicz ψ(z) dz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='20) with ψ(z) = Cf(z) � Ry g1(z + y)g2(y)dy, z = x1 − x2 and y = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Our goal is to study the last integral as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We start with a simple identity that gives us a way to integrate the function (e2πictz −1)/(2πicz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that this function is not absolutely integrable in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, its integral does converge A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 13 conditionally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' the final result might depend on how we integrate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' More precisely, we recall that for all t > 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='21) � R eitz − 1 iz dz := lim R→∞ � R −R eitz − 1 iz dz = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As a result of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='21), we have that lim R→∞ � R −R e−2πictz − 1 2πicz ψ(z)dz − ψ(0) 2c = lim R→∞ � R −R e−2πictz − 1 2πicz [ψ(z) − ψ(0)] dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In view of ψ(0) = Cf(0) � Ry g1(y)g2(y)dy, it suffices to prove the following in order to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10): lim t→∞ lim R→∞ � R −R e−2πictz − 1 2πicz [ψ(z) − ψ(0)] dzdy = 1 2πc � ∞ 0 ψ(z) − ψ(−z) iz dz = 1 2πic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' � Rz ψ(z) z dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='22) To prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='22), we rewrite the left-hand side as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='23) lim R→∞ � R −R e−2πictz − 1 2πc ψ(z) − ψ(0) iz dz = 1 2πic � lim R→∞ � R −R e−2πictz ψ(z) − ψ(0) z dz − lim R→∞ � R −R ψ(z) − ψ(0) z dz � Note that the last term gives the desired limit after using the fact that � R −R ψ(z) − ψ(0) z dz = � R 0 ψ(z) − ψ(−z) z dz and taking R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The final step is to show that the first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='23) tends to zero as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Note that F(z) := ψ(z) − ψ(0) z is not integrable in Rz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However the following lemma shows that its derivatives have better prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Its proof is postponed to the end of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Recall that ψ was defined in terms of g1, g2 right after (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' If g1 ∈ Hn+1(R) and g2 ∈ L2(R) then F ∈ W n,1(Rz) and lim |z|→∞(∂m z F)(z) = 0, for any 0 ≤ m ≤ n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='24) ∥∂m z F∥L1(Rz) ≲ ∥g1∥Hm+1 ∥g2∥L2 for any 1 ≤ m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='25) We integrate by parts the first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='23) � R −R e−ictz ψ(z) − ψ(0) z dz = − � R −R 1 ict ∂ze−ictz F(z)dz = − 1 ict e−ictz F(z) ��� R z=−R + 1 ict � R −R e−ictz∂zF(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8, we are able to take the limit R → ∞: lim R→∞ � R −R e−ictz ψ(z) − ψ(0) z dz = 1 ict � Rz e−ictz∂zF(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Next, we continue to integrate by parts using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='24) (which gets rid of the boundary terms) lim R→∞ � R −R e−ictz ψ(z) − ψ(0) z dz = � 1 ict �n � Rz e−ictz∂n z F(z) dz, thus we have ���� lim R→∞ � R −R e−ictz ψ(z) − ψ(0) z dz ���� ≤ � 1 ct �n ∥∂n z F∥L1(R2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='25) to finish the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Now, we need to prove the technical Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' First we can easily show that ∂n z F(z) = � 1 0 sn∂n+1 z ψ(zs)ds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='26) = � z 0 sn zn+1 ∂n+1 z ψ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='27) Then we show that for all 0 ≤ m ≤ n + 1 and all p ∈ [1, ∞] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='28) ∥∂m z ψ∥Lp(R) ≲ ||Cf||W m,p(Rz) ∥g1∥Hm ∥g2∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, the Leibniz rule yields ∂m z ψ(z) = m � j=0 � m j � ∂j zCf(z) � Ry ∂m−j z g1(y + z)g2(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' By Cauchy-Schwarz inequality, |∂m z ψ(z)| ≤ � � m � j=0 � m j � |∂j zCf(z)| � � ∥g1∥Hm ∥g2∥L2 thus one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Then we show that for all 0 ≤ m ≤ n, lim |z|→∞ ∂m z F(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, by the Cauchy-Schwarz inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='28), |∂m z F(z)| ≤ ��∂m+1 z ψ �� L2(Rz) �� |z| 0 s2m |z|2(m+1) ds � 1 2 = ��∂m+1 z ψ �� L2(Rz) |z| 1 2 −→ |z|→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Finally, for all 1 ≤ m ≤ n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='26) implies ∥∂m z F(z)∥L1(R2) ≤ � 1 0 sm ��∂m+1 z ψ(zs) �� L1(R2) ds ≤ ��∂m+1 z ψ �� L1(R2) �� 1 0 sm−1ds � and thus we conclude with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' □ A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 15 Natural generalizations of the dynamics in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) to higher dimensions could be obtained in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Firstly, the product c ∂k�u in the transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) could be generalized to a scalar product of a given unit vector e ∈ Rd with the gradient ∇k�u(t, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Unfortunately, this puts too much weight on the constant vector e and results in an obvious statistical anisotropy in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For applications to turbulence, we must require that statistical laws are not only invariant by translation, but also under rotation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' statistical isotropy), as commonly observed in laboratory and numerical experiments, and as expected from a physical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Another option would be to replace the multiplication by ix in the physical space formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) by the multiplication by i|x|, where |x| is the modulus of x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Once again, this would introduce anisotropy in the system, and more importantly, it would break statistical homogeneity even in dimension d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' One may check these claims directly using the exact solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) to compute the covariance function at a given time t and any two positions x, y (see [3] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, this covariance function eventually depends on the difference |x| − |y|, and not on x − y as would be desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Beyond these issues, none of these propositions would ensure that a given real-valued initial condition u(0, x) ∈ R gives rise to a real-valued solution u(t, x) ∈ R at all future times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In other words, in order to construct a real-valued solution in physical space, one needs to propose a dynamical picture able to preserve the Hermitian symmetry of the Fourier transform �u(t, k), as does the dynamics in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Higher dimensional real fractional gaussian fields: heuristic In the previous section we gave a rigorous proof of the construction of a dynamical complex white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This proof was carried out in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' For the dynamical real white noise of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3), on the other hand, it is more convenient to think of its wavenumber formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, even in the case of a dynamical complex white noise, the solution u(t, x) is not in Lp(Rx) for any 1 ≤ p < ∞ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='18)), hence its Fourier transform is not defined pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In conclusion, it is difficult to make sense of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) in wavenumber space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' However, we will not concern ourselves with such difficulties in this section, and we will work as if the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) were well-defined pointwise: we refer to Section 4 for a rigorous analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we will later see, working in wavenumber space is very convenient to formally show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) builds a dynamical real white noise, as well as to extend this construction to d-dimensional fractional Gaussian fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A transport equation in wavenumber space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We propose the following initial value problem as a generalization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) � � � � � � � � � ∂t�u(t, k) + divk � ck |k| �u(t, k) � + cH + 1 2 |k| �u(t, k) = �f(t, k) t > 0, k ∈ Rd, |k| > κ > 0, �u(t, k) = 0 t > 0, k ∈ Rd, |k| ≤ κ, �u(0, k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' where H is a real constant (which will be eventually connected with the H¨older exponent of the solution), and divk stands for the usual divergence operator in wavenumber space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As one can easily verify with a few vector calculus identities, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2) divk � ck |k| �u(k) � + c H + 1 2 |k| �u(k) = c ∂|k| �u(k) + c H + d − 1 2 |k| �u(k) with ∂|k| := k |k| · ∇k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' When d = 1 and H = −1/2, one recovers (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) from the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The initial value problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) can be regarded as a conservation law in wavenumber space with a source term �f(t, k), a damping term (H + 1/2) �u(t, k)/|k| and Dirichlet boundary conditions at the sphere |k| = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) has been thoroughly studied when �f(t, k) is regular enough (see [26], [33]), but in our case �f(t, k) is too rough for such classical results to be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In particular, we will 16 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE assume that the forcing term �f satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) E[ �f(t, k1) �f(s, k2)] = � Cf(k1) δt−s δk1−k2, where � Cf(k) is radial and null in the ball |k| < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) formally follows from considering the Fourier transform of a white noise in time satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='4) E[f(t, x)f(s, y)] = δt−s Cf(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As part of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) one has the following technical condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5) �u(t, k) = 0 t > 0, k ∈ Rd, |k| ≤ κ, for κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Indeed, if |k| = 0, then divk � k |k| · � and H+ 1 2 |k| are not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5) implies in particular that F−1�u has null spatial average, whenever defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' It might be possible to make sense of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) for κ = 0 by adequately changing condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5), imposing �f(t, k = 0) = 0 and an appropriate behaviour near k = 0, but this is outside the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Asymptotic behavior: power-law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In this section, we show that the two-point correlation of the solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) displays a power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In our first result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2), we obtain this asymptotic behavior as t → ∞ under fairly mild assumptions on the forcing �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Under stronger assumptions on �f, we derive a second result (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='5) showing this power law behavior in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Among other things, such power laws are important because their exponent determines the H¨older regularity of the solution in physical space (should it be possible to take the inverse Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The main idea of the heuristic proof of our desired results is to perform a change of variables to rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) as a 1D transport equation with respect to |k| and parametrized by the “angular variable” k |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Such an equation admits an explicit solution that we will exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We will further discuss such consequences in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='2 (Heuristic version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Let the forcing �f satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='3) in such a way that � Cf(k) = ψ(|k|) is radial, non-negative, non identically null, with s2H+d ψ(s) ∈ L1(R+ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Furthermore, we assume that ψ is null when |k| < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Then, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) admits a solution that satisfies the following asymptotic behavior lim t→∞ E[�u(t, k)�u(t, k′)] = |k|−(2H+d) (C(d, H) − Ψd,H(|k|)) δk−k′ where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6) Ψd,H(|k|) := 1 c � ∞ |k| s2H+d ψ(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' is a positive non-increasing absolutely continuous function and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='7) C(d, H) := Ψd,H(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As we have already pointed out in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6, the function Ψd,H can be seen as lower-order correction in comparison with the Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, if we assume that ψ is fast- decaying, then Ψd,H will also be fast-decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Heuristic proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' In order to find a formal solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1), we let �v(t, k) := |k|H+d− 1 2 �u(t, k), �g(t, k) := |k|H+d− 1 2 �f(t, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' A LINEAR STOCHASTIC MODEL OF TURBULENT CASCADES AND FRACTIONAL FIELDS 17 Next we rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='1) in terms of �v, namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8) � � � � � ∂t�v(t, k) + c ∂|k|�v(t, k) = �g(t, k) t > 0, |k| > κ, �v(t, k) = 0 t > 0, |k| ≤ κ, �v(0, k) = 0, where we use the fact that ∂|k| = k |k| · ∇k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='8) can be regarded as a 1D transport equation with respect to |k| and parametrized by the “angular variable” k/|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Thus it is easy to give an explicit solution: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='9) �v(t, k) = � t � t− |k|−κ c � + �g � s, (|k| − ct + cs) k |k| � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Next we compute the correlations of �v(t) using those of �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We have E[�v(t, k1)�v(t, k2)] = � t � t− |k1|−κ c � + � t � t− |k2|−κ c � + δcs1−cs2δ |k1|−ct+s1 |k1| k1− |k2|−t+cs2 |k2| k2 � Cg �|k1| − ct + cs1 |k1| k1 � ds1ds2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' We assume that we can write δcs1−cs2δ |k1|−ct+cs1 |k1| k1− |k2|−ct+cs2 |k2| k2 = δcs1−cs2δ |k1|−ct+cs1 |k1| k1− |k2|−ct+cs1 |k2| k2 even if this not mathematically rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Moreover, by the change of variables from cartesian to polar coordinates δ |k1|−ct+cs |k1| k1− |k2|−ct+cs |k2| k2 = �|k1| − ct + cs |k1| �−(d−1) δk1−k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This is possible since the Jacobian � |k1|−ct+cs |k1| �−(d−1) has no singularities in the region of integration thanks to κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' As a result, one obtains E[�v(t, k)�v(t, k′)] = � � � t � t− |k|−κ c � + �|k| − ct + cs |k| �−(d−1) � Cg �|k| − ct + cs |k| k � ds � � δk−k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' This immediately implies E[�u(t, k)�u(t, k′)] = |k|−(2H+d) � � � t � t− |k|−κ c � + (|k| − ct + cs)2H+d � Cf �|k| − ct + cs |k| k � ds � � δk−k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' The change of variables s �→ |k1| − ct + cs yields E[�u(t, k)�u(t, k′)] = |k|−(2H+d) � χ|k|>ct+κ � |k| |k|−ct s2H+d � Cf � s |k| k � ds c � δk−k′ + |k|−(2H+d) � χ|k|≤ct+κ � |k| κ s2H+d � Cf � s |k| k � ds c � δk−k′ where χA denotes the characteristic function of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Remember that ψ(|k|) := � Cf(k) since Cf is radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='6), we may rewrite E[�u(t, k)�u(t, k′)] = |k|−(2H+d) � χ|k|>ct+κ [Ψd,H(|k| − ct) − Ψd,H(|k|)] + χ|k|≤ct+κ [Ψd,H(κ) − Ψd,H(|k|)] � δk−k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content='10) 18 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' APOLIN´ARIO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' BECK, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' CHEVILLARD, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GALLAGHER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQf3_rL/content/2301.00780v1.pdf'} +page_content=' GRANDE ' 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relaxed micromorphic model to antiplane shear and its behaviour in the limit +cases of its material parameters is depicted in Fig. 2.2. +relaxed micromorphic +Cosserat elasticity +linear elasticity +with Cmacro +antiplane relaxed +micromorphic +antiplane Cosserat +elasticity +antiplane linear +elasticity +with µmacro +Lc → 0 +Cmicro → +∞ , +µc > 0 +Lc → 0 , +µc ≡ 0 +µmicro → +∞ , +µc > 0 +(contradiction) +antiplane +shear +antiplane +shear +antiplane +shear +antiplane linear +elasticity +with µmicro +linear elasticity +with Cmicro +Lc → +∞ +Lc → +∞ +two-scale +model +two-scale +model +non- +commutative +Figure 2.2: Kinematic reduction of the relaxed micromorphic model to antiplane shear and consistency at limit +cases according to Remark 2.1 and Section 2.1. The two-scale nature of the relaxed micromorphic model can +be clearly observed. +3 +Polynomial basis +In this section we briefly introduce Bernstein polynomials and dual numbers. Bernstein polynomials are used +to construct both the H 1-conforming subspace and, in conjunction with the polytopal template methodology, +the N´ed´elec elements. The computation of derivatives of the Bernstein base functions is achieved by employing +dual numbers, thus enabling the calculation of the value and the derivative of a base function simultaneously. +3.1 +Bernstein polynomials +Bernstein polynomials of order p are given by the binomial expansion of the barycentric representation of the +unit line +1 = (λ1 + λ2)p = ((1 − ξ) + ξ)p = +p +� +i=0 +�p +i +� +ξi(1 − ξ)p−i = +p +� +i=0 +p! +i!(p − i)!ξi(1 − ξ)p−i , +(3.1) +9 + +b4 +0(ξ) +b4 +1(ξ) +b4 +2(ξ) +b4 +3(ξ) +b4 +4(ξ) +ξ +1 +1 +0 +1 +1/2 +1/2 +Figure 3.1: Bernstein base functions of degree p = 4 on the unit domain. Their sum forms a partition of unity. +The base functions are symmetric for ξ = 0.5 with respect to their indices and always positive. +where ξ ∈ [0, 1]. The Bernstein polynomial reads +bp +i (ξ) = +�p +i +� +ξi(1 − ξ)p−i . +(3.2) +A direct result of the binomial expansion is that Bernstein polynomials form a partition of unity, see also Fig. 3.1 +p +� +i=0 +bp +i (ξ) = 1 . +(3.3) +Another consequence is that Bernstein polynomials are non-negative and less than or equal to 1 +0 ≤ bp +i (ξ) ≤ 1 , +ξ ∈ [0, 1] . +(3.4) +A necessary condition for the use of Bernstein polynomials in finite element approximations is for them to span +the entire polynomial space. +Theorem 3.1 (Span of Bernstein polynomials) +The span of Bernstein polynomials forms a basis of the one-dimensional polynomial space +Pp(ξ) = span{bp +i } , +ξ ⊆ R . +(3.5) +Proof. First we observe +dim(span{bp +i }) = dim Pp(ξ) = p + 1 . +(3.6) +The proof of linear independence is achieved by contradiction. Let the set span{bp +i } with 0 < i ≤ p be linearly +dependent, then there exists some combination with at least one non-zero constant ci ̸= 0 such that +p +� +i=1 +cibp +i (ξ) = 0 , +d +dξ +p +� +i=1 +cibp +i (ξ) = 0 . +(3.7) +However, by the partition of unity property Eq. (3.3), only the full combination (0 ≤ i ≤ p) generates a constant +and by the exact sequence property the kernel of the differentiation operator is exactly the space of constants +ker(∂) = R. The linear independence of the full span also follows from the partition of unity property, since +constants cannot be constructed otherwise. +10 + +Bernstein polynomials can be evaluated efficiently using the recursive formula +bp +0(ξ) = (1 − ξ)p , +bp +i+1(ξ) = +(p − i)ξ +(p + 1)(1 − ξ)bp +i (ξ) , +i ∈ {0, 1, ..., p − 1} , +(3.8) +which allows for fast evaluation of the base functions. +Remark 3.1 +Note that the formula Eq. (3.8) implies limξ→1 bp +i+1(ξ) = ∞. As such, evaluations using the formula are required +to use ξ < 1 preferably with additional tolerance. The limit case ξ = 1 is zero for all Bernstein base functions +aside from the last function belonging to the vertex, which simply returns one +bp +i (1) = 0 +∀ i ̸= p , +bp +p(1) = 1 . +(3.9) +3.2 +Dual numbers +Dual numbers [16] can be used to define define an augmented algebra, where the derivative of a function can +be computed simultaneously with the evaluation of the function. This enhancement is also commonly used +in forward automatic differentiation [8, 37], not to be confused with numerical differentiation, since unlike in +numerical differentiation, automatic differentiation is no approximation and yields the exact derivative. The +latter represents an alternative method to finding the derivatives of base functions, as opposed to explicit +formulas or approximations. Dual numbers augment the classical numbers by adding a non-zero number ε with +a zero square ε2 = 0. +Definition 3.1 (Dual number) +The dual number is defined by +x + x′ε , +ε ≪ 1 , +(3.10) +where x′ is the derivative (only in automatic differentiation), ε is an abstract number (infinitesimal) and formally +ε2 = 0. +The augmented algebra results automatically from the definition of the dual number. +Definition 3.2 (Augmented dual algebra) +The standard algebraic operations take the following form for dual numbers +1. Addition and subtraction +(x + x′ε) ± (y + y′ε) = x ± y + (x′ ± y′)ε . +(3.11) +2. Multiplication +(x + x′ε)(y + y′ε) = xy + (xy′ + x′y)ε , +(3.12) +since formally ε2 = 0. +3. Division is achieved by first defining the inverse element +(x + x′ε)(y + y′ε) = 1 +⇐⇒ +y = 1 +x, +y′ = − x′ +x2 , +(3.13) +such that +(x + x′ε)/(y + y′ε) = x/y + (x′/y − xy′/y2)ε . +(3.14) +Application of the above definitions to polynomials +p(x + ε) = +∞ +� +i=0 +ci(x + ε)i = +∞ +� +i=0 +1 +� +j=0 +ci +�i +j +� +xi−jεj = +∞ +� +i=0 +cixi + ε +∞ +� +i=1 +i cixi−1 = p(x) + p′(x)ε , +(3.15) +allows the extension to various types of analytical functions with a power-series representation (such as trigono- +metric or hyperbolic). +11 + +v1 +v3 +v2 +Γ +ν +τ +ξ +η +x1 +x3 +x2 +Ae +t +n +x +y +x : Γ → Ae +Figure 4.1: Barycentric mapping of the reference triangle to an element in the physical domain. +Definition 3.3 (General dual numbers function) +A function of a dual number is defined in general by +f(x + ε) = f(x) + f ′(x)ε , +(3.16) +being a fundamental formula for forward automatic differentiation. +The definition of dual numbers makes them directly applicable to the general rules of differentiation, such as +the chain rule or product rule, in which case the derivative is simply the composition of previous computations +with ε. The logic of dual numbers can be understood intuitively by the directional derivative +d +dxf(x) = ∂x′f(x) = d +dεf(x + x′ε) +���� +ε=0 += lim +ε→0 +f(x + x′ε) − f(x) +ε +, +(3.17) +where dividing by ε and setting ε = 0 are deferred to the last step of the computation, being the extraction of +the derivative and equivalent to the operation f(x + ε) − f(x) with the augmented algebra of dual numbers. +In this work we apply dual numbers for the computation of Bernstein polynomials using the recursive formula +Eq. (3.8), thus allowing to iteratively compute each base function simultaneously with its derivative. +4 +Triangular elements +The triangle elements are mapped from the reference element Γ to the physical domain Ae via barycentric +coordinates +x(ξ, η) = (1 − ξ − η)x1 + η x2 + ξ x3 , +x : Γ → Ae , +Γ = {(ξ, η) ∈ [0, 1]2 | ξ + η ≤ 1} , +(4.1) +where xi represent the coordinates of the vertices of one triangle in the physical domain, see Fig. 4.1. The +corresponding Jacobi matrix reads +J = Dx = +�x3 − x1, +x2 − x1 +� +∈ R2×2 . +(4.2) +4.1 +The Bernstein-B´ezier basis for triangles +The base functions on the triangle reference element are defined using the binomial expansion of the barycentric +coordinates on the domain Γ +1 = (λ1 + λ2 + λ3)p = ([1 − ξ − η] + η + ξ)p . +(4.3) +As such, the B´ezier base functions read +bp +ij(λ1, λ2, λ3) = +� +p +i +� � +p − i +j +� +λp−i−j +1 +λj +2λi +3 , +(4.4) +12 + +(a) +(b) +(c) +Figure 4.2: Cubic vertex (a), edge (b) and cell (c) B´ezier base functions on the reference triangle. +(0,0) +(1,0) +(1,1) +(0,1) +α +β +Γ +(0,0) +(1,0) +(0,1) +ξ +η +ξ : α → Γ +Figure 4.3: Duffy transformation from a quadrilateral to a triangle by collapse of the coordinate system. +with the equivalent bivariate form +bp +ij(ξ, η) = +�p +i +� �p − i +j +� +(1 − ξ − η)p−i−jηjξi , +(4.5) +of which some examples are depicted in Fig. 4.2. The Duffy transformation +ξ : [0, 1]2 → Γ , +{α, β} �→ {ξ, η} , +(4.6) +given by the relations +ξ = α , +α = ξ , +η = (1 − α)β , +β = +η +1 − ξ , +(4.7) +allows to view the triangle as a collapsed quadrilateral, see Fig. 4.3. Inserting the Duffy map into the definition +of the B´ezier base function yields the split +bp +ij(ξ, η) = +�p +i +� �p − i +j +� +(1 − ξ − η)p−i−jηjξi += +�p +i +� �p − i +j +� +(1 − α − [1 − α]β)p−i−j(1 − α)jβjαi += +�p +i +� �p − i +j +� +(1 − α)p−i−j(1 − β)p−i−j(1 − α)jβjαi +(4.8) += +�p +i +� +(1 − α)p−iαi +�p − i +j +� +(1 − β)p−i−jβj += bp +i (α) bp−i +j +(β) . +In other words, the Duffy transformation results in a natural factorization of the B´ezier triangle into Bernstein +base functions [1]. The latter allows for fast evaluation using sum factorization. Further, it is now clear that +B´ezier triangles are given by the interpolation of B´ezier curves, where the degree of the polynomial decreases +13 + +ξ +η +outer B´ezier curve with p = 3 +inner B´ezier curves with p < 3 +control polygon of η-curves +outer B´ezier curves with p = 3 +inner B´ezier curves with p = 3 +Figure 4.4: B´ezier triangle built by interpolating B´ezier curves with an ever decreasing polynomial degree. +v1 +v3 +v2 +ξ +η +Figure 4.5: Traversal order of base functions. The purple lines represent the order in which the base functions +are constructed by the factorized evaluation. Note that the traversal order on each edge is intrinsically from +the lower to the higher vertex index. +between each curve, see Fig. 4.4. In order to compute gradients on the reference domain one applies the chain +rule +∇ξbp +ij = (Dαξ)−T ∇αbp +ij , +Dαξ = +� 1 +0 +−β +1 − α +� +, +(Dαξ)−T = +1 +1 − α +�1 − α +β +0 +1 +� +. +(4.9) +The factorization is naturally suited for the use of dual numbers since the α-gradient of a base function reads +∇αbp +ij(α, β) = +� +��� +bp−i +j +d +dαbp +i +bp +i +d +dβ bp−i +j +� +��� , +(4.10) +such that only the derivatives of the Bernstein base functions with respect to their parameter are required. +The Duffy transformation induces an intrinsic optimal order of traversal of the base functions, compare +Fig. 4.5, namely +(i, j) = (0, 0) → (0, 1) → ... → (2, 2) → ... → (i, p − i) → ... → (p, 0) , +(4.11) +which respects a clockwise orientation of the element, compare [52]. Thus, the order of the sequence of discrete +values on common edges is determined by the global orientation. In order to relate a base function to a polytopal +piece of the element, one observes the following result. +Observation 4.1 (Triangle base functions) +The polytope of each base function bp +ij(ξ, η) can be determined as follows: +14 + +• The indices (0, 0), (0, p) and (p, 0) represent the first, second and last vertex base functions, respectively. +• The indices (0, j) with 0 < j < p and (i, 0) with 0 < i < p represent the first and second edge base +functions, respectively. Base functions of the slanted edge are given by (i, p − i) with 0 < i < p. +• The remaining index combinations are cell base functions. +With the latter observation, the construction of vertex-, edge- and cell base functions follows the intrinsic +traversal order induced by the Duffy transformation and relates to a specific polytope via index-pairs. +4.2 +N´ed´elec elements of the second type +We construct the base functions for the N´ed´elec element of the second type using the polytopal template +methodology introduced in [50]. The template sets read +T1 = {e2, e1} , +T2 = {e1 + e2, e1} , +T3 = {e1 + e2, −e2} , +T12 = {e2, −e1} , +T13 = {e1, e2} , +T23 = {(1/2)(e1 − e2), e1 + e2} , +T123 = {e1, e2} . +(4.12) +The space of B´ezier polynomials is split across the polytopes of the reference triangle into +Bp(Γ) = +� 3 +� +i=1 +Vp +i (Γ) +� +⊕ +� +� +� +� +j∈J +Ep +j (Γ) +� +� +� ⊕ Cp +123(Γ) , +J = {(1, 2), (1, 3), (2, 3)} , +(4.13) +where Vp +i are the sets of the vertex base functions, Ep +j are the sets of edge base functions, Cp +123 is the set of cell +base functions, and the ⊕ indicates summation over non-overlapping spaces. Consequently, the N´ed´elec basis +is given by +N p +II = +� 3 +� +i=1 +Vp +i ⊗ Ti +� +⊕ +� +� +� +� +j∈J +Ep +j ⊗ Tj +� +� +� ⊕ {Cp +123 ⊗ T123} , +J = {(1, 2), (1, 3), (2, 3)} . +(4.14) +Using the B´ezier basis one finds the following base functions, which inherit the optimal complexity of the +underlying basis. +Definition 4.1 (B´ezier-N´ed´elec II triangle basis) +The following base functions are defined on the reference triangle. +• On the edges the base function reads +e12 : +ϑ(ξ, η) = bp +00e2 , +ϑ(ξ, η) = bp +0p(e1 + e2) , +ϑ(ξ, η) = bp +0je2 , +0 < j < p , +e13 : +ϑ(ξ, η) = bp +00e1 , +ϑ(ξ, η) = bp +p0(e1 + e2) , +ϑ(ξ, η) = bp +i0e1 , +0 < i < p , +e23 : +ϑ(ξ, η) = bp +0pe1 , +ϑ(ξ, η) = −bp +p0e2 , +ϑ(ξ, η) = (1/2) bp +i,p−i(e1 − e2) , +0 < i < p , +(4.15) +where the first two base functions for each edge are the vertex-edge base functions and the third equation +generates pure edge base functions. +• The cell base functions read +c123 : +ϑ(ξ, η) = −bp +0je1 , +0 < j < p , +ϑ(ξ, η) = bp +i0e2 , +0 < i < p , +ϑ(ξ, η) = bp +i,p−i(e1 + e2) , +0 < i < p , +ϑ(ξ, η) = bp +ije2 , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η) = bp +ije1 , +0 < i < p , +0 < j < p − i , +(4.16) +15 + +where the first three are the respective edge-cell base functions. The remaining two are pure cell base +functions. +4.3 +N´ed´elec elements of the first type +In order to construct the N´ed´elec element of the first type we rely on the construction of the kernel introduced +in [58] via the exact de Rham sequence and the polytopal template for the non-kernel base functions following +[50]. The complete N´ed´elec space reads +N p +I = N 0 +I ⊕ +� +� +� +� +j∈J +∇Ep+1 +j +� +� +� ⊕ ∇Cp+1 +123 ⊕ +� 2 +� +i=1 +Vp +i ⊗ Ti +� +⊕ +� +� +� +� +j∈J +Ep +j ⊗ Tj +� +� +� ⊕ {Cp +123 ⊗ T123} , +J = {(1, 2), (1, 3), (2, 3)} , +(4.17) +where we relied on the decomposition Eq. (4.14). Applying the construction to the B´ezier basis yields the +following base functions. +Definition 4.2 (B´ezier-N´ed´elec I triangle basis) +We define the base functions on the reference triangle. +• On the edges we employ the lowest order N´ed´elec base functions and the edge gradients +e12 : +ϑ(ξ, η) = ϑI +1 , +ϑ(ξ, η) = ∇ξbp+1 +0j +, +0 < j < p + 1 , +e13 : +ϑ(ξ, η) = ϑI +2 , +ϑ(ξ, η) = ∇ξbp+1 +i0 +, +0 < i < p + 1 , +e23 : +ϑ(ξ, η) = ϑI +3 , +ϑ(ξ, η) = ∇ξbp+1 +i,p+1−i , +0 < i < p + 1 . +(4.18) +• The cell functions read +c123 : +ϑ(ξ, η) = bp +00ϑI +3 , +ϑ(ξ, η) = bp +0pϑI +2 , +ϑ(ξ, η) = bp +0j(ϑI +3 − ϑI +2) , +0 < j < p , +ϑ(ξ, η) = bp +i0(ϑI +1 + ϑI +3) , +0 < i < p , +ϑ(ξ, η) = bp +i,p−i(ϑI +1 − ϑI +2) , +0 < i < p , +ϑ(ξ, η) = bp +ij(ϑI +1 − ϑI +2 + ϑI +3) , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η) = ∇ξbp+1 +ij +, +0 < i < p + 1 , +0 < j < p + 1 − i , +(4.19) +where the last formula gives the cell gradients and the remaining base functions are non-gradients. +The definition relies on the base functions of the lowest order N´ed´elec element of the first type [5,50] +ϑI +1(ξ, η) = +� +η +1 − ξ +� +, +ϑI +2(ξ, η) = +�1 − η +ξ +� +, +ϑI +3(ξ, η) = +� η +−ξ +� +. +(4.20) +5 +Tetrahedral elements +The tetrahedral elements are mapped from the reference tetrahedron Ω by the three-dimensional barycentric +coordinates onto the physical domain Ve, see Fig. 5.1 +x(ξ, η, ζ) = (1 − ξ − η − ζ)x1 + ζ x2 + η x3 + ξ x4 , +x : Ω → Ve , +Ω = {(ξ, η, ζ) ∈ [0, 1]3 | ξ + η + ζ ≤ 1} . +(5.1) +16 + +ξ +η +ζ +Ω +v1 +v4 +v3 +v2 +τ +ν +Ve +x2 +x1 +x3 +x4 +x +y +z +t +n +x : Ω → Ve +Figure 5.1: Barycentric mapping of the reference tetrahedron to an element in the physical domain. +The corresponding Jacobi matrix reads +J = Dx = +�x4 − x1, +x3 − x1, +x2 − x1 +� +∈ R3×3 . +(5.2) +5.1 +The Bernstein-B´ezier basis for tetrahedra +Analogously to triangle elements, the B´ezier tetrahedra on the unit tetrahedron Ω are defined using the barycen- +tric coordinates by expanding the coefficients of +(λ1 + λ2 + λ3 + λ4)p = ([1 − ξ − η − ζ] + ζ + η + ξ)p = 1 , +(5.3) +thus finding +bp +ijk(λ1, λ2, λ3, λ4) = +�p +i +� �p − i +j +� �p − i − j +k +� +λp−i−j−k +1 +λk +2λj +3λk +4 , +(5.4) +with the equivalent trivariate form +bp +ijk(ξ, η, ζ) = +� +p +i +� � +p − i +j +� �p − i − j +k +� +(1 − ξ − η − ζ)p−i−j−kζkηjξi . +(5.5) +We construct the Duffy transformation by mapping the unit tetrahedron as a collapsed hexahedron +ξ : [0, 1]3 → Ω , +{α, β, γ} �→ {ξ, η, ζ} , +(5.6) +using the relations +ξ = α , +η = (1 − α)β , +ζ = (1 − α)(1 − β)γ , +α = ξ , +β = +η +1 − ξ , +γ = +ζ +1 − ξ − η , +(5.7) +as depicted in Fig. 5.2. Applying the Duffy transformation to B´ezier tetrahedra +bp +ijk(ξ, η, ζ) = +�p +i +� �p − i +j +� �p − i − j +k +� +(1 − ξ − η − ζ)p−i−j−kζkηjξi += +� +p +i +� � +p − i +j +� � +p − i − j +k +� +(1 − α − (1 − α)β − (1 − α)(1 − β)γ)p−i−j−k +· (1 − α)k(1 − β)kγk(1 − α)jβjαi += +�p +i +� �p − i +j +� �p − i − j +k +� +(1 − α)p−i−j−k(1 − β)p−i−j−k(1 − γ)p−i−j−k +(5.8) +· (1 − α)k(1 − β)kγk(1 − α)jβjαi += +�p +i +� +(1 − α)p−iαi +�p − i +j +� +(1 − β)p−i−jβj +�p − i − j +k +� +(1 − γ)p−i−j−kγk += bp +i (α)bp−i +j +(β)bp−i−j +k +(γ) , +17 + +α +β +γ +(0,0,0) +(1,0,0) +(0,0,1) +(1,1,0) +(1,1,1) +(0,1,1) +ξ +η +ζ +Ω +(0,0,0) +(1,0,0) +(0,1,0) +(0,0,1) +ξ : α → Ω +Figure 5.2: Duffy mapping of the unit hexahedron to the unit tetrahedron. +leads to an intrinsic factorization via univariate Bernstein base functions, which allow for fast evaluations +using sum factorization [1]. Further, since the pair bp−i +j +(β)bp−i−j +k +(γ) spans a B´ezier triangle, it is clear that +the multiplication with bp +i (α) interpolates between that triangle and a point in space, effectively spanning a +tetrahedron. In order to compute gradients the chain rule is employed with respect to the Duffy transformation +∇ξbp +ijk = (Dαξ)−T ∇αbp +ijk , +Dαξ = +� +� +1 +0 +0 +−β +1 − α +0 +(β − 1)γ +(α − 1)γ +(1 − α)(1 − β) +� +� , +(Dαξ)−T = +1 +(1 − α)(1 − β) +� +� +(1 − α)(1 − β) +(1 − β)β +γ +0 +1 − β +γ +0 +0 +1 +� +� . +(5.9) +We use dual numbers to compute the derivative of each Bernstein base function and construct the α-gradient +∇αbp +ijk(α, β, γ) = +� +������� +bp−i +j +bp−i−j +k +d +dαbp +i +bp +i bp−i−j +k +d +dβ bp−i +j +bp +i bp−i +j +d +dγ bp−i−j +k +� +������� +. +(5.10) +The Duffy transformation results in the optimal order of traversal of the base functions depicted in Fig. 5.3. +Note that the traversal order agrees with the oriental definitions introduced in [52] and each oriented face has +the same order of traversal as the triangle Fig. 4.5. We relate the base functions to their respective polytopes +using the index triplets. +Observation 5.1 (Tetrahedron base functions) +The polytope of each base function bp +ijk(ξ, η, ζ) is determined as follows. +• the indices (0, 0, 0), (0, 0, p), (0, p, 0) and (p, 0, 0) represent the respective vertex base functions; +• the first edge is associated with the triplet (0, 0, k) where 0 < k < p, the second with (0, j, 0) where 0 < j < p +and the third with (i, 0, 0) where 0 < i < p. The slated edges are given by (0, j, p − j) with 0 < j < p, +(i, 0, p − i) with 0 < i < p and (i, p − i, 0) with 0 < i < p, respectively; +• the base functions of the first face are given by (0, j, k) with 0 < j < p and 0 < k < p − j. The second face +is associated with the base functions given by the triplets (i, 0, k) with 0 < i < p and 0 < k < p − i. The +base functions of the third face are related to the indices (i, j, 0) with 0 < i < p and 0 < j < p − i. Lastly, +the base functions of the slated face are given by (i, j, p − i − j) with 0 < i < p and 0 < j < p − i; +• the remaining indices correspond to the cell base functions. +Examples of B´ezier base functions on their respective polytopes are depicted in Fig. 5.4. +18 + +ξ +η +ζ +v1 +v4 +v3 +v2 +Figure 5.3: Order of traversal of tetrahedral B´ezier base functions on the unit tetrahedron. The traversal order +on each face agrees with an orientation of the vertices fijk = {vi, vj, vk} such that i < j < k. The traversal +order on each edge is from the lower index vertex to the higher index vertex. +(a) +(b) +(c) +(d) +Figure 5.4: Quartic B´ezier vertex (a), edge (b), face (c), and cell (c) base functions on the reference tetrahedron. +19 + +5.2 +N´ed´elec elements of the second type +The B´ezier polynomial space is split according to the polytopes of the reference tetrahedron +Bp(Ω) = +� 4 +� +i=1 +Vp +i (Ω) +� +⊕ +� +� +� +� +j∈J +Ep +j (Ω) +� +� +� ⊕ +�� +k∈K +Fp +k(Ω) +� +⊕ Cp +1234(Ω) , +J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} , +K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} , +(5.11) +where Vp +i are the sets of vertex base functions, Ep +j are the sets of edge base functions, Fp +k are the sets of face +base functions and Cp +1234 is the set of cell base functions. We apply the template sets from [50] +T1 = {e3, e2, e1} , +T2 = {e1 + e2 + e3, e2, e1} , +T3 = {e1 + e2 + e3, −e3, e1} , +T4 = {e1 + e2 + e3, −e3, −e2} , +T12 = {e3, −e2, −e1} , +T13 = {e2, e3, −e1} , +T14 = {e1, e3, e2} , +T23 = {e2, e1 + e2 + e3, −e1} , +T24 = {e1, e1 + e2 + e3, e2} , +T34 = {e1, e1 + e2 + e3, −e3} , +T123 = {e3, e2, −e1} , +T124 = {e3, e1, e2} , +T134 = {e2, e1, −e3} , +T234 = {e2, e1, e1 + e2 + e3} , +T1234 = {e3, e2, e1} , +(5.12) +to span the N´ed´elec element of the second type +N p +II = +� 4 +� +i=1 +Vp +i ⊗ Ti +� +⊕ +� +� +� +� +j∈J +Ep +j ⊗ Tj +� +� +� ⊕ +�� +k∈K +Fp +k ⊗ Tk +� +⊕ {Cp +1234 ⊗ T1234} , +J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} , +K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} . +(5.13) +We can now define the B´ezier-N´ed´elec element of the second type for arbitrary powers while inheriting optimal +complexity. +Definition 5.1 (B´ezier-N´ed´elec II tetrahedral basis) +We define the base functions on the reference tetrahedron: +• on the edges the base functions read +e12 : +ϑ(ξ, η, ζ) = bp +000e3 , +ϑ(ξ, η, ζ) = bp +00p(e1 + e2 + e3) , +ϑ(ξ, η, ζ) = bp +00ke3 , +0 < k < p , +e13 : +ϑ(ξ, η, ζ) = bp +000e2 , +ϑ(ξ, η, ζ) = bp +0p0(e1 + e2 + e3) , +ϑ(ξ, η, ζ) = bp +0j0e2 , +0 < j < p , +e14 : +ϑ(ξ, η, ζ) = bp +000e1 , +ϑ(ξ, η, ζ) = bp +p00(e1 + e2 + e3) , +ϑ(ξ, η, ζ) = bp +i00e1 , +0 < i < p , +e23 : +ϑ(ξ, η, ζ) = bp +00pe2 , +ϑ(ξ, η, ζ) = −bp +0p0e3 , +ϑ(ξ, η, ζ) = bp +0j,p−je2 , +0 < j < p , +e24 : +ϑ(ξ, η, ζ) = bp +00pe1 , +ϑ(ξ, η, ζ) = −bp +p00e3 , +ϑ(ξ, η, ζ) = bp +i0,p−ie1 , +0 < i < p , +e34 : +ϑ(ξ, η, ζ) = bp +0p0e1 , +ϑ(ξ, η, ζ) = −bp +p00e2 , +ϑ(ξ, η, ζ) = bp +i,p−i,0e1 , +0 < i < p , +(5.14) +where the first two base functions on each edge are the vertex-edge base functions; +20 + +• the face base functions are given by +f123 : +ϑ(ξ, η, ζ) = −bp +00ke2 , +0 < k < p , +ϑ(ξ, η, ζ) = bp +0j0e3 , +0 < j < p , +ϑ(ξ, η, ζ) = bp +0j,p−j(e1 + e2 + e3) , +0 < j < p , +ϑ(ξ, η, ζ) = bp +0jke3 , +0 < j < p , +0 < k < p − j , +ϑ(ξ, η, ζ) = bp +0jke2 , +0 < j < p , +0 < k < p − j , +f124 : +ϑ(ξ, η, ζ) = −bp +00ke1 , +0 < k < p , +ϑ(ξ, η, ζ) = bp +i00e3 , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i0,p−i(e1 + e2 + e3) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i0ke3 , +0 < i < p , +0 < k < p − i , +ϑ(ξ, η, ζ) = bp +i0ke1 , +0 < i < p , +0 < k < p − i , +f134 : +ϑ(ξ, η, ζ) = −bp +0j0e1 , +0 < j < p , +ϑ(ξ, η, ζ) = bp +i00e2 , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i,p−i,0(e1 + e2 + e3) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +ij0e2 , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η, ζ) = bp +ij0e1 , +0 < i < p , +0 < j < p − i , +f234 : +ϑ(ξ, η, ζ) = −bp +0j,p−je1 , +0 < j < p , +ϑ(ξ, η, ζ) = bp +i0,p−ie2 , +0 < i < p , +ϑ(ξ, η, ζ) = −bp +i,p−i,0e3 , +0 < i < p , +ϑ(ξ, η, ζ) = bp +ij,p−i−je2 , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η, ζ) = bp +ij,p−i−je1 , +0 < i < p , +0 < j < p − i , +(5.15) +where the first three formulas for each face are the edge-face base functions; +• finally, the cell base functions read +c1234 : +ϑ(ξ, η, ζ) = −bp +0jke1 , +0 < j < p , +0 < k < p − j , +ϑ(ξ, η, ζ) = bp +i0ke2 , +0 < i < p , +0 < k < p − i , +ϑ(ξ, η, ζ) = −bp +ij0e3 , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η, ζ) = bp +ij,p−i−j(e1 + e2 + e3) , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η, ζ) = bp +ijke3 , +0 < i < p , +0 < j < p − i , +0 < k < p − i − j , +ϑ(ξ, η, ζ) = bp +ijke2 , +0 < i < p , +0 < j < p − i , +0 < k < p − i − j , +ϑ(ξ, η, ζ) = bp +ijke1 , +0 < i < p , +0 < j < p − i , +0 < k < p − i − j , +(5.16) +where the first four formulas are the face-cell base functions. +5.3 +N´ed´elec elements of the first type +In order to construct the N´ed´elec element of first type on tetrahedra we introduce the template sets +T1 = {ϑI +4, ϑI +5, ϑI +6} , +T2 = {−ϑI +2, −ϑI +3, ϑI +6} , +T3 = {−ϑI +3, −ϑI +5} , +T12 = {ϑI +4 − ϑI +2, ϑI +5 − ϑI +3} , +T13 = {ϑI +1 + ϑI +4, ϑI +6 − ϑI +3} , +T14 = {ϑI +1 + ϑI +5, ϑI +2 + ϑI +6} , +T23 = {ϑI +1 − ϑI +2, ϑI +6 − ϑI +5} , +T24 = {ϑI +1 − ϑI +3, ϑI +4 + ϑI +6} , +T34 = {ϑI +2 − ϑI +3, ϑI +4 − ϑI +5} , +T123 = {ϑI +1 − ϑI +2 + ϑI +4} , +T124 = {ϑI +1 − ϑI +3 + ϑI +5} , +T134 = {ϑI +2 − ϑI +3 + ϑI +6} , +T234 = {ϑI +4 − ϑI +5 + ϑI +6} , +(5.17) +21 + +which are based on the lowest order N´ed´elec base functions on the unit tetrahedron +ϑ1(ξ, η, ζ) = +� +� +ζ +ζ +1 − ξ − η +� +� , +ϑ2(ξ, η, ζ) = +� +� +η +1 − ξ − ζ +η +� +� , +ϑ3(ξ, η, ζ) = +� +� +1 − η − ζ +ξ +ξ +� +� , +ϑ4(ξ, η, ζ) = +� +� +0 +ζ +−η +� +� , +ϑ5(ξ, η, ζ) = +� +� +ζ +0 +−ξ +� +� , +ϑ6(ξ, η, ζ) = +� +� +η +−ξ +0 +� +� . +(5.18) +For the non-gradient cell functions we use the construction introduced in [2] +Rp = +� +(p + 1)bp +i−ej∇λj − +ij +p + 1∇ξbp+1 +i +| i ∈ Io +� +, +(5.19) +where Io is the set of multi-indices of cell functions, ej is the unit multi-index with the value one at position +j and ij is the value of the i-multi-index at position j. Note that only the first term in the cell functions is +required to span the next space in the sequence due to +curl +� +[p + 1]bp +i−ej∇ξλj − +ij +p + 1∇ξbp+1 +i +� += curl([p + 1]bp +i−ej∇ξλj) . +(5.20) +However, without the added gradient the function would not belong to [Pp]3 ⊕ξ ×[�P]3 and consequently, would +not be part of the N´ed´elec space. By limiting Rp to Rp +∗ such that Rp +∗ contains only the surface permutations +with ∇λj = ej and the cell permutations with j ∈ {1, 2}, one retrieves the necessary base functions. The +sum of the lowest order N´ed´elec base functions, the template base functions, gradient base functions, and the +non-gradient cell base functions yields exactly (p+4)(p+3)(p+1)/2, thus satisfying the required dimensionality +of the N´ed´elec space. The complete space reads +N p +I = N 0 +I ⊕ +�� +i∈I +∇Ep+1 +i +� +⊕ +� +� +� +� +j∈J +∇Fp+1 +j +� +� +� ⊕ ∇Cp+1 +1234 ⊕ +� 3 +� +k=1 +Vp +k ⊗ Tk +� +⊕ +�� +i∈I +Ep +i ⊗ Ti +� +⊕ +� +� +� +� +j∈J +Fp +j ⊗ Tj +� +� +� ⊕ Rp+1 +∗ +, +I = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} , +J = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} +. +(5.21) +Here, the B´ezier basis is used to construct the higher order N´ed´elec base functions of the first type. +Definition 5.2 (B´ezier-N´ed´elec I tetrahedral basis) +The base functions are defined on the reference tetrahedron: +• for the edges we use the lowest order base functions from Eq. (5.18). The remaining edge base functions +are given by the gradients +e12 : +ϑ(ξ, η, ζ) = ϑI +1 , +ϑ(ξ, η, ζ) = ∇ξbp+1 +00k , +0 < k < p + 1 , +e13 : +ϑ(ξ, η, ζ) = ϑI +2 , +ϑ(ξ, η, ζ) = ∇ξbp+1 +0j0 , +0 < j < p + 1 , +e14 : +ϑ(ξ, η, ζ) = ϑI +3 , +ϑ(ξ, η, ζ) = ∇ξbp+1 +i00 , +0 < i < p + 1 , +e23 : +ϑ(ξ, η, ζ) = ϑI +4 , +ϑ(ξ, η, ζ) = ∇ξbp+1 +0j,p+1−j , +0 < j < p + 1 , +e24 : +ϑ(ξ, η, ζ) = ϑI +5 , +ϑ(ξ, η, ζ) = ∇ξbp+1 +i0,p+1−i , +0 < i < p + 1 , +e34 : +ϑ(ξ, η, ζ) = ϑI +6 , +ϑ(ξ, η, ζ) = ∇ξbp+1 +00k , +0 < i < p + 1 ; +(5.22) +22 + +• on faces we employ both template base functions and gradients +f123 : +ϑ(ξ, η, ζ) = bp +000ϑI +4 , +ϑ(ξ, η, ζ) = −bp +00pϑI +2 , +ϑ(ξ, η, ζ) = bp +00k(ϑI +4 − ϑI +2) , +0 < k < p , +ϑ(ξ, η, ζ) = bp +0j0(ϑI +1 + ϑI +4) , +0 < j < p , +ϑ(ξ, η, ζ) = bp +0j,p−j(ϑI +1 − ϑI +2) , +0 < j < p , +ϑ(ξ, η, ζ) = bp +0jk(ϑI +1 − ϑI +2 + ϑI +4) , +0 < j < p , +0 < k < p − j , +ϑ(ξ, η, ζ) = ∇ξbp+1 +0jk , +0 < j < p + 1 , +0 < k < p + 1 − j , +f124 : +ϑ(ξ, η, ζ) = bp +000ϑI +5 , +ϑ(ξ, η, ζ) = −bp +00pϑI +3 , +ϑ(ξ, η, ζ) = bp +00k(ϑI +5 − ϑI +3) , +0 < k < p , +ϑ(ξ, η, ζ) = bp +i00(ϑI +1 + ϑI +5) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i0,p−i(ϑI +1 − ϑI +3) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i0k(ϑI +1 − ϑI +3 + ϑI +5) , +0 < i < p , +0 < k < p − i , +ϑ(ξ, η, ζ) = ∇ξbp+1 +i0k , +0 < i < p + 1 , +0 < k < p + 1 − i , +f134 : +ϑ(ξ, η, ζ) = bp +000ϑI +6 , +ϑ(ξ, η, ζ) = −bp +0p0ϑI +3 , +ϑ(ξ, η, ζ) = bp +0j0(ϑI +6 − ϑI +3) , +0 < j < p , +ϑ(ξ, η, ζ) = bp +i00(ϑI +2 + ϑI +6) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i,p−i,0(ϑI +2 − ϑI +3) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +ij0(ϑI +2 − ϑI +3 + ϑI +6) , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η, ζ) = ∇ξbp+1 +ij0 , +0 < i < p + 1 , +0 < j < p + 1 − i , +f234 : +ϑ(ξ, η, ζ) = bp +00pϑI +6 , +ϑ(ξ, η, ζ) = −bp +0p0ϑI +5 , +ϑ(ξ, η, ζ) = bp +0j,p−j(ϑI +6 − ϑI +5) , +0 < j < p , +ϑ(ξ, η, ζ) = bp +i0,p−i(ϑI +4 + ϑI +6) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +i,p−i,0(ϑI +4 − ϑI +5) , +0 < i < p , +ϑ(ξ, η, ζ) = bp +ij,p−i−j(ϑI +4 − ϑI +5 + ϑI +6) , +0 < i < p , +0 < j < p − i , +ϑ(ξ, η, ζ) = ∇ξbp+1 +ij,p−i.j , +0 < i < p + 1 , +0 < j < p + 1 − i ; +(5.23) +• the cell base functions read +c1234 : +ϑ(ξ, η, ζ) = (p + 2)bp+1 +i−1,jke1 − +i +p + 2∇ξbp+2 +ijk , +0 < i < p + 2 , +0 < j < p + 2 − i , +0 < k < p + 2 − i − j +, +ϑ(ξ, η, ζ) = (p + 2)bp+1 +i,j−1,ke2 − +j +p + 2∇ξbp+2 +ijk , +0 < i < p + 2 , +0 < j < p + 2 − i , +0 < k < p + 2 − i − j +, +ϑ(ξ, η, ζ) = (p + 2)bp+1 +ij0 e3 − +1 +p + 2∇ξbp+2 +ij1 , +0 < i < p + 2 , +0 < j < p + 2 − i , +ϑ(ξ, η, ζ) = ∇ξbp+1 +ijk , +0 < i < p + 1 , +0 < j < p + 1 − i , +0 < k < p + 1 − i − j +. +(5.24) +23 + +6 +Numerical quadrature +Although the base functions are expressed using (α, β, γ) the domain is either the reference triangle or the +reference tetrahedron, which require fewer quadrature points than their counterparts given by the Duffy trans- +formation (quad or hexahedron). As such, we employ a mixture of the efficient quadrature points introduced +in [14,19,39,56,57] for triangles and tetrahedra, where we avoid quadrature schemes with points on the edges +or faces of the reference domain due to the recursion formula of the Bernstein polynomials Eq. (3.8). The +quadrature points are mapped to their equivalent expression in (α, β, γ). Consequently, the integration over the +reference triangle or tetrahedron reads +� +Ae +f(x, y) dA = +� +Γ +(f ◦ (ξ, η))(α, β) | det J| dΓ , +� +Ve +f(x, y, z) dV = +� +Ω +(f ◦ (ξ, η, ζ))(α, β, γ) | det J| dΩ . +(6.1) +For the lower order elements we use the Lagrangian-N´ed´elec construction from [52,53]. +7 +Boundary conditions +The degrees of freedom in [12] commute between the continuous and discrete spaces. +As such, they allow +to exactly satisfy the consistent coupling condition [11]. +We note that the functionals can be viewed as a +hierarchical system of Dirichlet boundary problems. In the case of hierarchical base functions [58], they can +be solved independently. However, here the boundary value of each polytope is required in advance due to the +non-hierarchical nature of Bernstein polynomials. In other words, one must first solve the problem for vertices, +then for edges, afterwards for faces, and finally for the cell. In our case the degrees of freedom for the cell are +irrelevant since a cell is never part of the boundary. +7.1 +Boundary vertices +The finite element mesh identifies each vertex with a tuple of coordinates. It suffices to evaluate the displacement +field at the vertex +ud +i = �u +���� +xi +. +(7.1) +If the field is vectorial, each component is evaluated at the designated vertex. The boundary conditions of the +microdistortion field are associated with tangential projections and as such do not have vertex-type degrees of +freedom. This is the case since a vertex does not define a unique tangential plane. +7.2 +Boundary edges +The edge functionals from [12] for the H 1-conforming subspace +lij(u) = +� +si +∂qj +∂s +∂u +∂s ds , +q ∈ Pp(s) , +(7.2) +can be reformulated for a reference edge on a unit domain α ∈ [0, 1]. We parametrize the edge via +x(α) = (1 − α)x1 + αx2 . +(7.3) +As such, the following relation exists between the unit parameter and the arc-length parameter +t = d +dαx = x2 − x1 , +ds = ∥dx∥ = ∥x2 − x1∥dα = ∥t∥dα . +(7.4) +24 + +α +0 +1 +ξ : α → Γ +ξ2 +ξ1 +Γ +τ +ξ +η +x2 +x1 +A +t +x +y +x : Γ → A +Figure 7.1: Barycentric mapping of edges from the unit domain to the reference triangle and onto the physical +domain. +By the chain rule we find +du +ds = du +dα +dα +ds = ∥t∥−1 du +dα , +(7.5) +for some function u. On edges, the test and trial functions are Bernstein polynomials parametrized by the unit +domain. The function representing the boundary condition �u(x) however, is parametrized by the Cartesian +coordinates of the physical space. We find its derivative with respect to the arc-length parameter by observing +d +ds �u = ⟨ d +dsx, ∇x�u⟩ . +(7.6) +The derivative of the coordinates with respect to the arc-length is simply the normed tangent vector +d +dsx = dx +dα +dα +ds = ∥t∥−1t . +(7.7) +Consequently, the edge boundary condition is given by +� +si +∂qj +∂s +∂u +∂s ds = +� 1 +0 +� +∥t∥−1 dqj +dα +� � +∥t∥−1 du +dα +� +∥t∥ dα += +� 1 +0 +� +∥t∥−1 dqj +dα +� +⟨∥t∥−1t, ∇x�u⟩∥t∥ dα = +� +si +∂qj +∂s +∂�u +∂s ds +∀ qj ∈ Pp(α) , +(7.8) +and can be solved by assembling the stiffness matrix of the edge and the load vector induced by the prescribed +displacement field �u, representing volume forces +kij = +� 1 +0 +� +∥t∥−1 dni +dα +� � +∥t∥−1 dnj +dα +� +∥t∥ dα , +fi = +� 1 +0 +⟨∥t∥−1t, ∇x�u⟩ +� +∥t∥−1 dni +dα +� +∥t∥ dα . +(7.9) +Next we consider the Dirichlet boundary conditions for the microdistortion with the N´ed´elec space of the +second type NII. The problem reads +� +si +qj⟨t, p⟩ ds = +� +si +qj⟨t, ∇x�u⟩ ds +∀ qj ∈ Pp(si) . +(7.10) +Observe that on the edge the test functions qj are chosen to be the Bernstein polynomials. Further, by the +polytopal template construction of the NII-space there holds ⟨t, θi⟩|s = ni(α). Therefore, the components of +the corresponding stiffness matrix and load vectors read +kij = +� 1 +0 +ni nj∥t∥ dα , +fi = +� 1 +0 +ni⟨t, ∇x�u⟩∥t∥ dα . +(7.11) +25 + +Note that in order to maintain the exactness property, the degree of the N´ed´elec spaces N p +I , N p +II is always one +less than the degree of the subspace Bp+1. +Lastly, we consider the N´ed´elec element of the first type. The problem is given by +� +si +qj⟨t, p⟩ ds = +� +si +qj⟨t, ∇x�u⟩ ds +∀ qj ∈ Pp(si) . +(7.12) +We define +qi = d +dαnp+1 +i +, +(7.13) +and observe that on the edges the N´ed´elec base functions yield +⟨t, θj⟩ = ⟨t, ∇xnp+1 +j +⟩ = d +dαnp+1 +j +. +(7.14) +Therefore, the components of the stiffness matrix and the load vector result in +kij = +� 1 +0 +dnp+1 +i +dα +dnp+1 +j +dα +∥t∥ dα , +fi = +� 1 +0 +dnp+1 +i +dα +⟨t, ∇x�u⟩∥t∥ dα . +(7.15) +7.3 +Boundary faces +We start with the face boundary condition for the H 1-conforming subspace. The problem reads +� +Ai +⟨∇fqj, ∇fu⟩ dA = +� +Ai +⟨∇fqj, ∇f �u⟩ dA +∀ qj ∈ Pp(Ai) . +(7.16) +The surface is parameterized by the barycentric mapping from the unit triangle Γ = {(ξ, η) ∈ [0, 1]2 | ξ +η ≤ 1}. +The surface gradient is given by +∇f �u = ∇x�u − +1 +∥n∥2 ⟨∇x�u, n⟩n , +(7.17) +where n is the surface normal. The surface gradient can also be expressed via +∇fu = ei∂x +i u = gβ∂ξ +βu , +β ∈ {1, 2} , +(7.18) +where ∂x +β are partial derivates with respect to the physical coordinates, ∂ξ +β are partial derivatives with respect +to the reference domain and gβ are the contravariant base vectors. The Einstein summation convention over +corresponding indices is implied. The covariant base vectors are given by +gβ = ∂x +∂ξβ . +(7.19) +One can find the contravariant vector orthogonal to the surface by +g3 = n = g1 × g2 . +(7.20) +We define the mixed transformation matrix +T = +� +g1 , g2 , g3� +. +(7.21) +Due to the orthogonality relation ⟨gi, gj⟩ = δ j +i the transposed inverse of T is clearly +T −T = +� +g1 , g2 , g3 +� +. +(7.22) +Thus, we can compute the surface gradient of functions parametrized by the reference triangle via +∇fu = +� +g1 , g2� +∇ξu = T −T +∗ +∇ξu , +T −T +∗ += +� +g1 , g2� +. +(7.23) +26 + +Further, there holds the following relation between the physical surface and the reference surface +dA = ∥n∥dΓ = ∥g3∥dΓ = +� +⟨g1 × g2, g3⟩ dΓ = +√ +det T dΓ . +(7.24) +Consequently, we can write the components of the stiffness matrix and load vector as +kij = +� +Γ +⟨T −T +∗ +∇ξni, T −T +∗ +∇ξnj⟩ +√ +det T dΓ , +fi = +� +Γ +⟨T −T +∗ +∇ξni, ∇x�u − (det T )−1⟨∇x�u, n⟩n⟩ +√ +det T dΓ = +� +Γ +⟨T −T +∗ +∇ξni, ∇x�u⟩ +√ +det T dΓ , +(7.25) +with the orthogonality ⟨gβ, n⟩ = 0 for β ∈ {1, 2}. +In order to embed the consistent coupling boundary condition to the microdistortion we deviate from the +degrees of freedom defined in [12] and apply the simpler H (divR)-projection +⟨qi, p, ⟩H(divR) = ⟨qi, ∇f �u⟩H(divR) +∀ qi ∈ N p +I (A) +or +∀ qi ∈ N p +II(A) . +(7.26) +Due to ker(curl) = ∇H 1 the problem reduces to +� +Ai +⟨qj, p⟩ + ⟨curl2Dqj, curl2Dp⟩ dA = +� +Ai +⟨qj, ∇f �u⟩ dA +∀ qj ∈ N p +I (A) +or +∀ qj ∈ N p +II(A) . +(7.27) +We express the co- and contravariant Piola transformation from the two-dimensional reference domain to the +three-dimensional physical domain using +θi = T −T +∗ +ϑi , +divx R θi = +1 +√ +det T +divξ R ϑi . +(7.28) +Thus, the stiffness matrix components and load vector components read +kij = +� +Γ +⟨T −T +∗ +ϑi, T −T +∗ +ϑj⟩ + ⟨(det T )−1/2 divξ R ϑi, (det T )−1/2 divξ R ϑj⟩ +√ +det T dΓ , +fi = +� +Γ +⟨T −T +∗ +ϑi, ∇x�u − (det T )−1⟨∇x�u, n⟩n⟩ +√ +det T dΓ = +� +Γ +⟨T −T +∗ +ϑi, ∇x�u⟩ +√ +det T dΓ , +(7.29) +where we again make use of the orthogonality between the surface tangent vectors and its normal vector. +8 +Numerical examples +In the following we test the finite element formulations with an artificial analytical solution in the antiplane shear +model and with an analytical solution for an infinite plane under cylindrical bending in the three dimensional +model. Finally, we benchmark the ability of the finite element formulations to correctly interpolate between +micro Cmicro and macro Cmacro stiffnesses as described by the characteristic length scale parameter Lc. The +majority of convergence results are presented by measuring the error in the Lebesgue norm over the domain +∥�u − uh∥L2 = +�� +V +∥�u − uh∥2 dV , +∥ �P − P h∥L2 = +�� +V +∥ �P − P h∥2 dV , +(8.1) +in which context {�u, �P } and {uh, P h} are the analytical and approximate subspace solutions, respectively. +8.1 +Compatible microdistortion +In [53] we explored the conditions for which the microdistortion p reduces to a gradient field, i.e. p is compatible. +By defining the micro-moment with a scalar potential +m = ∇100 − x2 − y2 +10 += −1 +5 +�x +y +� +, +(8.2) +27 + +and constructing an analytical solution for the displacement field +�u = sin +�x2 + y2 +5 +� +, +(8.3) +we can recover the analytical solution of the microdistortion +p = +1 +µe + µmicro +(m + µe∇�u) = 1 +2 +� +−1 +5 +�x +y +� ++ 2 +5 +�x cos([x2 + y2]/5) +y cos([x2 + y2]/5) +�� += 1 +5 +�x cos([x2 + y2]/5) +y cos([x2 + y2]/5) +� +− 1 +10 +�x +y +� +, +(8.4) +where for simplicity we set all material constants to one. Since m is a gradient field, the microdistortion p is +also reduced to a gradient field and curl2Dp = 0, see [53]. Note that this result is specific to antiplane shear +and does not generalize to the full three-dimensional model, compare [52]. We note that the microdistortion +is not equal to the gradient of the displacement field and as such, their tangential projections on an arbitrary +boundary are not automatically the same. However, for both the gradient of the displacement field and the +micro-moment is the tangential projection on the boundary of the circular domain A = {x ∈ R2 | ∥x∥ ≤ 10} +equal to zero +⟨∇t, �u⟩ +���� +∂A += ⟨t, m⟩ +���� +∂A += 0 , +(8.5) +and as such the microdistortion belongs to p ∈ H0(curl, A). +Consequently, we can set sD = ∂A and the +consistent coupling condition remains compatible. +With the displacement and the microdistortion fields at +hand we derive the corresponding forces +f = 1 +25 +� +2x2 sin +�x2 + y2 +5 +� ++ 2y2 sin +�x2 + y2 +5 +� +− 10 cos +�x2 + y2 +5 +� +− 5 +� +. +(8.6) +The approximation of the displacement and microdistortion fields using linear and higher order elements is +shown in Fig. 8.1. We note that even with almost 3000 finite elements and 6000 degrees of freedom the linear +formulation is incapable of finding an adequate approximation. On the other side of the spectrum, the higher +order approximation (degree 7) with 57 elements and 4097 degrees of freedom yields very accurate results in +the interior of the domain. However, the exterior of the domain is captured rather poorly. This is the case since +the geometry of the circular domain is being approximated by linear triangles. Thus, in this setting, a finer +mesh captures the geometry in a more precise manner. The effects of the geometry on the approximation of the +solution are also clearly visible in the convergence graphs in Fig. 8.2; only after a certain accuracy in the domain +description is achieved do the finite elements retrieve their predicted convergence rates, compare [52,53]. This +is clearly observable when comparing the convergence curves of the linear and seventh order elements. The +linear element generates quadratic convergence p + 1 = 1 + 1 = 2, whereas the seventh-order element yields +the convergence slope 7 (where 8 is expected). +Although the seventh-order formulation encompasses more +degrees of freedom, it employs a coarser mesh and as such, generates higher errors at the boundary. The errors +themselves can be traced back to the consistent coupling condition since, for a non-perfect circle the gradient +of the displacement field induces tangential projections on the imperfect boundary. The influence of the latter +effect is even more apparent in the convergence of the microdistortion, where the higher order formulations are +unable to perform optimally on coarse meshes. +8.2 +Cylindrical bending +In order to test the capability of the finite element formulations to capture the intrinsic behaviour of the relaxed +micromorphic model, we compare with analytical solutions of boundary-value problems. The first example +considers the displacement and microdistortion fields under cylindrical bending [43] for infinitely extended +plates. Let the plates be defined as V = (−∞, ∞)2 × [−1/2, 1/2], than the analytical solution for cylindrical +bending reads +u = κ +� +� +−xz +0 +x2/2 +� +� , +P = −κ +� +� +[41z + 20 +√ +82 sech( +� +41/2) sinh( +√ +82z)]/1681 +0 +x +0 +0 +0 +−x +0 +0 +� +� , +(8.7) +28 + +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +(j) +(k) +(l) +Figure 8.1: Depiction of the displacement field (a)-(c) and the microdistortion field (d)-(f) for the antiplane +shear problem, for the linear element under h-refinement with 225, 763 and 2966 elements, corresponding to +485, 1591 and 6060 degrees of freedom. The p-refinement of the displacement field on the coarsest mesh of 57 +elements is visualized in (g)-(l) with p ∈ {3, 5, 7}, corresponding to 731, 2072 and 4097 degrees of freedom. +29 + +11 +NA144 +44103 +104 +105 +10−3 +10−1 +101 +degrees of freedom +∥�u − uh∥L2 +L1 × N 0 +I +L2 × N 1 +II +B3 × N 2 +II +B5 × N 4 +II +B7 × N 6 +II +O(h2) +O(h7) +(a) +103 +104 +105 +10−3 +10−1 +101 +degrees of freedom +∥�p − ph∥L2 +L1 × N 0 +I +L2 × N 1 +II +B3 × N 2 +II +B5 × N 4 +II +B7 × N 6 +II +O(h) +O(h2) +(b) +Figure 8.2: Convergence of displacement (a) and the microdistortion (b) under h-refinement for multiple poly- +nomial degrees for the antiplane shear problem. +where sech(x) = 1/ cosh(x), and for the following values of material constants +λe = λmicro = 0 , +µe = µmacro = 1/2 , +µc = 0 , +Lc = 1 , +µmicro = 20 . +(8.8) +The intensity of the curvature parameter κ of the plate is chosen to be κ = 14/200. +Remark 8.1 +The particular case of the cylindrical bending for which λe = λmicro = 0 (equivalent to a zero micro-Poisson’s +ratio) has been solved, along with its more general case (λe ̸= λmicro ̸= 0), in [43]. The advantage of considering +this particular case is that a cut out finite plate of the infinite domain automatically exhibits the consistent +coupling boundary conditions on its side surfaces. +Remark 8.2 +Note that the general analytical solution for cylindrical bending does not depend on µc, so we can set µc = 0 +without loss of generality, compare [43]. +We define the finite domain V = [−10, 10]2 × [−1/2, 1/2] and the boundaries +AD1 = {−10} × [−10, 10] × [−1/2, 1/2] , +AD2 = {10} × [−10, 10] × [−1/2, 1/2] , +AN = ∂V \ {AD1 ⊕ AD2} . +(8.9) +Additionally, on the Dirichlet boundary we impose the translated analytical solution �u = u − +�0 +0 +3.5�T . +The displacement field and the last row of the microdistortion are depicted in Fig. 8.3. The displacement +field is dominated by its quadratic term and captured correctly. +The last row of the microdistortion is a +linear function and easily approximated even with linear elements. On the contrary, the P11 component of +the microdistortion is a hyperbolic function of the z-axis. +The results of its approximation at x = y = 0 +(the centre of the plane) are given in Fig. 8.4. We observe that even increasing the number of linear finite +elements to the extreme only results in better oscillations around the analytical solution. In comparison, higher +order formulations converge towards the expected hyperbolic behaviour. The approximation of the quadratic +N´ed´elec element of the first type is nearly perfect, whereas its second type counterpart clearly deviates from +the analytical solution at z ≈ −0.25. Taking the cubic second type element yields the desired result. This +phenomenon is an evident indicator of the prominent role of the Curl of the microdistortion in this type of +problems. Firstly, the microdistortion is a non-gradient field. Secondly, the Curl of the analytical solution +induces an hyperbolic sine term. Such functions are often approximated using at least cubic terms in power +series, thus explaining the necessity of such high order elements for correct computations. +30 + +(a) +(b) +Figure 8.3: Displacement (a) and last row of the microdistortion (b) for the quadratic formulation using the +N´ed´elec element of the first type. +−0.5 +0 +0.5 +−1 +0 +1 +·10−2 +z-axis +P11(z) +ne = 5640 +ne = 44592 +ne = 354720 +(a) +−0.5 +0 +0.5 +−1 +0 +1 +·10−2 +z-axis +P11(z) +B2 × N 1 +I +B3 × N 2 +I +L2 × N 1 +II +B3 × N 2 +II +B4 × N 3 +II +(b) +Figure 8.4: Convergence of the lowest order formulation under h-refinement with 732, 5640 and 44592 elements +(a) and of the higher order formulations under p-refinement using 732 elements(b) towards the analytical solution +(dashed curve) of the P11(z) component at x = y = 0. +31 + +8.3 +Bounded stiffness property +The characteristic length scale parameter Lc allows the relaxed micromorphic model to capture the transition +from highly homogeneous materials to materials with a pronounced micro-structure by governing the influence +of the micro-structure on the overall behaviour of the model. We demonstrate this property of the model with +an example, where we vary Lc and measure the resulting energy. +Let the domain be given by the axis-symmetric cube V = [−1, 1]3 with a total Dirichlet boundary +AD1 = {(x, y, z) ∈ [−1, 1]3 | x = ±1} , +AD2 = {(x, y, z) ∈ [−1, 1]3 | y = ±1} , +AD3 = {(x, y, z) ∈ [−1, 1]3 | z = ±1} , +(8.10) +we embed the periodic boundary conditions +�u +���� +AD1 += +� +� +(1 − y2) sin(π[1 − z2])/10 +0 +0 +� +� , +�u +���� +AD2 += +� +� +0 +(1 − x2) sin(π[1 − z2])/10 +0 +� +� , +�u +���� +AD3 += +� +� +0 +0 +(1 − y2) sin(π[1 − x2])/10 +� +� . +(8.11) +The material parameters are chosen as +λmacro = 2 , +µmacro = 1 , +λmicro = 10 , +µmicro = 5 , +µc = 1 , +(8.12) +thus giving rise to the following meso-parameters via Eq. (2.19) +λe = 2.5 , +µe = 1.25 . +(8.13) +The displacement field as well as some examples of the employed meshes are shown in Fig. 8.5. In order to +compute the upper and lower bound on the energy we utilize the equivalent Cauchy model formulation with +the micro- and macro elasticity parameters. In order to assert the high accuracy of the solution of the bounds +we employ tenth order finite elements. The progression of the energy in dependence of the characteristic length +parameter Lc is given in Fig. 8.6. We observe the high mesh dependency of the lower order formulations, where +the energy is clearly overestimated. The higher order formulations all capture the upper bound correctly but +diverge with respect to the result of the lower bound. Notably, the approximation using the N´ed´elec element +of the first type is more accurate than the equivalent formulation with the N´ed´elec element of the second type, +thus indicating the non-negligible involvement of the micro-dislocation in the energy. Using standard mesh +coarseness the cubic element formulation with N´ed´elec elements of the first type yields satisfactory results. In +order to achieve the same on highly coarse meshes, one needs to employ seventh order elements. +9 +Conclusions and outlook +The intrinsic behaviour of the relaxed micromorphic model is revealed by the analytical solutions to boundary +value problems. Clearly, the continuum exhibits hyperbolic and trigonometric solutions, which are not easily +approximated by low order finite elements. The example provided in Section 8.2 demonstrates that cubic and +higher order finite elements yield excellent results in approximate solutions of the model. +The polytopal template methodology introduced in [50] allows to easily and flexibly construct H (curl)- +conforming vectorial finite elements that inherit many of the characteristics of an underlying H 1-conforming +basis, which can be chosen independently. In this work, we made use of Bernstein-B´ezier polynomials. The +latter boast optimal complexity properties manifesting in the form of sum factorization. The natural decom- +position of their multi-variate versions into multiplications of univariate Bernstein base functions via the Duffy +transformation allows to construct optimal iterators for their evaluation using recursion formulas. Further, this +characteristic makes the use of dual numbers in the computation of their derivatives ideal. Finally, the intrinsic +order of traversal induced by the factorization is exploited optimally by the choice of clock-wise orientation +of the reference element. The consequence of these combined features is a high-performance hp-finite element +program. +32 + +(a) +(b) +(c) +Figure 8.5: Displacement field of the Cauchy model on the coarsest mesh of 48 finite elements of the tenth order +(a) and depictions of the meshes with 384 (b) and 3072 (c) elements, respectively. +The ability of the relaxed micromorphic model to interpolate between the energies of homogeneous materials +and materials with an underlying micro-structure using the characteristic length scale parameter Lc is demon- +strated in Section 8.3. It is also shown that in order to correctly capture the span of energies for the values of +Lc either fine-discretizations or higher order elements are required. +The excellent performance of the proposed higher order finite elements in the linear static case is a precur- +sor for their application in the dynamic setting, which is important since the relaxed micromorphic model is +often employed in the computation of elastic waves (e.g., for acoustic metamaterials), where solutions for high +frequency ranges are commonly needed. +The proposed computational scheme is lacking in its description of curved geometries. Due to the consistent +coupling condition, this can easily lead to errors emanating from the boundary. Consequently, a topic for future +works would be the investigation of curved finite elements [20,21] and their behaviour with respect to the model. +Acknowledgements +Angela Madeo and Gianluca Rizzi acknowledge support from the European Commission through the funding +of the ERC Consolidator Grant META-LEGO, N◦ 101001759.00 +Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational +Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project- +No. 440935806. +10 +References +[1] Ainsworth, M., Andriamaro, G., Davydov, O.: Bernstein–B´ezier finite elements of arbitrary order and optimal assembly +procedures. SIAM Journal on Scientific Computing 33(6), 3087–3109 (2011) +[2] Ainsworth, M., Fu, G.: Bernstein–B´ezier bases for tetrahedral finite elements. 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Computational Mechanics (2022) +[50] Sky, A., Muench, I.: Polytopal templates for the formulation of semi-continuous vectorial finite elements of arbitrary order. +arXiv:2210.03525 (2022) +[51] Sky, A., Muench, I., Neff, P.: On [H 1]3×3, [H (curl)]3 and H (symCurl) finite elements for matrix-valued Curl problems. Journal +of Engineering Mathematics 136(1), 5 (2022) +[52] Sky, A., Neunteufel, M., Muench, I., Sch¨oberl, J., Neff, P.: Primal and mixed finite element formulations for the relaxed +micromorphic model. Computer Methods in Applied Mechanics and Engineering 399, 115298 (2022) +[53] Sky, A., Neunteufel, M., M¨unch, I., Sch¨oberl, J., Neff, P.: A hybrid H 1 × H (curl) finite element formulation for a relaxed +micromorphic continuum model of antiplane shear. Computational Mechanics 68(1), 1–24 (2021) +[54] Solin, P., Segeth, K., Dolezel, I.: Higher-Order Finite Element Methods (1st ed.). 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URL https://www.numerik.math.tugraz.at/~zaglmayr/pub/szthesis.pdf +36 + diff --git a/BdAzT4oBgHgl3EQfh_0J/content/tmp_files/load_file.txt b/BdAzT4oBgHgl3EQfh_0J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc02b0c7caf899e8ecc28561345b54e1ceeff51f --- /dev/null +++ b/BdAzT4oBgHgl3EQfh_0J/content/tmp_files/load_file.txt @@ -0,0 +1,1483 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf,len=1482 +page_content='Higher order Bernstein-B´ezier and N´ed´elec finite elements for the relaxed micromorphic model Adam Sky1, Ingo Muench2, Gianluca Rizzi3 and Patrizio Neff4 January 5, 2023 Abstract The relaxed micromorphic model is a generalized continuum model that is well-posed in the space X = [H 1]3 × [H (curl)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, finite element formulations of the model rely on H 1-conforming subspaces and N´ed´elec elements for discrete solutions of the corresponding variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This work applies the recently introduced polytopal template methodology for the construction of N´ed´elec elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This is done in conjunction with Bernstein-B´ezier polynomials and dual numbers in order to compute hp- FEM solutions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Bernstein-B´ezier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The application of the polytopal template methodology for the construction of the N´ed´elec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein- B´ezier basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We introduce the Bernstein-B´ezier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of N´ed´elec elements based on Bernstein-B´ezier polynomials with the polytopal template methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The performance of the elements is shown in examples of the relaxed micromorphic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Key words: N´ed´elec elements, Bernstein-B´ezier elements, relaxed micromorphic model, dual numbers, au- tomatic differentiation, hp-FEM, generalized continua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 1 Introduction One challenge that arises in the computation of materials with a pronounced micro-structure is the necessity of modelling the complex geometry of the domain as a whole, in order to correctly capture its intricate kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In other words, unit-cell geometries in metamaterials or various hole-shapes in porous media have to be accounted for in order to assert the viability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Naturally, this correlates with the resolution of the discretization in finite element simulations, resulting in longer computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The relaxed micromorphic model [35] offers an alternative approach by introducing a continuum model with enriched kinematics, accounting for the independent distortion arising from the micro-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As such, for each material point, the model introduces the microdistortion field P in addition to the standard displacement field u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, each material point is endowed with twelve degrees of freedom, effectively turning into an affine-deformable micro-body with its own orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In contrast to the classical micromorphic model [17] 1Corresponding author: Adam Sky, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8, 44227 Dortmund, Germany, email: adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='sky@tu-dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='de 2Ingo Muench, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8, 44227 Dortmund, Germany, email: ingo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='muench@tu-dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='de 3Gianluca Rizzi, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt- Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8, 44227 Dortmund, Germany, email: gianluca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='rizzi@tu-dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='de 4Patrizio Neff, Chair for Nonlinear Analysis and Modelling, Faculty of Mathematics, Universit¨at Duisburg-Essen, Thea- Leymann Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 9, 45127 Essen, Germany, email: patrizio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='neff@uni-due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='01491v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='NA] 4 Jan 2023 by Eringen [15] and Mindlin [29], the relaxed micromorphic model does not employ the full gradient of the microdistortion DP in its energy functional but rather its skew-symmetric part Curl P , designated as the micro-dislocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Therefore, the micro-dislocation Curl P remains a second-order tensor, whereas DP is a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, the model allows the transition between materials with a pronounced micro- structure and homogeneous materials using the characteristic length scale parameter Lc, which governs the influence of the micro-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In highly homogeneous materials the characteristic length scale parameter approaches zero Lc → 0, and for materials with a pronounced micro-structure its value is related to the size of the underlying unit-cell geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Recent works demonstrate the effectiveness of the model in the simulation of band-gap metamaterials [7, 10, 13, 27, 28] and shielding against elastic waves [4, 40, 41, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Furthermore, analytical solutions are already available for bending [43], torsion [42], shear [44], and extension [45] kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We note that the usage of the curl operator in the free energy functional directly influences the appropriate Hilbert spaces for existence and uniqueness of the related variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Namely, the relaxed micromorphic model is well-posed in {u, P } ∈ X = [H 1]3 × [H (curl)]3 [18,34], although the regularity of the microdistortion can be improved to P ∈ [H 1]3×3 for certain smoothness of the data [22, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As shown in [52], the X -space asserts well-posedness according to the Lax-Milgram theorem, such that H 1-conforming subspaces and N´ed´elec elements [9,30,31] inherit the well-posedness property as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In this work we apply the polytopal template methodology introduced in [50] in order to construct higher order N´ed´elec elements based on Bernstein polynomials [23] and apply the formulation to the relaxed micro- morphic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Bernstein polynomials are chosen due to their optimal complexity property in the assembly procedure [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We further enhance this feature by employing dual numbers [16] in order to compute the values of the base functions and their derivatives simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The polytopal template methodology allows to extend this property to the assembly of the N´ed´elec base functions, resulting in fast computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Alternatively, the formulation of higher order elements on the basis of Legendre polynomials can be found in [48, 54, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The construction of low order N´ed´elec elements can be found in [5, 51] and specifically in the context of the the relaxed micromorphic model in [47,49,52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' First, we introduce the relaxed micromorphic model and its limit cases with respect to the characteristic length scale parameter Lc, after which we reduce it to a model of antiplane shear [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Next, we shortly discuss Bernstein polynomials and dual numbers for automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The B´ezier polynomial basis for triangles and tetrahedra is introduced, along with its factorization, highlighting its compatibility with dual numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We consider a numerical example in antiplane shear for two-dimensional elements, a three-dimensional example for convergence of cylindrical bending, and a benchmark for the behaviour of the model with respect to the characteristic length scale parameter Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Lastly, we present our conclusions and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The following definitions are employed throughout this work: vectors are indicated by bold letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Non-bold letters represent scalars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' in general, formulas are defined using the Cartesian basis, where the base vectors are denoted by e1, e2 and e3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' three-dimensional domains in the physical space are denoted with V ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The corresponding reference domain is given by Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' analogously, in two dimensions we employ A ⊂ R2 for the physical domain and Γ for the reference domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' curves on the physical domain are denoted by s, whereas curves in the reference domain by µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' the tangent and normal vectors in the physical domain are given by t and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Their counter- parts in the reference domain are τ for tangent vectors and ν for normal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2 2 The relaxed micromorphic model The relaxed micromorphic model [35] is governed by a free energy functional, incorporating the gradient of the displacement field Du, the microdistortion P and the Curl of the microdistortion I(u, P ) = 1 2 � V ⟨sym(Du − P ), Ce sym(Du − P )⟩ + ⟨sym P , Cmicro sym P ⟩ + ⟨skew(Du − P ), Cc skew(Du − P )⟩ + µmacroL2 c⟨Curl P , L Curl P ⟩ dV − � V ⟨u, f⟩ + ⟨P , M⟩ dV → min w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' {u, P } , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) where the Curl operator for second order tensors is defined row-wise as Curl P = � � curl( �P11 P12 P13 � ) curl( �P21 P22 P23 � ) curl( � P31 P32 P33 � ) � � = � � P13,y − P12,z P11,z − P13,x P12,x − P11,y P23,y − P22,z P21,z − P23,x P22,x − P21,y P33,y − P32,z P31,z − P33,x P32,x − P31,y � � , curl p = ∇ × p , p : V ⊂ R3 → R3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2) and curl(·) is the vectorial curl operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The displacement field and the microdistortion field are functions of the reference domain u : V ⊂ R3 → R3 , P : V ⊂ R3 → R3×3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3) The tensors Ce, Cmicro, L ∈ R3×3×3×3 are standard positive definite fourth order elasticity tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' For isotropic materials they take the form Ce = λe1 ⊗ 1 + 2µe J , Cmicro = λmicro1 ⊗ 1 + 2µmicro J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4) where 1 is the second order identity tensor and J is the fourth order identity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The fourth order tensor Cc ∈ R3×3×3×3 is a positive semi-definite material tensor related to Cosserat micro-polar continua and accounts for infinitesimal rotations Cc : so(3) → so(3), where so(3) is the space of skew-symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' For isotropic materials there holds Cc = 2µc J, where µc ≥ 0 is called the Cosserat couple modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, for simplicity, we assume L = J in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The macroscopic shear modulus is denoted by µmacro and Lc represents the characteristic length scale motivated by the geometry of the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The forces and micro-moments are given by f and M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Equilibrium is found at minima of the energy functional, which is strictly convex (also for Cc ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As such, we consider variations with respect to its parameters, namely the displacement and the microdistortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Taking variations of the energy functional with respect to the displacement field u yields δuI = � V ⟨sym Dδu, Ce sym(Du − P )⟩ + ⟨skew Dδu, Cc skew(Du − P )⟩ − ⟨δu, f⟩ dV = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) The variation with respect to the microdistortion P results in δP I = � V ⟨sym δP , Ce sym(Du − P )⟩ + ⟨skew δP , Cc skew(Du − P )⟩ − ⟨sym δP , Cmicro sym P ⟩ − µmacroL2 c⟨Curl δP , Curl P ⟩ + ⟨δP , M⟩ dV = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) From the total variation we extract the bilinear form a({δu, δP }, {u, P }) = � V ⟨sym(Dδu − δP ), Ce sym(Du − P )⟩ + ⟨sym δP , Cmicro sym P ⟩ + ⟨skew(Dδu − δP ), Cc skew(Du − P )⟩ + µmacroL2 c⟨Curl δP , Curl P ⟩ dV , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='7) and linear form of the loads l({δu, δP }) = � V ⟨δu, f⟩ + ⟨δP , M⟩ dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) 3 Applying integration by parts to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) yields � ∂V ⟨δu , [Ce sym(Du − P ) + Cc skew(Du − P )] n⟩ dA − � V ⟨δu , Div[Ce sym(Du − P ) + Cc skew(Du − P )] − f⟩ dV = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) Likewise, integration by parts of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) results in � V ⟨δP , Ce sym(Du − P ) + Cc skew(Du − P ) − Cmicro sym P − µmacroL2 c Curl Curl P + M⟩ dV − µmacroL2 c � ∂V ⟨δP , Curl P × n⟩ dA = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) The strong form is extracted from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) by splitting the boundary A = AD ∪ AN , AD ∩ AN = ∅ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='11) into a Dirichlet boundary with embedded boundary conditions and a Neumann boundary with natural boundary conditions, such that no tractions are imposed on the Neumann boundary − Div[Ce sym(Du − P ) + Cc skew(Du − P )] = f in V , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12a) −Ce sym(Du − P ) − Cc skew(Du − P ) + Cmicro sym P + µmacro L2 c Curl Curl P = M in V , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12b) u = �u on Au D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12c) P × n = �P × n on AP D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12d) [Ce sym(Du − P ) + Cc skew(Du − P )] n = 0 on Au N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12e) Curl P × n = 0 on AP N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12f) The force stress tensor �σ := Ce sym(Du − P ) + Cc skew(Du − P ) is symmetric if and only if Cc ≡ 0, a case which is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12 represents a tensorial Maxwell-problem coupled to linear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We observe that the Dirichlet boundary condition for the microdistortion controls only its tangential components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' It is unclear, how to control the micro-movements of a material point without also affecting the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Therefore, the relaxed micromorphic model introduces the so called consistent coupling condition [11] P × n = D�u × n on AP D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) where the prescribed displacement on the Dirichlet boundary �u automatically dictates the tangential component of the microdistortion on that same boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, the consistent coupling condition enforces the definitions AD = Au D = AP D and AN = Au N = AP N (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, the consistent coupling condition substitutes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The set of equations in Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12 remains well-posed for Cc ≡ 0 due to the generalized Korn inequality for incompatible tensor fields [24–26,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The inequality relies on a non-vanishing Dirichlet boundary for the microdistortion field AP D ̸= ∅, which the consistent coupling condition guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 Limits of the characteristic length scale parameter - a true two scale model In the relaxed micromorphic model the characteristic length Lc takes the role of a scaling parameter between the well-defined macro and the micro scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This property, unique to the relaxed micromorphic model, allows the theory to interpolate between materials with a pronounced micro-structure and homogeneous materials, thus relating the characteristic length scale parameter Lc to the size of the micro-structure in metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In the lower limit Lc → 0 the continuum is treated as homogeneous and the solution of the classical Cauchy continuum theory is retrieved [3,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This can be observed by reconsidering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12b) for Lc = 0, −Ce sym(Du − P ) − Cc skew(Du − P ) + Cmicro sym P = M , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='14) which can now be used to express the microdistortion P algebraically sym P = (Ce + Cmicro)−1(sym M + Ce sym Du) , skew P = C−1 c skew M + skew Du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='15) 4 x y n f M V AD = Au D = AP D AN = Au N = AP N Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1: The domain in the relaxed micromorphic model with Dirichlet and Neumann boundaries under internal forces and micro-moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The Dirichlet boundary of the microdistortion is given by the consistent coupling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The model can capture the complex kinematics of an underlying micro-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Setting M = 0 corresponds to Cauchy continua, where micro-moments are not accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Thus, one finds Cc skew(Du − P ) = 0 , Ce sym(Du − P ) = Cmicro sym P , sym P = (Ce + Cmicro)−1Ce sym Du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='16) Applying the former results to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12a) yields − Div[Ce sym(Du − P )] = − Div[Cmicro(Ce + Cmicro)−1Ce sym Du] = − Div[Cmacro sym Du] = f , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='17) where the definition Cmacro = Cmicro(Ce + Cmicro)−1Ce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='18) relates the meso- and micro-elasticity tensors to the classical macro-elasticity tensor of the Cauchy continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In fact, Cmacro contains the material constants that arise from standard homogenization for large periodic structures [3,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' For isotropic materials one can directly express the macro parameters [33] µmacro = µe µmicro µe + µmicro , 2µmacro + 3λmacro = (2µe + 3λe)(2µmicro + 3λmicro) (2µe + 3λe) + (2µmicro + 3λmicro) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) in terms of the parameters of the relaxed micromorphic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In the upper limit Lc → +∞, the stiffness of the micro-body becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As the characteristic length Lc can be viewed as a zoom-factor into the microstructure, the state Lc → +∞ can be interpreted as the entire domain being the micro-body itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' However, this is only theoretically possible as in practice, the limit is given by the size of one unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Since the energy functional being minimized contains µmacroL2 c∥ Curl P ∥2, on contractible domains and bounded energy this implies the reduction of the microdistortion to a gradient field P → Dv due to the classical identity Curl Dv = 0 ∀ v ∈ [C ∞(V )]3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='20) thus asserting finite energies of the relaxed micromorphic model for arbitrarily large characteristic length values Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The corresponding energy functional in terms of the reduced kinematics {u, v} : V → R3 now reads I(u, v) = 1 2 � V ⟨sym(Du − Dv), Ce sym(Du − Dv)⟩ + ⟨sym Dv, Cmicro sym Dv⟩ + ⟨skew(Du − Dv), Cc skew(Du − Dv)⟩ dV − � V ⟨u, f⟩ + ⟨Dv, M⟩ dV , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='21) such that variation with respect to the two vector fields u and v leads to δuI = � V ⟨sym Dδu, Ce sym(Du − Dv)⟩ + ⟨skew Dδu, Cc skew(Du − Dv)⟩ − ⟨δu, f⟩ dV = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='22a) δvI = � V ⟨sym Dδv, Ce sym(Du − Dv)⟩ + ⟨skew Dδv, Cc skew(Du − Dv)⟩ − ⟨sym Dδv, Cmicro sym Dv⟩ + ⟨Dδv, M⟩ dV = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='22b) 5 The resulting bilinear form is given by a({δu, δv}, {u, v}) = � V ⟨sym(Dδu − Dδv), Ce sym(Du − Dv)⟩ + ⟨sym Dδv, Cmicro sym Dv⟩ + ⟨skew(Dδu − Dδv), Cc skew(Du − Dv)⟩ dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='23) By partial integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='22a) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='22b) one finds the equilibrium equations − Div[Ce sym(Du − Dv) + Cc skew(Du − Dv)] = f in V , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24a) − Div[Ce sym(Du − Dv) + Cc skew(Du − Dv)] + Div[Cmicro sym Dv] = Div M in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24b) We can now substitute the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24a) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24b) to find − Div(Cmicro sym Dv) = f − Div M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='25) Clearly, setting v = u satisfies both local equilibrium equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24a) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24b) for f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, the consistent coupling condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) is also automatically satisfied, asserting the equivalence of the tangential projections of both fields on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Since, as shown in [32, 52] using the extended Brezzi theorem, the case Lc → +∞ is well-posed (including Cc ≡ 0), the solution v = u is the unique solution to the bilinear form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='23) with the right-hand side l({δu, δv}) = ⟨Dδv, M⟩ dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='26) Effectively, equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='25) implies that the limit Lc → +∞ defines a classical Cauchy continuum with a finite stiffness governed by Cmicro, representing the upper limit of the stiffness for the relaxed micromorphic continuum [32], where the corresponding forces read m = Div M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We emphasize that this interpretation of Cmicro is impossible in the classical micromorphic model since there the limit Lc → +∞ results in a constant microdistortion field P : V → R3×3 as its full gradient DP is incorporated via µmacroL2 c∥DP ∥2 into the energy functional [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 Antiplane shear We introduce the relaxed micromorphic model of antiplane shear1 [55] by reducing the displacement field to u = �0, 0, u�T , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='27) such that u = u(x, y) is a function of the x − y-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, its gradient reads Du = � � 0 0 0 0 0 0 u,x u,y 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='28) The structure of the microdistortion tensor is chosen accordingly P = � � 0 0 0 0 0 0 p1 p2 0 � � , Curl P = � � 0 0 0 0 0 0 0 0 p2,x − p1,y � � = � � 0 0 0 0 0 0 0 0 curl2Dp � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='29) 1Note that the antiplane shear model encompasses 1 + 2 = 3 degrees of freedom and is the simplest non-trivial active version for the relaxed micromorphic model, as the one-dimensional elongation ansatz features only 1 + 1 = 2 degrees of freedom and eliminates the curl operator I(u, p) = 1 2 � s (λe + 2µe)|u′ − p|2 + (λmicro + 2µmicro)|p|2 ds − � s u f + p m ds → min w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' {u, p} , since Du = u′ e1 ⊗ e1 and P = p e1 ⊗ e1, such that skew(Du − P ) = 0 and Curl P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This is not to be confused with uniaxial extension, which entails 1 + 3 = 4 degrees of freedom [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 6 Analogously to the displacement field u, the microdistortion P is also set to be a function of the {x, y}-variables P = P (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We observe the following sym-skew decompositions of the gradient and microdistortion tensors sym P = 1 2 � � 0 0 p1 0 0 p2 p1 p2 0 � � , sym(Du − P ) = 1 2 � � 0 0 u,x − p1 0 0 u,y − p2 u,x − p1 u,y − p2 0 � � , skew(Du − P ) = 1 2 � � 0 0 p1 − u,x 0 0 p2 − u,y u,x − p1 u,y − p2 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='30) Clearly, there holds tr[sym P ] = tr[sym(Du − P )] = tr[skew(Du − P )] = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='31) such that the contraction with the material tensors reduces to Ce sym(Du − P ) = 2µe sym(Du − P ) , Cmicro sym(Du − P ) = 2µmicro sym P , Cc skew(Du − P ) = 2µc skew(Du − P ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='32) As such, the quadratic forms of the energy functional are given by ⟨sym(Du − P ), Ce sym(Du − P )⟩ = µe∥∇u − p∥2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='33a) ⟨skew(Du − P ), Cc skew(Du − P )⟩ = µc∥∇u − p∥2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='33b) ⟨sym P , Cmicro sym P ⟩ = µmicro∥p∥2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='33c) with the definitions ∇u = �u,x u,y � , p = �p1 p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='34) The resulting energy functional for antiplane shear reads therefore I(u, p) = 1 2 � A (µe + µc)∥∇u − p∥2 + µmicro∥p∥2 + µmacroL2 c∥curl2Dp∥2 dA − � A u f + ⟨p, m⟩ dA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='35) In order to maintain consistency with the three-dimensional model we must choose µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The reasoning for this choice is explained upon in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, the energy functional is given by I(u, p) = 1 2 � A µe∥∇u − p∥2 + µmicro∥p∥2 + µmacroL2 c∥curl2Dp∥2 dA − � A u f + ⟨p, m⟩ dA → min w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' {u, p} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='36) Note that on two-dimensional domains the differential operators are reduced to ∇u = �u,x u,y � , R∇u = � u,y −u,x � , R = � 0 1 −1 0 � , curl2Dp = div(Rp) = p2,x − p1,y , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='37) where we note that curl2D is just a rotated divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Taking variations of the energy functional with respect to the displacement field results in δuI = � A µe⟨∇δu, ∇u − p⟩ − δu f dA = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='38) and variation with respect to the microdistortion yields δpI = � A µe⟨δp, ∇u − p⟩ − µmicro⟨δp, p⟩ − µmacroL2 c(curl2Dδp)curl2Dp + ⟨δp, m⟩ dA = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='39) 7 Consequently, one finds the bilinear and linear forms a({δu, δp}, {u, p}) = � A µe⟨∇δu − δp, ∇u − p⟩ + µmicro⟨δp, p⟩ + µmacroL2 c(curl2Dδp)curl2Dp dA , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='40a) l({δu, δp}) = � A δu f + ⟨δp, m⟩ dA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='40b) Partial integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='38) results in � ∂A δu ⟨µe(∇u − p), n⟩ ds − � A δu [µe div(∇u − p) + f] dA = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='41) and analogously for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='39), yielding � A ⟨δp, µe(∇u − p) − µmicro p − µmacroL2 cR∇curl2Dp + m⟩ dA − � ∂A ⟨δp, µmacroL2 c(curl2Dp) t⟩ ds = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='42) Consequently, the strong form reads −µe div(∇u − p) = f in A , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43a) −µe(∇u − p) + µmicro p + µmacroL2 cR∇curl2Dp = m in A , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43b) u = �u on su D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43c) ⟨p, t⟩ = ⟨�p, t⟩ on sP D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43d) ⟨∇u, n⟩ = ⟨p, n⟩ on su N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43e) curl2Dp = 0 on sP N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43f) The consistent coupling condition accordingly reduces to ⟨p, t⟩ = ⟨∇�u, t⟩ on sD = sP D = su D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='44) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 Note that without setting µc = 0 in the antiplane shear model, the analogous result to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='17) in the limit Lc → 0 would read − � µmicro [µe + µc] µe + µc + µmicro � � �� � ̸=µmacro ∆u = f , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='45) where the relation to the macro parameter µmacro in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, the limit defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='16) with M = 0 yields the contradiction sym P = (Ce + Cmicro)−1Ce sym Du , Cc skew P = Cc skew Du , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='46) since the equations degenerate to p = µe µe + µmicro ∇u , µcp = µc∇u , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='47) due to the equivalent three-dimensional forms for antiplane shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Choosing µmicro = 0 leads to a loss of structure in the strong form Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='43, while satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As such, we must set the Cosserat couple modulus µc = 0 to preserve the structure of the equations and satisfy both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Although the relaxed micromorphic model includes the Cosserat model as a singular limit for Cmicro → +∞ (µmicro → +∞), it is impossible to deduce the Cosserat model of antiplane shear as a limit of the antiplane relaxed micromorphic model, since one needs to satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='47) for µc > 0 and µmicro → +∞, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8 The kinematic reduction of the relaxed micromorphic model to antiplane shear and its behaviour in the limit cases of its material parameters is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' relaxed micromorphic Cosserat elasticity linear elasticity with Cmacro antiplane relaxed micromorphic antiplane Cosserat elasticity antiplane linear elasticity with µmacro Lc → 0 Cmicro → +∞ , µc > 0 Lc → 0 , µc ≡ 0 µmicro → +∞ , µc > 0 (contradiction) antiplane shear antiplane shear antiplane shear antiplane linear elasticity with µmicro linear elasticity with Cmicro Lc → +∞ Lc → +∞ two-scale model two-scale model non- commutative Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2: Kinematic reduction of the relaxed micromorphic model to antiplane shear and consistency at limit cases according to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The two-scale nature of the relaxed micromorphic model can be clearly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 3 Polynomial basis In this section we briefly introduce Bernstein polynomials and dual numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Bernstein polynomials are used to construct both the H 1-conforming subspace and, in conjunction with the polytopal template methodology, the N´ed´elec elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The computation of derivatives of the Bernstein base functions is achieved by employing dual numbers, thus enabling the calculation of the value and the derivative of a base function simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 Bernstein polynomials Bernstein polynomials of order p are given by the binomial expansion of the barycentric representation of the unit line 1 = (λ1 + λ2)p = ((1 − ξ) + ξ)p = p � i=0 �p i � ξi(1 − ξ)p−i = p � i=0 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (p − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='ξi(1 − ξ)p−i , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) 9 b4 0(ξ) b4 1(ξ) b4 2(ξ) b4 3(ξ) b4 4(ξ) ξ 1 1 0 1 1/2 1/2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1: Bernstein base functions of degree p = 4 on the unit domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Their sum forms a partition of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The base functions are symmetric for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 with respect to their indices and always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' where ξ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The Bernstein polynomial reads bp i (ξ) = �p i � ξi(1 − ξ)p−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2) A direct result of the binomial expansion is that Bernstein polynomials form a partition of unity, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 p � i=0 bp i (ξ) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3) Another consequence is that Bernstein polynomials are non-negative and less than or equal to 1 0 ≤ bp i (ξ) ≤ 1 , ξ ∈ [0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4) A necessary condition for the use of Bernstein polynomials in finite element approximations is for them to span the entire polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (Span of Bernstein polynomials) The span of Bernstein polynomials forms a basis of the one-dimensional polynomial space Pp(ξ) = span{bp i } , ξ ⊆ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' First we observe dim(span{bp i }) = dim Pp(ξ) = p + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) The proof of linear independence is achieved by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Let the set span{bp i } with 0 < i ≤ p be linearly dependent, then there exists some combination with at least one non-zero constant ci ̸= 0 such that p � i=1 cibp i (ξ) = 0 , d dξ p � i=1 cibp i (ξ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='7) However, by the partition of unity property Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3), only the full combination (0 ≤ i ≤ p) generates a constant and by the exact sequence property the kernel of the differentiation operator is exactly the space of constants ker(∂) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The linear independence of the full span also follows from the partition of unity property, since constants cannot be constructed otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 10 Bernstein polynomials can be evaluated efficiently using the recursive formula bp 0(ξ) = (1 − ξ)p , bp i+1(ξ) = (p − i)ξ (p + 1)(1 − ξ)bp i (ξ) , i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', p − 1} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) which allows for fast evaluation of the base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 Note that the formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) implies limξ→1 bp i+1(ξ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As such, evaluations using the formula are required to use ξ < 1 preferably with additional tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The limit case ξ = 1 is zero for all Bernstein base functions aside from the last function belonging to the vertex, which simply returns one bp i (1) = 0 ∀ i ̸= p , bp p(1) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 Dual numbers Dual numbers [16] can be used to define define an augmented algebra, where the derivative of a function can be computed simultaneously with the evaluation of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This enhancement is also commonly used in forward automatic differentiation [8, 37], not to be confused with numerical differentiation, since unlike in numerical differentiation, automatic differentiation is no approximation and yields the exact derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The latter represents an alternative method to finding the derivatives of base functions, as opposed to explicit formulas or approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Dual numbers augment the classical numbers by adding a non-zero number ε with a zero square ε2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (Dual number) The dual number is defined by x + x′ε , ε ≪ 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) where x′ is the derivative (only in automatic differentiation), ε is an abstract number (infinitesimal) and formally ε2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The augmented algebra results automatically from the definition of the dual number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 (Augmented dual algebra) The standard algebraic operations take the following form for dual numbers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Addition and subtraction (x + x′ε) ± (y + y′ε) = x ± y + (x′ ± y′)ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Multiplication (x + x′ε)(y + y′ε) = xy + (xy′ + x′y)ε , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12) since formally ε2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Division is achieved by first defining the inverse element (x + x′ε)(y + y′ε) = 1 ⇐⇒ y = 1 x, y′ = − x′ x2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) such that (x + x′ε)/(y + y′ε) = x/y + (x′/y − xy′/y2)ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='14) Application of the above definitions to polynomials p(x + ε) = ∞ � i=0 ci(x + ε)i = ∞ � i=0 1 � j=0 ci �i j � xi−jεj = ∞ � i=0 cixi + ε ∞ � i=1 i cixi−1 = p(x) + p′(x)ε , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='15) allows the extension to various types of analytical functions with a power-series representation (such as trigono- metric or hyperbolic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 11 v1 v3 v2 Γ ν τ ξ η x1 x3 x2 Ae t n x y x : Γ → Ae Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1: Barycentric mapping of the reference triangle to an element in the physical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3 (General dual numbers function) A function of a dual number is defined in general by f(x + ε) = f(x) + f ′(x)ε , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='16) being a fundamental formula for forward automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The definition of dual numbers makes them directly applicable to the general rules of differentiation, such as the chain rule or product rule, in which case the derivative is simply the composition of previous computations with ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The logic of dual numbers can be understood intuitively by the directional derivative d dxf(x) = ∂x′f(x) = d dεf(x + x′ε) ���� ε=0 = lim ε→0 f(x + x′ε) − f(x) ε , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='17) where dividing by ε and setting ε = 0 are deferred to the last step of the computation, being the extraction of the derivative and equivalent to the operation f(x + ε) − f(x) with the augmented algebra of dual numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In this work we apply dual numbers for the computation of Bernstein polynomials using the recursive formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8), thus allowing to iteratively compute each base function simultaneously with its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4 Triangular elements The triangle elements are mapped from the reference element Γ to the physical domain Ae via barycentric coordinates x(ξ, η) = (1 − ξ − η)x1 + η x2 + ξ x3 , x : Γ → Ae , Γ = {(ξ, η) ∈ [0, 1]2 | ξ + η ≤ 1} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) where xi represent the coordinates of the vertices of one triangle in the physical domain, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The corresponding Jacobi matrix reads J = Dx = �x3 − x1, x2 − x1 � ∈ R2×2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 The Bernstein-B´ezier basis for triangles The base functions on the triangle reference element are defined using the binomial expansion of the barycentric coordinates on the domain Γ 1 = (λ1 + λ2 + λ3)p = ([1 − ξ − η] + η + ξ)p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3) As such, the B´ezier base functions read bp ij(λ1, λ2, λ3) = � p i � � p − i j � λp−i−j 1 λj 2λi 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4) 12 (a) (b) (c) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2: Cubic vertex (a), edge (b) and cell (c) B´ezier base functions on the reference triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (0,0) (1,0) (1,1) (0,1) α β Γ (0,0) (1,0) (0,1) ξ η ξ : α → Γ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3: Duffy transformation from a quadrilateral to a triangle by collapse of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' with the equivalent bivariate form bp ij(ξ, η) = �p i � �p − i j � (1 − ξ − η)p−i−jηjξi , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) of which some examples are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The Duffy transformation ξ : [0, 1]2 → Γ , {α, β} �→ {ξ, η} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) given by the relations ξ = α , α = ξ , η = (1 − α)β , β = η 1 − ξ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='7) allows to view the triangle as a collapsed quadrilateral, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Inserting the Duffy map into the definition of the B´ezier base function yields the split bp ij(ξ, η) = �p i � �p − i j � (1 − ξ − η)p−i−jηjξi = �p i � �p − i j � (1 − α − [1 − α]β)p−i−j(1 − α)jβjαi = �p i � �p − i j � (1 − α)p−i−j(1 − β)p−i−j(1 − α)jβjαi (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) = �p i � (1 − α)p−iαi �p − i j � (1 − β)p−i−jβj = bp i (α) bp−i j (β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In other words, the Duffy transformation results in a natural factorization of the B´ezier triangle into Bernstein base functions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The latter allows for fast evaluation using sum factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, it is now clear that B´ezier triangles are given by the interpolation of B´ezier curves, where the degree of the polynomial decreases 13 ξ η outer B´ezier curve with p = 3 inner B´ezier curves with p < 3 control polygon of η-curves outer B´ezier curves with p = 3 inner B´ezier curves with p = 3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4: B´ezier triangle built by interpolating B´ezier curves with an ever decreasing polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' v1 v3 v2 ξ η Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5: Traversal order of base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The purple lines represent the order in which the base functions are constructed by the factorized evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Note that the traversal order on each edge is intrinsically from the lower to the higher vertex index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' between each curve, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to compute gradients on the reference domain one applies the chain rule ∇ξbp ij = (Dαξ)−T ∇αbp ij , Dαξ = � 1 0 −β 1 − α � , (Dαξ)−T = 1 1 − α �1 − α β 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) The factorization is naturally suited for the use of dual numbers since the α-gradient of a base function reads ∇αbp ij(α, β) = � ��� bp−i j d dαbp i bp i d dβ bp−i j � ��� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) such that only the derivatives of the Bernstein base functions with respect to their parameter are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The Duffy transformation induces an intrinsic optimal order of traversal of the base functions, compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5, namely (i, j) = (0, 0) → (0, 1) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' → (2, 2) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' → (i, p − i) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' → (p, 0) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='11) which respects a clockwise orientation of the element, compare [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Thus, the order of the sequence of discrete values on common edges is determined by the global orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to relate a base function to a polytopal piece of the element, one observes the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (Triangle base functions) The polytope of each base function bp ij(ξ, η) can be determined as follows: 14 The indices (0, 0), (0, p) and (p, 0) represent the first, second and last vertex base functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The indices (0, j) with 0 < j < p and (i, 0) with 0 < i < p represent the first and second edge base functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Base functions of the slanted edge are given by (i, p − i) with 0 < i < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The remaining index combinations are cell base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' With the latter observation, the construction of vertex-, edge- and cell base functions follows the intrinsic traversal order induced by the Duffy transformation and relates to a specific polytope via index-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 N´ed´elec elements of the second type We construct the base functions for the N´ed´elec element of the second type using the polytopal template methodology introduced in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The template sets read T1 = {e2, e1} , T2 = {e1 + e2, e1} , T3 = {e1 + e2, −e2} , T12 = {e2, −e1} , T13 = {e1, e2} , T23 = {(1/2)(e1 − e2), e1 + e2} , T123 = {e1, e2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12) The space of B´ezier polynomials is split across the polytopes of the reference triangle into Bp(Γ) = � 3 � i=1 Vp i (Γ) � ⊕ � � � � j∈J Ep j (Γ) � � � ⊕ Cp 123(Γ) , J = {(1, 2), (1, 3), (2, 3)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) where Vp i are the sets of the vertex base functions, Ep j are the sets of edge base functions, Cp 123 is the set of cell base functions, and the ⊕ indicates summation over non-overlapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, the N´ed´elec basis is given by N p II = � 3 � i=1 Vp i ⊗ Ti � ⊕ � � � � j∈J Ep j ⊗ Tj � � � ⊕ {Cp 123 ⊗ T123} , J = {(1, 2), (1, 3), (2, 3)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='14) Using the B´ezier basis one finds the following base functions, which inherit the optimal complexity of the underlying basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (B´ezier-N´ed´elec II triangle basis) The following base functions are defined on the reference triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' On the edges the base function reads e12 : ϑ(ξ, η) = bp 00e2 , ϑ(ξ, η) = bp 0p(e1 + e2) , ϑ(ξ, η) = bp 0je2 , 0 < j < p , e13 : ϑ(ξ, η) = bp 00e1 , ϑ(ξ, η) = bp p0(e1 + e2) , ϑ(ξ, η) = bp i0e1 , 0 < i < p , e23 : ϑ(ξ, η) = bp 0pe1 , ϑ(ξ, η) = −bp p0e2 , ϑ(ξ, η) = (1/2) bp i,p−i(e1 − e2) , 0 < i < p , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='15) where the first two base functions for each edge are the vertex-edge base functions and the third equation generates pure edge base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The cell base functions read c123 : ϑ(ξ, η) = −bp 0je1 , 0 < j < p , ϑ(ξ, η) = bp i0e2 , 0 < i < p , ϑ(ξ, η) = bp i,p−i(e1 + e2) , 0 < i < p , ϑ(ξ, η) = bp ije2 , 0 < i < p , 0 < j < p − i , ϑ(ξ, η) = bp ije1 , 0 < i < p , 0 < j < p − i , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='16) 15 where the first three are the respective edge-cell base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The remaining two are pure cell base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3 N´ed´elec elements of the first type In order to construct the N´ed´elec element of the first type we rely on the construction of the kernel introduced in [58] via the exact de Rham sequence and the polytopal template for the non-kernel base functions following [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The complete N´ed´elec space reads N p I = N 0 I ⊕ � � � � j∈J ∇Ep+1 j � � � ⊕ ∇Cp+1 123 ⊕ � 2 � i=1 Vp i ⊗ Ti � ⊕ � � � � j∈J Ep j ⊗ Tj � � � ⊕ {Cp 123 ⊗ T123} , J = {(1, 2), (1, 3), (2, 3)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='17) where we relied on the decomposition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Applying the construction to the B´ezier basis yields the following base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 (B´ezier-N´ed´elec I triangle basis) We define the base functions on the reference triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' On the edges we employ the lowest order N´ed´elec base functions and the edge gradients e12 : ϑ(ξ, η) = ϑI 1 , ϑ(ξ, η) = ∇ξbp+1 0j , 0 < j < p + 1 , e13 : ϑ(ξ, η) = ϑI 2 , ϑ(ξ, η) = ∇ξbp+1 i0 , 0 < i < p + 1 , e23 : ϑ(ξ, η) = ϑI 3 , ϑ(ξ, η) = ∇ξbp+1 i,p+1−i , 0 < i < p + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='18) The cell functions read c123 : ϑ(ξ, η) = bp 00ϑI 3 , ϑ(ξ, η) = bp 0pϑI 2 , ϑ(ξ, η) = bp 0j(ϑI 3 − ϑI 2) , 0 < j < p , ϑ(ξ, η) = bp i0(ϑI 1 + ϑI 3) , 0 < i < p , ϑ(ξ, η) = bp i,p−i(ϑI 1 − ϑI 2) , 0 < i < p , ϑ(ξ, η) = bp ij(ϑI 1 − ϑI 2 + ϑI 3) , 0 < i < p , 0 < j < p − i , ϑ(ξ, η) = ∇ξbp+1 ij , 0 < i < p + 1 , 0 < j < p + 1 − i , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) where the last formula gives the cell gradients and the remaining base functions are non-gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The definition relies on the base functions of the lowest order N´ed´elec element of the first type [5,50] ϑI 1(ξ, η) = � η 1 − ξ � , ϑI 2(ξ, η) = �1 − η ξ � , ϑI 3(ξ, η) = � η −ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='20) 5 Tetrahedral elements The tetrahedral elements are mapped from the reference tetrahedron Ω by the three-dimensional barycentric coordinates onto the physical domain Ve, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 x(ξ, η, ζ) = (1 − ξ − η − ζ)x1 + ζ x2 + η x3 + ξ x4 , x : Ω → Ve , Ω = {(ξ, η, ζ) ∈ [0, 1]3 | ξ + η + ζ ≤ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) 16 ξ η ζ Ω v1 v4 v3 v2 τ ν Ve x2 x1 x3 x4 x y z t n x : Ω → Ve Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1: Barycentric mapping of the reference tetrahedron to an element in the physical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The corresponding Jacobi matrix reads J = Dx = �x4 − x1, x3 − x1, x2 − x1 � ∈ R3×3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 The Bernstein-B´ezier basis for tetrahedra Analogously to triangle elements, the B´ezier tetrahedra on the unit tetrahedron Ω are defined using the barycen- tric coordinates by expanding the coefficients of (λ1 + λ2 + λ3 + λ4)p = ([1 − ξ − η − ζ] + ζ + η + ξ)p = 1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3) thus finding bp ijk(λ1, λ2, λ3, λ4) = �p i � �p − i j � �p − i − j k � λp−i−j−k 1 λk 2λj 3λk 4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4) with the equivalent trivariate form bp ijk(ξ, η, ζ) = � p i � � p − i j � �p − i − j k � (1 − ξ − η − ζ)p−i−j−kζkηjξi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) We construct the Duffy transformation by mapping the unit tetrahedron as a collapsed hexahedron ξ : [0, 1]3 → Ω , {α, β, γ} �→ {ξ, η, ζ} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) using the relations ξ = α , η = (1 − α)β , ζ = (1 − α)(1 − β)γ , α = ξ , β = η 1 − ξ , γ = ζ 1 − ξ − η , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='7) as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Applying the Duffy transformation to B´ezier tetrahedra bp ijk(ξ, η, ζ) = �p i � �p − i j � �p − i − j k � (1 − ξ − η − ζ)p−i−j−kζkηjξi = � p i � � p − i j � � p − i − j k � (1 − α − (1 − α)β − (1 − α)(1 − β)γ)p−i−j−k (1 − α)k(1 − β)kγk(1 − α)jβjαi = �p i � �p − i j � �p − i − j k � (1 − α)p−i−j−k(1 − β)p−i−j−k(1 − γ)p−i−j−k (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) (1 − α)k(1 − β)kγk(1 − α)jβjαi = �p i � (1 − α)p−iαi �p − i j � (1 − β)p−i−jβj �p − i − j k � (1 − γ)p−i−j−kγk = bp i (α)bp−i j (β)bp−i−j k (γ) , 17 α β γ (0,0,0) (1,0,0) (0,0,1) (1,1,0) (1,1,1) (0,1,1) ξ η ζ Ω (0,0,0) (1,0,0) (0,1,0) (0,0,1) ξ : α → Ω Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2: Duffy mapping of the unit hexahedron to the unit tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' leads to an intrinsic factorization via univariate Bernstein base functions, which allow for fast evaluations using sum factorization [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, since the pair bp−i j (β)bp−i−j k (γ) spans a B´ezier triangle, it is clear that the multiplication with bp i (α) interpolates between that triangle and a point in space, effectively spanning a tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to compute gradients the chain rule is employed with respect to the Duffy transformation ∇ξbp ijk = (Dαξ)−T ∇αbp ijk , Dαξ = � � 1 0 0 −β 1 − α 0 (β − 1)γ (α − 1)γ (1 − α)(1 − β) � � , (Dαξ)−T = 1 (1 − α)(1 − β) � � (1 − α)(1 − β) (1 − β)β γ 0 1 − β γ 0 0 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) We use dual numbers to compute the derivative of each Bernstein base function and construct the α-gradient ∇αbp ijk(α, β, γ) = � ������� bp−i j bp−i−j k d dαbp i bp i bp−i−j k d dβ bp−i j bp i bp−i j d dγ bp−i−j k � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) The Duffy transformation results in the optimal order of traversal of the base functions depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Note that the traversal order agrees with the oriental definitions introduced in [52] and each oriented face has the same order of traversal as the triangle Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We relate the base functions to their respective polytopes using the index triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (Tetrahedron base functions) The polytope of each base function bp ijk(ξ, η, ζ) is determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' the indices (0, 0, 0), (0, 0, p), (0, p, 0) and (p, 0, 0) represent the respective vertex base functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' the first edge is associated with the triplet (0, 0, k) where 0 < k < p, the second with (0, j, 0) where 0 < j < p and the third with (i, 0, 0) where 0 < i < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The slated edges are given by (0, j, p − j) with 0 < j < p, (i, 0, p − i) with 0 < i < p and (i, p − i, 0) with 0 < i < p, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' the base functions of the first face are given by (0, j, k) with 0 < j < p and 0 < k < p − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The second face is associated with the base functions given by the triplets (i, 0, k) with 0 < i < p and 0 < k < p − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The base functions of the third face are related to the indices (i, j, 0) with 0 < i < p and 0 < j < p − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Lastly, the base functions of the slated face are given by (i, j, p − i − j) with 0 < i < p and 0 < j < p − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' the remaining indices correspond to the cell base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Examples of B´ezier base functions on their respective polytopes are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 18 ξ η ζ v1 v4 v3 v2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3: Order of traversal of tetrahedral B´ezier base functions on the unit tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The traversal order on each face agrees with an orientation of the vertices fijk = {vi, vj, vk} such that i < j < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The traversal order on each edge is from the lower index vertex to the higher index vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4: Quartic B´ezier vertex (a), edge (b), face (c), and cell (c) base functions on the reference tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 N´ed´elec elements of the second type The B´ezier polynomial space is split according to the polytopes of the reference tetrahedron Bp(Ω) = � 4 � i=1 Vp i (Ω) � ⊕ � � � � j∈J Ep j (Ω) � � � ⊕ �� k∈K Fp k(Ω) � ⊕ Cp 1234(Ω) , J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} , K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='11) where Vp i are the sets of vertex base functions, Ep j are the sets of edge base functions, Fp k are the sets of face base functions and Cp 1234 is the set of cell base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We apply the template sets from [50] T1 = {e3, e2, e1} , T2 = {e1 + e2 + e3, e2, e1} , T3 = {e1 + e2 + e3, −e3, e1} , T4 = {e1 + e2 + e3, −e3, −e2} , T12 = {e3, −e2, −e1} , T13 = {e2, e3, −e1} , T14 = {e1, e3, e2} , T23 = {e2, e1 + e2 + e3, −e1} , T24 = {e1, e1 + e2 + e3, e2} , T34 = {e1, e1 + e2 + e3, −e3} , T123 = {e3, e2, −e1} , T124 = {e3, e1, e2} , T134 = {e2, e1, −e3} , T234 = {e2, e1, e1 + e2 + e3} , T1234 = {e3, e2, e1} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12) to span the N´ed´elec element of the second type N p II = � 4 � i=1 Vp i ⊗ Ti � ⊕ � � � � j∈J Ep j ⊗ Tj � � � ⊕ �� k∈K Fp k ⊗ Tk � ⊕ {Cp 1234 ⊗ T1234} , J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} , K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) We can now define the B´ezier-N´ed´elec element of the second type for arbitrary powers while inheriting optimal complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 (B´ezier-N´ed´elec II tetrahedral basis) We define the base functions on the reference tetrahedron: on the edges the base functions read e12 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 000e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00p(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00ke3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e13 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 000e2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0p0(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j0e2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e14 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 000e1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp p00(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i00e1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e23 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00pe2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 0p0e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−je2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e24 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00pe1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp p00e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−ie1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e34 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0p0e1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp p00e2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='0e1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='14) where the first two base functions on each edge are the vertex-edge base functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 20 the face base functions are given by f123 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 00ke2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j0e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−j(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0jke3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0jke2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' f124 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 00ke1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i00e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0ke3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0ke1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' f134 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 0j0e1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i00e2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='0(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij0e2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij0e1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' f234 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 0j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−je1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−ie2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='0e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i−je2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i−je1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='15) where the first three formulas for each face are the edge-face base functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' the cell base functions read c1234 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 0jke1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0ke2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp ij0e3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i−j(e1 + e2 + e3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ijke3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ijke2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ijke1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='16) where the first four formulas are the face-cell base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3 N´ed´elec elements of the first type In order to construct the N´ed´elec element of first type on tetrahedra we introduce the template sets T1 = {ϑI 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 6} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T2 = {−ϑI 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' −ϑI 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 6} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T3 = {−ϑI 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' −ϑI 5} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T12 = {ϑI 4 − ϑI 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 5 − ϑI 3} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T13 = {ϑI 1 + ϑI 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 6 − ϑI 3} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T14 = {ϑI 1 + ϑI 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 2 + ϑI 6} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T23 = {ϑI 1 − ϑI 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 6 − ϑI 5} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T24 = {ϑI 1 − ϑI 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 4 + ϑI 6} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T34 = {ϑI 2 − ϑI 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑI 4 − ϑI 5} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T123 = {ϑI 1 − ϑI 2 + ϑI 4} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T124 = {ϑI 1 − ϑI 3 + ϑI 5} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T134 = {ϑI 2 − ϑI 3 + ϑI 6} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' T234 = {ϑI 4 − ϑI 5 + ϑI 6} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='17) 21 which are based on the lowest order N´ed´elec base functions on the unit tetrahedron ϑ1(ξ, η, ζ) = � � ζ ζ 1 − ξ − η � � , ϑ2(ξ, η, ζ) = � � η 1 − ξ − ζ η � � , ϑ3(ξ, η, ζ) = � � 1 − η − ζ ξ ξ � � , ϑ4(ξ, η, ζ) = � � 0 ζ −η � � , ϑ5(ξ, η, ζ) = � � ζ 0 −ξ � � , ϑ6(ξ, η, ζ) = � � η −ξ 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='18) For the non-gradient cell functions we use the construction introduced in [2] Rp = � (p + 1)bp i−ej∇λj − ij p + 1∇ξbp+1 i | i ∈ Io � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) where Io is the set of multi-indices of cell functions, ej is the unit multi-index with the value one at position j and ij is the value of the i-multi-index at position j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Note that only the first term in the cell functions is required to span the next space in the sequence due to curl � [p + 1]bp i−ej∇ξλj − ij p + 1∇ξbp+1 i � = curl([p + 1]bp i−ej∇ξλj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='20) However, without the added gradient the function would not belong to [Pp]3 ⊕ξ ×[�P]3 and consequently, would not be part of the N´ed´elec space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' By limiting Rp to Rp ∗ such that Rp ∗ contains only the surface permutations with ∇λj = ej and the cell permutations with j ∈ {1, 2}, one retrieves the necessary base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The sum of the lowest order N´ed´elec base functions, the template base functions, gradient base functions, and the non-gradient cell base functions yields exactly (p+4)(p+3)(p+1)/2, thus satisfying the required dimensionality of the N´ed´elec space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The complete space reads N p I = N 0 I ⊕ �� i∈I ∇Ep+1 i � ⊕ � � � � j∈J ∇Fp+1 j � � � ⊕ ∇Cp+1 1234 ⊕ � 3 � k=1 Vp k ⊗ Tk � ⊕ �� i∈I Ep i ⊗ Ti � ⊕ � � � � j∈J Fp j ⊗ Tj � � � ⊕ Rp+1 ∗ , I = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} , J = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='21) Here, the B´ezier basis is used to construct the higher order N´ed´elec base functions of the first type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 (B´ezier-N´ed´elec I tetrahedral basis) The base functions are defined on the reference tetrahedron: for the edges we use the lowest order base functions from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The remaining edge base functions are given by the gradients e12 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ϑI 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 00k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e13 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ϑI 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 0j0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e14 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ϑI 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 i00 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e23 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ϑI 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 0j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p+1−j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e24 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ϑI 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p+1−i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' e34 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ϑI 6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 00k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p + 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='22) 22 on faces we employ both template base functions and gradients f123 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 000ϑI 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 00pϑI 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00k(ϑI 4 − ϑI 2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j0(ϑI 1 + ϑI 4) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−j(ϑI 1 − ϑI 2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0jk(ϑI 1 − ϑI 2 + ϑI 4) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 0jk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p + 1 − j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' f124 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 000ϑI 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 00pϑI 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00k(ϑI 5 − ϑI 3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i00(ϑI 1 + ϑI 5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i(ϑI 1 − ϑI 3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0k(ϑI 1 − ϑI 3 + ϑI 5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 i0k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < k < p + 1 − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' f134 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 000ϑI 6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 0p0ϑI 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j0(ϑI 6 − ϑI 3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i00(ϑI 2 + ϑI 6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='0(ϑI 2 − ϑI 3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij0(ϑI 2 − ϑI 3 + ϑI 6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 ij0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p + 1 − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' f234 : ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 00pϑI 6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = −bp 0p0ϑI 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp 0j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−j(ϑI 6 − ϑI 5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i(ϑI 4 + ϑI 6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='0(ϑI 4 − ϑI 5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = bp ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i−j(ϑI 4 − ϑI 5 + ϑI 6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < i < p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 0 < j < p − i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ϑ(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' ζ) = ∇ξbp+1 ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='p−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='j , 0 < i < p + 1 , 0 < j < p + 1 − i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='23) the cell base functions read c1234 : ϑ(ξ, η, ζ) = (p + 2)bp+1 i−1,jke1 − i p + 2∇ξbp+2 ijk , 0 < i < p + 2 , 0 < j < p + 2 − i , 0 < k < p + 2 − i − j , ϑ(ξ, η, ζ) = (p + 2)bp+1 i,j−1,ke2 − j p + 2∇ξbp+2 ijk , 0 < i < p + 2 , 0 < j < p + 2 − i , 0 < k < p + 2 − i − j , ϑ(ξ, η, ζ) = (p + 2)bp+1 ij0 e3 − 1 p + 2∇ξbp+2 ij1 , 0 < i < p + 2 , 0 < j < p + 2 − i , ϑ(ξ, η, ζ) = ∇ξbp+1 ijk , 0 < i < p + 1 , 0 < j < p + 1 − i , 0 < k < p + 1 − i − j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24) 23 6 Numerical quadrature Although the base functions are expressed using (α, β, γ) the domain is either the reference triangle or the reference tetrahedron, which require fewer quadrature points than their counterparts given by the Duffy trans- formation (quad or hexahedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As such, we employ a mixture of the efficient quadrature points introduced in [14,19,39,56,57] for triangles and tetrahedra, where we avoid quadrature schemes with points on the edges or faces of the reference domain due to the recursion formula of the Bernstein polynomials Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The quadrature points are mapped to their equivalent expression in (α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, the integration over the reference triangle or tetrahedron reads � Ae f(x, y) dA = � Γ (f ◦ (ξ, η))(α, β) | det J| dΓ , � Ve f(x, y, z) dV = � Ω (f ◦ (ξ, η, ζ))(α, β, γ) | det J| dΩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) For the lower order elements we use the Lagrangian-N´ed´elec construction from [52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 7 Boundary conditions The degrees of freedom in [12] commute between the continuous and discrete spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' As such, they allow to exactly satisfy the consistent coupling condition [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We note that the functionals can be viewed as a hierarchical system of Dirichlet boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In the case of hierarchical base functions [58], they can be solved independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' However, here the boundary value of each polytope is required in advance due to the non-hierarchical nature of Bernstein polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In other words, one must first solve the problem for vertices, then for edges, afterwards for faces, and finally for the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In our case the degrees of freedom for the cell are irrelevant since a cell is never part of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 Boundary vertices The finite element mesh identifies each vertex with a tuple of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' It suffices to evaluate the displacement field at the vertex ud i = �u ���� xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) If the field is vectorial, each component is evaluated at the designated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The boundary conditions of the microdistortion field are associated with tangential projections and as such do not have vertex-type degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This is the case since a vertex does not define a unique tangential plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 Boundary edges The edge functionals from [12] for the H 1-conforming subspace lij(u) = � si ∂qj ∂s ∂u ∂s ds , q ∈ Pp(s) , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2) can be reformulated for a reference edge on a unit domain α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We parametrize the edge via x(α) = (1 − α)x1 + αx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3) As such, the following relation exists between the unit parameter and the arc-length parameter t = d dαx = x2 − x1 , ds = ∥dx∥ = ∥x2 − x1∥dα = ∥t∥dα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4) 24 α 0 1 ξ : α → Γ ξ2 ξ1 Γ τ ξ η x2 x1 A t x y x : Γ → A Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1: Barycentric mapping of edges from the unit domain to the reference triangle and onto the physical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' By the chain rule we find du ds = du dα dα ds = ∥t∥−1 du dα , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) for some function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' On edges, the test and trial functions are Bernstein polynomials parametrized by the unit domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The function representing the boundary condition �u(x) however, is parametrized by the Cartesian coordinates of the physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We find its derivative with respect to the arc-length parameter by observing d ds �u = ⟨ d dsx, ∇x�u⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) The derivative of the coordinates with respect to the arc-length is simply the normed tangent vector d dsx = dx dα dα ds = ∥t∥−1t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='7) Consequently, the edge boundary condition is given by � si ∂qj ∂s ∂u ∂s ds = � 1 0 � ∥t∥−1 dqj dα � � ∥t∥−1 du dα � ∥t∥ dα = � 1 0 � ∥t∥−1 dqj dα � ⟨∥t∥−1t, ∇x�u⟩∥t∥ dα = � si ∂qj ∂s ∂�u ∂s ds ∀ qj ∈ Pp(α) , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) and can be solved by assembling the stiffness matrix of the edge and the load vector induced by the prescribed displacement field �u, representing volume forces kij = � 1 0 � ∥t∥−1 dni dα � � ∥t∥−1 dnj dα � ∥t∥ dα , fi = � 1 0 ⟨∥t∥−1t, ∇x�u⟩ � ∥t∥−1 dni dα � ∥t∥ dα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) Next we consider the Dirichlet boundary conditions for the microdistortion with the N´ed´elec space of the second type NII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The problem reads � si qj⟨t, p⟩ ds = � si qj⟨t, ∇x�u⟩ ds ∀ qj ∈ Pp(si) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) Observe that on the edge the test functions qj are chosen to be the Bernstein polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, by the polytopal template construction of the NII-space there holds ⟨t, θi⟩|s = ni(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Therefore, the components of the corresponding stiffness matrix and load vectors read kij = � 1 0 ni nj∥t∥ dα , fi = � 1 0 ni⟨t, ∇x�u⟩∥t∥ dα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='11) 25 Note that in order to maintain the exactness property, the degree of the N´ed´elec spaces N p I , N p II is always one less than the degree of the subspace Bp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Lastly, we consider the N´ed´elec element of the first type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The problem is given by � si qj⟨t, p⟩ ds = � si qj⟨t, ∇x�u⟩ ds ∀ qj ∈ Pp(si) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12) We define qi = d dαnp+1 i , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) and observe that on the edges the N´ed´elec base functions yield ⟨t, θj⟩ = ⟨t, ∇xnp+1 j ⟩ = d dαnp+1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='14) Therefore, the components of the stiffness matrix and the load vector result in kij = � 1 0 dnp+1 i dα dnp+1 j dα ∥t∥ dα , fi = � 1 0 dnp+1 i dα ⟨t, ∇x�u⟩∥t∥ dα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='15) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3 Boundary faces We start with the face boundary condition for the H 1-conforming subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The problem reads � Ai ⟨∇fqj, ∇fu⟩ dA = � Ai ⟨∇fqj, ∇f �u⟩ dA ∀ qj ∈ Pp(Ai) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='16) The surface is parameterized by the barycentric mapping from the unit triangle Γ = {(ξ, η) ∈ [0, 1]2 | ξ +η ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The surface gradient is given by ∇f �u = ∇x�u − 1 ∥n∥2 ⟨∇x�u, n⟩n , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='17) where n is the surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The surface gradient can also be expressed via ∇fu = ei∂x i u = gβ∂ξ βu , β ∈ {1, 2} , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='18) where ∂x β are partial derivates with respect to the physical coordinates, ∂ξ β are partial derivatives with respect to the reference domain and gβ are the contravariant base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The Einstein summation convention over corresponding indices is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The covariant base vectors are given by gβ = ∂x ∂ξβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) One can find the contravariant vector orthogonal to the surface by g3 = n = g1 × g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='20) We define the mixed transformation matrix T = � g1 , g2 , g3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='21) Due to the orthogonality relation ⟨gi, gj⟩ = δ j i the transposed inverse of T is clearly T −T = � g1 , g2 , g3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='22) Thus, we can compute the surface gradient of functions parametrized by the reference triangle via ∇fu = � g1 , g2� ∇ξu = T −T ∗ ∇ξu , T −T ∗ = � g1 , g2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='23) 26 Further, there holds the following relation between the physical surface and the reference surface dA = ∥n∥dΓ = ∥g3∥dΓ = � ⟨g1 × g2, g3⟩ dΓ = √ det T dΓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='24) Consequently, we can write the components of the stiffness matrix and load vector as kij = � Γ ⟨T −T ∗ ∇ξni, T −T ∗ ∇ξnj⟩ √ det T dΓ , fi = � Γ ⟨T −T ∗ ∇ξni, ∇x�u − (det T )−1⟨∇x�u, n⟩n⟩ √ det T dΓ = � Γ ⟨T −T ∗ ∇ξni, ∇x�u⟩ √ det T dΓ , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='25) with the orthogonality ⟨gβ, n⟩ = 0 for β ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to embed the consistent coupling boundary condition to the microdistortion we deviate from the degrees of freedom defined in [12] and apply the simpler H (divR)-projection ⟨qi, p, ⟩H(divR) = ⟨qi, ∇f �u⟩H(divR) ∀ qi ∈ N p I (A) or ∀ qi ∈ N p II(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='26) Due to ker(curl) = ∇H 1 the problem reduces to � Ai ⟨qj, p⟩ + ⟨curl2Dqj, curl2Dp⟩ dA = � Ai ⟨qj, ∇f �u⟩ dA ∀ qj ∈ N p I (A) or ∀ qj ∈ N p II(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='27) We express the co- and contravariant Piola transformation from the two-dimensional reference domain to the three-dimensional physical domain using θi = T −T ∗ ϑi , divx R θi = 1 √ det T divξ R ϑi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='28) Thus, the stiffness matrix components and load vector components read kij = � Γ ⟨T −T ∗ ϑi, T −T ∗ ϑj⟩ + ⟨(det T )−1/2 divξ R ϑi, (det T )−1/2 divξ R ϑj⟩ √ det T dΓ , fi = � Γ ⟨T −T ∗ ϑi, ∇x�u − (det T )−1⟨∇x�u, n⟩n⟩ √ det T dΓ = � Γ ⟨T −T ∗ ϑi, ∇x�u⟩ √ det T dΓ , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='29) where we again make use of the orthogonality between the surface tangent vectors and its normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8 Numerical examples In the following we test the finite element formulations with an artificial analytical solution in the antiplane shear model and with an analytical solution for an infinite plane under cylindrical bending in the three dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Finally, we benchmark the ability of the finite element formulations to correctly interpolate between micro Cmicro and macro Cmacro stiffnesses as described by the characteristic length scale parameter Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The majority of convergence results are presented by measuring the error in the Lebesgue norm over the domain ∥�u − uh∥L2 = �� V ∥�u − uh∥2 dV , ∥ �P − P h∥L2 = �� V ∥ �P − P h∥2 dV , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1) in which context {�u, �P } and {uh, P h} are the analytical and approximate subspace solutions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 Compatible microdistortion In [53] we explored the conditions for which the microdistortion p reduces to a gradient field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' p is compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' By defining the micro-moment with a scalar potential m = ∇100 − x2 − y2 10 = −1 5 �x y � , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2) 27 and constructing an analytical solution for the displacement field �u = sin �x2 + y2 5 � , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3) we can recover the analytical solution of the microdistortion p = 1 µe + µmicro (m + µe∇�u) = 1 2 � −1 5 �x y � + 2 5 �x cos([x2 + y2]/5) y cos([x2 + y2]/5) �� = 1 5 �x cos([x2 + y2]/5) y cos([x2 + y2]/5) � − 1 10 �x y � , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4) where for simplicity we set all material constants to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Since m is a gradient field, the microdistortion p is also reduced to a gradient field and curl2Dp = 0, see [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Note that this result is specific to antiplane shear and does not generalize to the full three-dimensional model, compare [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We note that the microdistortion is not equal to the gradient of the displacement field and as such, their tangential projections on an arbitrary boundary are not automatically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' However, for both the gradient of the displacement field and the micro-moment is the tangential projection on the boundary of the circular domain A = {x ∈ R2 | ∥x∥ ≤ 10} equal to zero ⟨∇t, �u⟩ ���� ∂A = ⟨t, m⟩ ���� ∂A = 0 , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5) and as such the microdistortion belongs to p ∈ H0(curl, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, we can set sD = ∂A and the consistent coupling condition remains compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' With the displacement and the microdistortion fields at hand we derive the corresponding forces f = 1 25 � 2x2 sin �x2 + y2 5 � + 2y2 sin �x2 + y2 5 � − 10 cos �x2 + y2 5 � − 5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6) The approximation of the displacement and microdistortion fields using linear and higher order elements is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We note that even with almost 3000 finite elements and 6000 degrees of freedom the linear formulation is incapable of finding an adequate approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' On the other side of the spectrum, the higher order approximation (degree 7) with 57 elements and 4097 degrees of freedom yields very accurate results in the interior of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' However, the exterior of the domain is captured rather poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This is the case since the geometry of the circular domain is being approximated by linear triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Thus, in this setting, a finer mesh captures the geometry in a more precise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The effects of the geometry on the approximation of the solution are also clearly visible in the convergence graphs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' only after a certain accuracy in the domain description is achieved do the finite elements retrieve their predicted convergence rates, compare [52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This is clearly observable when comparing the convergence curves of the linear and seventh order elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The linear element generates quadratic convergence p + 1 = 1 + 1 = 2, whereas the seventh-order element yields the convergence slope 7 (where 8 is expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Although the seventh-order formulation encompasses more degrees of freedom, it employs a coarser mesh and as such, generates higher errors at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The errors themselves can be traced back to the consistent coupling condition since, for a non-perfect circle the gradient of the displacement field induces tangential projections on the imperfect boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The influence of the latter effect is even more apparent in the convergence of the microdistortion, where the higher order formulations are unable to perform optimally on coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 Cylindrical bending In order to test the capability of the finite element formulations to capture the intrinsic behaviour of the relaxed micromorphic model, we compare with analytical solutions of boundary-value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The first example considers the displacement and microdistortion fields under cylindrical bending [43] for infinitely extended plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Let the plates be defined as V = (−∞, ∞)2 × [−1/2, 1/2], than the analytical solution for cylindrical bending reads u = κ � � −xz 0 x2/2 � � , P = −κ � � [41z + 20 √ 82 sech( � 41/2) sinh( √ 82z)]/1681 0 x 0 0 0 −x 0 0 � � , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='7) 28 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1: Depiction of the displacement field (a)-(c) and the microdistortion field (d)-(f) for the antiplane shear problem, for the linear element under h-refinement with 225, 763 and 2966 elements, corresponding to 485, 1591 and 6060 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The p-refinement of the displacement field on the coarsest mesh of 57 elements is visualized in (g)-(l) with p ∈ {3, 5, 7}, corresponding to 731, 2072 and 4097 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 29 11 NA144 44103 104 105 10−3 10−1 101 degrees of freedom ∥�u − uh∥L2 L1 × N 0 I L2 × N 1 II B3 × N 2 II B5 × N 4 II B7 × N 6 II O(h2) O(h7) (a) 103 104 105 10−3 10−1 101 degrees of freedom ∥�p − ph∥L2 L1 × N 0 I L2 × N 1 II B3 × N 2 II B5 × N 4 II B7 × N 6 II O(h) O(h2) (b) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2: Convergence of displacement (a) and the microdistortion (b) under h-refinement for multiple poly- nomial degrees for the antiplane shear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' where sech(x) = 1/ cosh(x), and for the following values of material constants λe = λmicro = 0 , µe = µmacro = 1/2 , µc = 0 , Lc = 1 , µmicro = 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='8) The intensity of the curvature parameter κ of the plate is chosen to be κ = 14/200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 The particular case of the cylindrical bending for which λe = λmicro = 0 (equivalent to a zero micro-Poisson’s ratio) has been solved, along with its more general case (λe ̸= λmicro ̸= 0), in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The advantage of considering this particular case is that a cut out finite plate of the infinite domain automatically exhibits the consistent coupling boundary conditions on its side surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 Note that the general analytical solution for cylindrical bending does not depend on µc, so we can set µc = 0 without loss of generality, compare [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We define the finite domain V = [−10, 10]2 × [−1/2, 1/2] and the boundaries AD1 = {−10} × [−10, 10] × [−1/2, 1/2] , AD2 = {10} × [−10, 10] × [−1/2, 1/2] , AN = ∂V \\ {AD1 ⊕ AD2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9) Additionally, on the Dirichlet boundary we impose the translated analytical solution �u = u − �0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The displacement field and the last row of the microdistortion are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The displacement field is dominated by its quadratic term and captured correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The last row of the microdistortion is a linear function and easily approximated even with linear elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' On the contrary, the P11 component of the microdistortion is a hyperbolic function of the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The results of its approximation at x = y = 0 (the centre of the plane) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We observe that even increasing the number of linear finite elements to the extreme only results in better oscillations around the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In comparison, higher order formulations converge towards the expected hyperbolic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The approximation of the quadratic N´ed´elec element of the first type is nearly perfect, whereas its second type counterpart clearly deviates from the analytical solution at z ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Taking the cubic second type element yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' This phenomenon is an evident indicator of the prominent role of the Curl of the microdistortion in this type of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Firstly, the microdistortion is a non-gradient field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Secondly, the Curl of the analytical solution induces an hyperbolic sine term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Such functions are often approximated using at least cubic terms in power series, thus explaining the necessity of such high order elements for correct computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 30 (a) (b) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3: Displacement (a) and last row of the microdistortion (b) for the quadratic formulation using the N´ed´elec element of the first type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 −1 0 1 10−2 z-axis P11(z) ne = 5640 ne = 44592 ne = 354720 (a) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 −1 0 1 10−2 z-axis P11(z) B2 × N 1 I B3 × N 2 I L2 × N 1 II B3 × N 2 II B4 × N 3 II (b) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='4: Convergence of the lowest order formulation under h-refinement with 732, 5640 and 44592 elements (a) and of the higher order formulations under p-refinement using 732 elements(b) towards the analytical solution (dashed curve) of the P11(z) component at x = y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3 Bounded stiffness property The characteristic length scale parameter Lc allows the relaxed micromorphic model to capture the transition from highly homogeneous materials to materials with a pronounced micro-structure by governing the influence of the micro-structure on the overall behaviour of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We demonstrate this property of the model with an example, where we vary Lc and measure the resulting energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Let the domain be given by the axis-symmetric cube V = [−1, 1]3 with a total Dirichlet boundary AD1 = {(x, y, z) ∈ [−1, 1]3 | x = ±1} , AD2 = {(x, y, z) ∈ [−1, 1]3 | y = ±1} , AD3 = {(x, y, z) ∈ [−1, 1]3 | z = ±1} , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='10) we embed the periodic boundary conditions �u ���� AD1 = � � (1 − y2) sin(π[1 − z2])/10 0 0 � � , �u ���� AD2 = � � 0 (1 − x2) sin(π[1 − z2])/10 0 � � , �u ���� AD3 = � � 0 0 (1 − y2) sin(π[1 − x2])/10 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='11) The material parameters are chosen as λmacro = 2 , µmacro = 1 , λmicro = 10 , µmicro = 5 , µc = 1 , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='12) thus giving rise to the following meso-parameters via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='19) λe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 , µe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='13) The displacement field as well as some examples of the employed meshes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to compute the upper and lower bound on the energy we utilize the equivalent Cauchy model formulation with the micro- and macro elasticity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to assert the high accuracy of the solution of the bounds we employ tenth order finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The progression of the energy in dependence of the characteristic length parameter Lc is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' We observe the high mesh dependency of the lower order formulations, where the energy is clearly overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The higher order formulations all capture the upper bound correctly but diverge with respect to the result of the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Notably, the approximation using the N´ed´elec element of the first type is more accurate than the equivalent formulation with the N´ed´elec element of the second type, thus indicating the non-negligible involvement of the micro-dislocation in the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Using standard mesh coarseness the cubic element formulation with N´ed´elec elements of the first type yields satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In order to achieve the same on highly coarse meshes, one needs to employ seventh order elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 9 Conclusions and outlook The intrinsic behaviour of the relaxed micromorphic model is revealed by the analytical solutions to boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Clearly, the continuum exhibits hyperbolic and trigonometric solutions, which are not easily approximated by low order finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The example provided in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='2 demonstrates that cubic and higher order finite elements yield excellent results in approximate solutions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The polytopal template methodology introduced in [50] allows to easily and flexibly construct H (curl)- conforming vectorial finite elements that inherit many of the characteristics of an underlying H 1-conforming basis, which can be chosen independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' In this work, we made use of Bernstein-B´ezier polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The latter boast optimal complexity properties manifesting in the form of sum factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The natural decom- position of their multi-variate versions into multiplications of univariate Bernstein base functions via the Duffy transformation allows to construct optimal iterators for their evaluation using recursion formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Further, this characteristic makes the use of dual numbers in the computation of their derivatives ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Finally, the intrinsic order of traversal induced by the factorization is exploited optimally by the choice of clock-wise orientation of the reference element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The consequence of these combined features is a high-performance hp-finite element program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 32 (a) (b) (c) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5: Displacement field of the Cauchy model on the coarsest mesh of 48 finite elements of the tenth order (a) and depictions of the meshes with 384 (b) and 3072 (c) elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The ability of the relaxed micromorphic model to interpolate between the energies of homogeneous materials and materials with an underlying micro-structure using the characteristic length scale parameter Lc is demon- strated in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' It is also shown that in order to correctly capture the span of energies for the values of Lc either fine-discretizations or higher order elements are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The excellent performance of the proposed higher order finite elements in the linear static case is a precur- sor for their application in the dynamic setting, which is important since the relaxed micromorphic model is often employed in the computation of elastic waves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', for acoustic metamaterials), where solutions for high frequency ranges are commonly needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The proposed computational scheme is lacking in its description of curved geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Due to the consistent coupling condition, this can easily lead to errors emanating from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Consequently, a topic for future works would be the investigation of curved finite elements [20,21] and their behaviour with respect to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Acknowledgements Angela Madeo and Gianluca Rizzi acknowledge support from the European Commission through the funding of the ERC Consolidator Grant META-LEGO, N◦ 101001759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='00 Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project- No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 440935806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 10 References [1] Ainsworth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Andriamaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Davydov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': Bernstein–B´ezier finite elements of arbitrary order and optimal assembly procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' SIAM Journal on Scientific Computing 33(6), 3087–3109 (2011) [2] Ainsworth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Fu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': Bernstein–B´ezier bases for tetrahedral finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Computer Methods in Applied Mechanics and Engineering 340, 178–201 (2018) [3] Aivaliotis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Tallarico, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', d’Agostino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Daouadji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Neff, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Madeo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Archive of Applied Mechanics 90(5), 1073–1096 (2020) [4] Alberdi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Robbins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Walsh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Dingreville, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': Exploring wave propagation in heterogeneous metastructures using the relaxed micromorphic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Journal of the Mechanics and Physics of Solids 155, 104540 (2021) [5] Anjam, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Valdman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': Fast MATLAB assembly of FEM matrices in 2d and 3d: Edge elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Applied Mathematics and Computation 267, 252–263 (2015) [6] Barbagallo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Madeo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', d’Agostino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Abreu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Ghiba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Neff, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' International Journal of Solids and Structures 120, 7–30 (2017) 33 10−3 100 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9 Lc I ne = 384 ne = 3072 ne = 24576 ne = 48000 Cmacro Cmicro (a) 10−3 100 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9 Lc I ne = 384 ne = 3072 ne = 24576 Cmacro Cmicro (b) 10−3 100 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9 Lc I ne = 384 , B3 × N 2 I ne = 3072 , B3 × N 2 I ne = 384 , B3 × N 2 II ne = 3072 , B3 × N 2 II Cmacro Cmicro (c) 10−3 100 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='9 Lc I B5 × N 4 I B7 × N 6 I B5 × N 4 II B7 × N 6 II Cmacro Cmicro (d) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='6: Energy progression of the relaxed micromorphic model with respect to Lc using the linear (a), quadratic (b) and cubic (c) finite element formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' The energy computed with the coarsest mesh of 48 elements is depicted in (d) for various polynomial powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' 34 [7] Barbagallo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Tallarico, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', D’Agostino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Aivaliotis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Neff, P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Martin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Neff, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': More on anti-plane shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Journal of Optimization Theory and Applications 184(1), 226–249 (2020) [56] Witherden, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Vincent, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': On the identification of symmetric quadrature rules for finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Computers & Mathematics with Applications 69(10), 1232–1241 (2015) [57] Xiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=', Gimbutas, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Computers & Mathematics with Applications 59(2), 663–676 (2010) [58] Zaglmayr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=': High order finite element methods for electromagnetic field computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' thesis, Johannes Kepler Universit¨at Linz (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='numerik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='tugraz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='at/~zaglmayr/pub/szthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} +page_content='pdf 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfh_0J/content/2301.01491v1.pdf'} diff --git a/CNAzT4oBgHgl3EQfTvwq/content/tmp_files/2301.01253v1.pdf.txt b/CNAzT4oBgHgl3EQfTvwq/content/tmp_files/2301.01253v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..34ee2c1b519851ed37cca4900ce1135592c2177b --- /dev/null +++ b/CNAzT4oBgHgl3EQfTvwq/content/tmp_files/2301.01253v1.pdf.txt @@ -0,0 +1,1062 @@ +Deep Learning for bias-correcting comprehensive +high-resolution Earth system models +Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3 +1Earth System Modelling, School of Engineering & Design, Technical University of Munich, +Munich, Germany +2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany +3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK +Key Points: +• A generative adversarial network is shown to improve daily precipitation fields from +a state-of-the-art Earth system model. +• Biases in long-term temporal distributions are strongly reduced by the generative +adversarial network. +• Our network-based approach can be complemented with quantile mapping to fur- +ther improve precipitation fields. +–1– +arXiv:2301.01253v1 [physics.ao-ph] 16 Dec 2022 + +Abstract +The accurate representation of precipitation in Earth system models (ESMs) is crucial for +reliable projections of the ecological and socioeconomic impacts in response to anthropogenic +global warming. The complex cross-scale interactions of processes that produces precipi- +tation are challenging to model, however, inducing potentially strong biases in ESM fields, +especially regarding extremes. State-of-the-art bias correction methods only address errors +in the simulated frequency distributions locally, at every individual grid cell. Improving +unrealistic spatial patterns of the ESM output, which would require spatial context, has +not been possible so far. Here, we show that a post-processing method based on physically +constrained generative adversarial networks (GANs) can correct biases of a state-of-the-art, +CMIP6-class ESM both in local frequency distributions and in the spatial patterns at once. +While our method improves local frequency distributions equally well as gold-standard bias- +adjustment frameworks it strongly outperforms any existing methods in the correction of +spatial patterns, especially in terms of the characteristic spatial intermittency of precipita- +tion extremes. +1 Introduction +Precipitation is a crucial climate variable and changing amounts, frequencies, or spatial +distributions have potentially severe ecological and socioeconomic impacts. +With global +warming projected to continue in the coming decades, assessing the impacts of changes +in precipitation characteristics is an urgent challenge (Wilcox & Donner, 2007; Boyle & +Klein, 2010; IPCC, 2021). Climate impact models are designed to assess the impacts of +global warming on, for example, ecosystems, crop yields, vegetation and other land-surface +characteristics, infrastructure, water resources, or the economy in general (Kotz et al., 2022), +using the output of climate or Earth system models (ESMs) as input. Especially for reliable +assessments of the ecological and socioeconomic impacts, accurate ESM precipitation fields +to feed the impact models are therefore crucial. +ESMs are integrated on spatial grids with finite resolution. The resolution is limited +by the computational resources that are necessary to perform simulations on decadal to +centennial time scales. Current state-of-the-art ESMs have a horizontal resolution on the +order of 100km, in exceptional cases going down to 50km. Smaller-scale physical processes +that are relevant for the generation of precipitation operate on scales below the size of +individual grid cells. These can therefore not be resolved explicitly in ESMs and have to +included as parameterizations of the resolved prognostic variables. These include droplet +interactions, turbulence, and phase transitions in clouds that play a central role in the +generation of precipitation. +The limited grid resolution hence introduces errors in the simulated precipitation fields, +leading to biases in short-term spatial patterns and long-term summary statistics. These +biases need to be addressed prior to passing the ESM precipitation fields to impact mod- +els. In particular, climate impact models are often developed and calibrated with input +data from reanalysis data rather than ESM simulations. These reanalyses are created with +data assimilation routines and combine various observations with high-resolution weather +models. They hence provide a much more realistic input than the ESM simulations and +statistical bias correction methods are necessary to remove biases in the ESM simulations +output and to make them more similar to the reanalysis data for which the impact models +are calibrated. Quantile mapping (QM) is a standard technique to correct systematic errors +in ESM simulations. QM estimates a mapping between distributions from historical sim- +ulations and observations that can thereafter be applied to future simulations in order to +provide more accurate simulated precipitation fields to impact models (D´equ´e, 2007; Tong +et al., 2021; Gudmundsson et al., 2012; Cannon et al., 2015). +State-of-the-art bias correction methods such as QM are, however, confined to address +errors in the simulated frequency distributions locally, i.e., at every grid cell individually. +–2– + +Unrealistic spatial patterns of the ESM output, which would require spatial context, have +therefore so far not been addressed by postprocessing methods. For precipitation this is +particularly important because it has characteristic high intermittency not only in time, +but also in its spatial patterns. Mulitvariate bias correction approaches have recently been +developed, aiming to improve spatial dependencies (Vrac, 2018; Cannon, 2018). However, +these approaches are typically only employed in regional studies, as the dimension of the +input becomes too large for global high-resolution ESM simulations. Moreover, such meth- +ods have been reported to suffer from instabilities and overfitting, while differences in their +applicability and assumptions make them challenging to use (Fran¸cois et al., 2020). +Here, we employ a recently introduced postprocessing method (Hess et al., 2022) based +on a cycle-consistent adversarial network (CycleGAN) to consistently improve both local +frequency distributions and spatial patterns of state-of-art high-resolution ESM precipita- +tion fields. Artificial neural networks from computer vision and image processing have been +successfully applied to various tasks in Earth system science, ranging from weather forecast- +ing (Weyn et al., 2020; Rasp & Thuerey, 2021) to post-processing (Gr¨onquist et al., 2021; +Price & Rasp, 2022), by extracting spatial features with convolutional layers (LeCun et al., +2015). Generative adversarial networks (Goodfellow et al., 2014) in particular have emerged +as a promising architecture that produces sharp images that are necessary to capture the +high-frequency variability of precipitation (Ravuri et al., 2021; Price & Rasp, 2022; Harris et +al., 2022). GANs have been specifically developed to be trained on unpaired image datasets +(Zhu et al., 2017). This makes them a natural choice for post-processing the output of cli- +mate projections, which – unlike weather forecasts – are not nudged to follow the trajectory +of observations; due to the chaotic nature of the atmosphere small deviations in the initial +conditions or parameters lead to exponentially diverging trajectories (Lorenz, 1996). As a +result, numerical weather forecasts lose their deterministic forecast skill after approximately +two weeks at most and century-scale climate simulations do not agree with observed daily +weather records. Indeed the task of climate models is rather to produce accurate long-term +statistics that to agree with observations. +We apply our CycleGAN approach to correct global high-resolution precipitation simu- +lations of the GFDL-ESM4 model (Krasting et al., 2018) as a representative ESM from the +Climate Model Intercomparison Project phase 6 (CMIP6). So far, GANs-based approaches +have only been applied to postprocess ESM simulations either in a regional context (Fran¸cois +et al., 2021), or to a very-low-resolution global ESM (Hess et al., 2022). We show here that +a suitably designed CycleGAN is capable of improving even the distributions and spatial +patterns of precipitation fields from a state-of-the-art comprehensive ESM, namely GFDL- +ESM4. In particular, in contrast to rather specific existing methods for postprocessing ESM +output for climate impact modelling, we will show that the CycleGAN is general and can +readily be applied to different ESMs and observational datasets used as ground truth. +In order to assure that physical conservation laws are not violated by the GAN-based +postprocessing, we include a suitable physical constraint, enforcing that the overall global +sum of daily precipitation values is not changed by the GAN-based transformations; es- +sentially, this assures that precipitation is only spatially redistributed (see Methods). By +framing bias correction as an image-to-image translation task, our approach corrects both +spatial patterns of daily precipitation fields on short time scales and temporal distributions +aggregated over decadal time scales. We evaluate the skill to improve spatial patterns and +temporal distributions against the gold-standard ISIMIP3BASD framework (Lange, 2019), +which relies strongly on QM. +Quantifying the “realisticness” of spatial precipitation patterns is a key problem in +current research (Ravuri et al., 2021). We use spatial spectral densities and the fractal +dimension of spatial patterns as a measure to quantify the similarity of intermittent and un- +paired precipitation fields. We will show that our CycleGAN is indeed spatial context-aware +and strongly improves the characteristic intermittency in spatial precipitation patterns. We +–3– + +will also show that our CycleGAN combined with a subseqeunt application of ISIMIP3BASD +routine leads to the best overall performance. +2 Results +We evaluate our CycleGAN method on two different tasks and time scales. First, the +correction of daily rainfall frequency distributions at each grid cell locally, aggregated from +decade-long time series. Second, we quantify the ability to improve spatial patterns on daily +time scales. Our GAN approach is compared to the raw GFDL-ESM4 model output, as well +as to the ISIMIP3BASD methodology applied to the GFDL-ESM4 output. +2.1 Temporal distributions +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +Histogram +a +0 +98.4 +99.7 +99.94 +99.98 +99.993 +99.997 +W5E5v2 precipitation percentiles +W5E5v2 +GFDL-ESM4 +ISIMIP3BASD +GAN +GAN (unconstrained) +GAN-ISIMIP3BASD +0 +25 +50 +75 +100 +125 +150 +Precipitation [mm/d] +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Absolute error +b +Figure +1: Histograms of relative precipitation frequencies over the entire globe and test +period (2004-2014). (a) The histograms are shown for the W5E5v2 ground truth (black), +GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), unconstrained GAN (orange), +and the constrained-GAN-ISIMIP3BASD combination (blue). +(b) Distances of the his- +tograms to the W5E5v2 ground truth are shown for the same models as in (a). Percentiles +corresponding to the W5E5v2 precipitation values are given on the second x-axis at the +top. Note that GFDL-ESM4 overestimates the frequencies of strong and extreme rainfall +events. All compared methods show similar performance in correcting the local frequency +distributions. +–4– + +We compute global histograms of relative precipitation frequencies using daily time +series (Fig. 1a). The GFDL-ESM4 model overestimates frequencies in the tail, namely for +events above 50 mm/day (i.e., the 99.7th percentile). Our GAN-based method as well as +ISIMIP3BASD and the GAN-ISIMIP3BASD combination correct the histogram to match +the W5E5v2 ground truth equally well, as can be also seen in the absolute error of the +histograms (Fig. 1b). +Comparing the differences in long-term averages of precipitation per grid cell (Fig. 2 +and Methods), large biases are apparent in the GFDL-ESM4 model output, especially in +the tropics. The double-peaked Intertropical Convergence Zone (ITCZ) bias is visible. The +double-ITCZ bias can also be inferred from the latitudinal profile of the precipitation mean +in Fig. 3. +Table 1 summarizes the annual biases shown in Fig. 2 as absolute averages, and addi- +tionally for the four seasons. The GAN alone reduces the annual bias of the GFDL-ESM4 +model by 38.7%. The unconstrained GAN performs better than the physically constrained +one, with bias reductions of 50.5%. As expected, the ISIMIP3BASD gives even better results +for correcting the local mean, since it is specifically designed to accurately transform the +local frequency distributions. It is therefore remarkable that applying the ISIMIP3BASD +procedure on the constrained GAN output improves the post-processing further, leading to +a local bias reduction of the mean by 63.6%, compared to ISIMIP3BASD with 59.4%. For +seasonal time series the order in which the methods perform is the same as for the annual +data. +Besides the error in the mean, we also compute differences in the 95th percentile for each +grid cell, shown in Fig. S1 and as mean absolute errors in Table 1. Also in this case of heavy +precipitation values we find that ISIMIP3BASD outperforms the GAN, but that combining +GAN and ISIMIP3BASD leads to best agreement of the locally computed quantiles. +Table 1: The globally averaged absolute value of the grid cell-wise difference in the long- +term precipitation average, as well as the 95th percentile, between the W5E5v2 ground truth +and GFDL-ESM4, ISIMIP3BASD, GAN, unconstrained GAN, and the GAN-ISIMIP3BASD +combination for annual and seasonal time series (in [mm/day]). The relative improvement +over the raw GFDL-ESM4 climate model output is shown as percentages for each method. +Season +Percentile +GFDL- +ESM4 +ISIMIP3- +BASD +% +GAN +% +GAN +(unconst.) +% +GAN- +ISIMIP3- +BASD +% +Annual +- +0.535 +0.217 +59.4 +0.328 +38.7 +0.265 +50.5 +0.195 +63.6 +DJF +- +0.634 +0.321 +49.4 +0.395 +37.7 +0.371 +41.5 +0.308 +51.4 +MAM +- +0.722 +0.314 +56.5 +0.419 +42.0 +0.378 +47.6 +0.285 +60.5 +JJA +- +0.743 +0.289 +61.1 +0.451 +39.3 +0.357 +52.0 +0.280 +62.3 +SON +- +0.643 +0.327 +49.1 +0.409 +36.4 +0.362 +43.7 +0.306 +52.4 +Annual +95th +2.264 +1.073 +52.6 +1.415 +37.5 +1.213 +46.4 +0.945 +58.3 +DJF +95th +2.782 +1.496 +46.2 +1.725 +38.0 +1.655 +40.5 +1.432 +48.5 +MAM +95th +2.948 +1.482 +49.7 +1.805 +38.8 +1.661 +43.7 +1.337 +54.6 +JJA +95th +2.944 +1.366 +53.6 +1.852 +37.1 +1.532 +48.0 +1.247 +57.6 +SON +95th +2.689 +1.495 +44.4 +1.741 +35.3 +1.592 +40.8 +1.366 +49.2 +–5– + +Figure +2: Bias in the long-term average precipitation over the entire test set between +the W5E5v2 ground truth (a) and GFDL-ESM4 (b), ISIMIP3BASD (c), GAN (d), uncon- +strained GAN (e) and the GAN-ISIMIP3BASD combination (f). +2.2 Spatial patterns +We compare the ability of the GAN to improve spatial patterns based on the W5E5v2 +ground truth, against the GFDL-ESM4 simulations and the ISIMIP3BASD method applied +to the GFDL-ESM4 simulations. To model realistic precipitation fields, the characteristic +spatial intermittency needs to be captured accurately. +We compute the spatial power spectral density (PSD) of global precipitation fields, +averaged over the test set for each method. GFDL-ESM4 shows noticeable deviations from +W5E5v2 in the PSD (Fig. 4). Our GAN can correct these over the entire range of wave- +–6– + +W5E5v2 mean [mm/d] +GFDL-ESM4 +a +b +N.09 +0° +S.09 +0 +ISIMIP3BASD +GAN +N.09 +0° +60°S +GAN (unconstrained) +GAN-ISIMIP3BASD +e +f +N.09 +0° +S.09 +120°W +60°W +0 +60°E +120°E +120°W +60°W +0° +60°E +120°E +7.5 +-7.5 -5.0 -2.5 +0.0 +2.5 +5.0 +Bias [mm/d]80 S +60 S +40 S +20 S +0 +20 N +40 N +60 N +80 N +Latitude +0 +1 +2 +3 +4 +5 +6 +7 +Mean precipitation [mm/d] +W5E5v2 +GFDL-ESM4: MAE = 0.241 +ISIMIP3BASD: MAE = 0.120 +GAN: MAE = 0.226 +GAN (unconstrained): MAE = 0.102 +GAN-ISIMIP3BASD: MAE = 0.068 +Figure +3: Precipitation averaged over longitudes and the entire test set period from the +W5E5v2 ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN +(cyan), unconstrained GAN (orange) and the GAN-ISIMIP3BASD combination (blue). To +quantify the differences between the shown lines, we show their mean absolute error w.r.t +the W5E5v2 ground truth in the legend. These values are different from the ones shown in +Table 1 as the average is taken here over the longitudes without their absolute value. The +GAN-ISIMIP3BASD approach shows the lowest error. +lengths, closely matching the W5E5v2 ground truth. Improvements over ISIMIP3BASD +are especially pronounced in the range of high frequencies (low wavelengths), which are +responsible for the intermittent spatial variability of daily precipitation fields. Adding the +physical constraint to the GAN does not affect the ability to produce realistic PSD distribu- +tions. After applying ISIMIP3BASD to the GAN-processed fields, most of the improvements +generated by the GAN are retained, as shown by the GAN-ISIMIP3BASD results. +For a second way to quantifying how realistic the simulated and post-processed pre- +cipitation fields are, with a focus on high-frequency spatial intermittency, we investigate +the fractal dimension (Edgar & Edgar, 2008) of the lines separating grid cells with daily +rainfall sums above and below a given quantile threshold (see Methods). For a sample and +qualitative comparison of precipitation fields over the South American continent see Fig. S2. +The daily spatial precipitation fields are first converted to binary images using a quantile +threshold. The respective quantiles are determined from the precipitation distribution over +the entire test set period and globe. The mean of the fractal dimension computed with box- +counting (see Methods) (Lovejoy et al., 1987; Meisel et al., 1992; Husain et al., 2021) for each +time slice is then investigated (Fig. 5). Both the GFDL-ESM4 simulations themselves and +the results of applying the ISIMIP3BASD post-processing to them exhibit spatial patterns +with a lower fractal dimension than the W5E5v2 ground truth, implying too low spatial +intermittency. In contrast, the GAN translates spatial fields simulated by GFDL-ESM4 in +a way that results in closely matching fractal dimensions over the entire range of quantiles. +3 Discussion +Postprocessing climate projections is a fundamentally different task from postprocessing +weather forecast simulations (Hess et al., 2022). In the latter case, data-driven postprocess- +ing methods, e.g. based on deep learning, to minimize differences between paired samples +–7– + +128 +256 +512 +1024 +2048 +4096 +8192 +Wavelength [km] +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +PSD [a.u] +W5E5v2 +GFDL-ESM4 +ISIMIP3BASD +GAN +GAN (unconstrained) +GAN-ISIMIP3BASD +Figure 4: The power spectral density (PSD) of the spatial precipitation fields is shown as +an average over all samples in the test set for the W5E5v2 ground truth (black) and GFDL- +ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan, dashed), unconstrained GAN (orange, +dashed-dotted) and the constrained-GAN-ISIMIP3BASD combination (blue, dotted). The +GANs and W5E5v2 ground truth agree so closely that they are indistinguishable. In contrast +to ISIMIP3BASD, the GAN can correct the intermittent spectrum accurately over the entire +range down to the smallest wavelengths. +of variables such as spatial precipitation fields (Hess & Boers, 2022). Beyond time scales of +a few days, however, the chaotic nature of the atmosphere leads to exponentially diverging +trajectories, and for climate or Earth system model output there is no observation-based +ground truth to directly compare to. We therefore frame the post-processing of ESM projec- +tions, with applications for subsequent 195 impact modelling in mind, as an image-to-image +translation task with unpaired samples. +To this end we apply a recently developed postprocessing method based on physically +constrained CycleGANs to global simulations of a state-of-the-art, high-resolution ESM +from the CMIP6 model ensemble, namely the GFDL-ESM4 (Krasting et al., 2018; O' Neill +et al., 2016). We evaluate our method against the gold-standard bias correction framework +ISIMIP3BASD. Our model can be trained on unpaired samples that are characteristic for +climate simulations. It is able to correct the ESM simulations in two regards: temporal +distributions over long time scales, including extremes in the distrivutions’ tails, as well +as spatial patterns of individual global snap shots of the model output. The latter is not +possible with established methods. +Our GAN-based approach is designed as a general +framework that can be readily applied to different ESMs and observational target datasets. +This is in contrast to existing bias-adjustment methods that are often tailored to specific +applications. +We chose to correct precipitation because it is arguably one of the hardest variables +to represent accurately in ESMs. So far, GANs have only been applied to regional studies +or low-resolution global ESMs (Fran¸cois et al., 2021; Hess et al., 2022). The GFDL-ESM4 +model simulations are hence chosen in order to test if our CycleGAN approach would lead +–8– + +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Quantile +1.3 +1.4 +1.5 +1.6 +1.7 +Fractal dimension +W5E5v2 +GFDL-ESM4: MAE = 0.048 +ISIMIP3BASD: MAE = 0.037 +GAN: MAE = 0.002 +GAN (unconstrained): MAE = 0.002 +GAN-ISIMIP3BASD: MAE = 0.004 +Figure 5: The fractal dimension (see Methods) of binary global precipitation fields is com- +pared as averages for different quantile thresholds. +Results are shown for the W5E5v2 +ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), un- +constrained GAN (orange, dashed), and the GAN-ISIMIP3BASD combination (blue). The +GAN can accurately reproduce the fractal dimension of the W5E5v2 ground truth spatial +precipitation fields over all quantile thresholds, clearly outperforming the ISIMIP3BASD +basline. +to improvements even when postprocessing global high-resolution simulations of one of the +most complex and sophisticated ESMs to date. In the same spirit, we evaluate our ap- +proach against a very strong baseline given by the state-of-the-art bias correction framework +ISIMIP3BASD, which is based on a trend-preserving QM method (Lange, 2019). +Comparing long-term summary statistics, our method yields histograms of relative pre- +cipitation frequencies that very closely agree with corresponding histograms from reanalysis +data (Fig. 1). The means that the extremes in the far end of the tail are accurately cap- +tured, with similar skill to the ISIMIP3BASD baseline that is mainly designed for this task. +Differences in the grid cell-wise long-term average show that the GAN skillfully reduces bi- +ases (Fig. 2); in particular, the often reported double-peaked ITCZ bias of the GFDL-ESM4 +simulations, which is a common feature of most climate models (Tian & Dong, 2020), is +strongly reduced (Fig. 3). The ISIMIP3BASD method - being specifically designed for this +- produces slightly lower biases for grid-cell-wise averages than the GAN; we show that +combining both methods by first applying the GAN and then the ISIMIP3BASD procedure +leads to the overall best performance. +Regarding the correction of spatial patterns of the modelled precipitation fields, we +compare the spectral density and fractal dimensions of the spatial precipitation fields. Our +results show that indeed only the GAN can capture the characteristic spatial intermittency +of precipitation closely (Figs. 4 and 5). We believe that the measure of fractal dimension +is also relevant for other fields such as nowcasting and medium-range weather forecasting, +where blurriness in deep learning-based predictions is often reported (Ravuri et al., 2021) +and needs to be further quantified. +–9– + +Post-processing methods for climate projections have to be able to preserve the trends +that result from the non-stationary dynamics of the Earth system on long-time scales. We +have therefore introduced the architecture constraint of preserving the global precipitation +amount on every day in the climate model output (Hess et al., 2022). We find that this does +not affect the quality of the spatial patterns that are produced by our CycleGAN method. +However, the skill of correcting mean error biases is slightly reduced by the constraint. This +can be expected in part as the constraint is constructed to follow the global mean of the +ESM. Hence, biases in the global ESM mean can influence the constrained GAN. This also +motivates our choice to demonstrate the combination of the constrained GAN with the QM- +based ISIMIP3BASD procedure, since it can be applied to future climate scenarios, making +it more suitable for actual applications than the unconstrained architecture. +There are several directions to further develop or approach. The architecture employed +here has been built for equally spaced two-dimensional images. Extending the CycleGAN +architecture to perform convolutions on the spherical surface, e.g. using graph neural net- +works, might lead to more efficient and accurate models. Moreover, GANs are comparably +difficult to train, which could make it challenging to identify suitable network architectures. +Using large ensembles of climate simulations could provide additional training data that +could further improve the performance. Another straightforward extension of our method +would be the inclusion of further input variables or the prediction additional high-impact +physical variables, such as near-surface temperatures that are also important for regional +impact models. +4 Methods +4.1 Training data +We use global fields of daily precipitation with a horizontal resolution of 1◦ from the +GFDL-ESM4 Earth system model (Krasting et al., 2018) and the W5E5v2 reanalysis prod- +uct (Cucchi et al., 2020; WFDE5 over land merged with ERA5 over the ocean (W5E5 v2.0), +2021) as observation-based ground truth. +The W5E5v2 dataset is based on the ERA5 +(Hersbach et al., 2020) reanalysis and has been bias-adjusted using the Global Precipitation +Climatology Centre (GPCC) full data monthly product v2020 (Schneider et al., 2011) over +land and the Global Precipitation Climatology Project (GPCP) v2.3 dataset (Huffman et +al., 1997) over the ocean. Both datasets have been regridded to the same 1◦ horizontal +resolution using bilinear interpolation following (Beck et al., 2019). We split the dataset +into three periods for training (1950-2000), validation (2001-2003), and testing (2004-2014). +This corresponds to 8030 samples for training, 1095 for validation, and 4015 for testing. +During pre-processing, the training data is log-transformed with ˜x = log(x+ϵ)−log(ϵ) with +ϵ = 0.0001, following Rasp and Thuerey (2021), to account for zeros in the transform. The +data is then normalized to the interval [−1, 1] following (Zhu et al., 2017). +4.2 Cycle-consistent generative adversarial networks +This section gives a brief overview of the CycleGAN used in this study. We refer to +(Zhu et al., 2017; Hess et al., 2022) for a more comprehensive description and discussion. +Generative adversarial networks learn to generate images that are nearly indistinguishable +from real-world examples through a two-player game (Goodfellow et al., 2014). +In this +set-up, a first network G, the so-called generator, produces images with the objective to +fool a second network D, the discriminator, which has to classify whether a given sample +is generated (“fake”) or drawn from a real-world dataset (“real”). Mathematically this can +be formalized as +G∗ = min +G +max +D +LGAN(D, G), +(1) +–10– + +with G∗ being the optimal generator network. The loss function LGAN(D, G) can be defined +as +LGAN(D, G) = Ey∼py(y)[log(D(y))] + Ex∼px(x)[log(1 − D(G(x)))], +(2) +where py(y) is the distribution of the real-world target data and samples from px(x) are +used as inputs by G to produce realistic images. The CycleGAN (Zhu et al., 2017) consists +of two generator-discriminator pairs, where the generators G and F learn inverse mappings +between two domains X and Y . This allows to define an additional cycle-consistency loss +that constraints the training of the networks, i.e. +Lcycle(G, F) = Ex∼px(x)[||F(G(x)) − x||1] +(3) ++ Ey∼py(y)[||G(F(y)) − y||1]. +It measures the error caused by a translation cycle of an image to the other domain and +back. Further, an additional loss term is introduced to regularize the networks to be close +to an identity mapping with, +Lident(G, F) = Ex∼px(x)[||G(x) − x||1] +(4) ++ Ey∼py(y)[||F(y) − y||1]. +In practice, the log-likelihood loss can be replaced by a mean squared error loss to facilitate +a more stable training. +Further, the generator loss is reformulated to be minimized by +inverting the labels, i.e. +LGenerator = Ex∼px(x)[(DX(G(x)) − 1)2] ++ Ey∼py(y)[(DY (F(y)) − 1)2] +(5) ++ λLcycle(G, F) + ˜λLident(G, F), +where λ and ˜λ are set to 10 and 5 respectively following (Zhu et al., 2017). The corresponding +loss term for the discriminator networks is given by +LDiscriminator = Ey∼py(y)[(DY (y) − 1)2] + Ex∼px(x)[(DX(G(x)))2] +(6) ++ Ex∼px(x)[(DX(x) − 1)2] + Ey∼py(y)[(DY (F(y)))2]. +(7) +The weights of the generator and discriminator networks are then optimized with the ADAM +(Kingma & Ba, 2014) optimizer using a learning rate of 2e−4 and updated in an alternating +fashion. We train the network for 350 epochs and a batch size of 1, saving model checkpoints +every other epoch. We evaluate the checkpoints on the validation dataset to determine the +best model instance. +4.3 Network Architectures +Both the generator and discriminator have fully convolutional architectures. The gen- +erator uses ReLU activation functions, instance normalization, and reflection padding. The +discriminator uses leaky ReLU activations with slope 0.2 instead, together with instance +normalization. For a more detailed description, we refer to our previous study (Hess et al., +2022). The network architectures in this study are the same, only with a change in the +number of residual layers in the generator network from 6 to 7. +The final layer of the generator can be constrained to preserve the global sum of the +input, i.e. by rescaling +˜yi = yi +�Ngrid +i +xi +�Ngrid +i +yi +, +(8) +–11– + +where xi and yi are grid cell values of the generator input and output respectively and +Ngrid is the number of grid cells. The generator without this constraint will be referred +to as unconstrained in this study. The global physical constraint enforces that the global +daily precipitation sum is not affected by the CycleGAN postprocessing and hence remains +identical to the original value from the GFDL-ESM4 simualtions. This is motivated by the +observation that large-scale average trends in precipitation follow the Clausius-Clapeyron +relation (Traxl et al., 2021), which is based on thermodynamic relations and hence can be +expected to be modelled well in GFDL-ESM4. +4.4 Quantile mapping-based bias adjustment +We compare the performance of our GAN-based method to the bias adjustment method +ISMIP3BASD v3.0.1 (Lange, 2019, 2022) that has been developed for phase 3 of the Inter- +Sectoral Impact Model Intercomparison Project (Warszawski et al., 2014; Frieler et al., +2017). This state-of-the-art bias-adjustment method is based on a trend-preserving quantile +mapping (QM) framework. It represents a very strong baseline for comparison as it has +been developed prior to this study and used not only in ISIMIP3 but also to prepare many +of the climate projections that went into the Interactive Atlas produced as part of the 6th +assessment report of working group 1 of the Intergovernmental Panel on Climate Change +(IPCC, https://interactive-atlas.ipcc.ch/). In QM, a transformation between the cumulative +distribution functions (CDFs) of the historical simulation and observations is fitted and then +applied to future simulations. The CDFs can either be empirical or parametric, the latter +being a Bernoulli-gamma distribution for the precipitation in this study. The CFDs are +fitted and mapped for each grid cell and day of the year separately. For bias-adjusting the +GFDL-ESM4 simulation, parametric QM was found to give the best results, while empirical +CDFs are used in combination with the GAN. +To evaluate the methods in this study we define the grid cell-wise bias as the difference +in long-term averages as, +Bias(ˆy, y) = 1 +T +T +� +t=1 +ˆyt − 1 +T +T +� +t=1 +yt, +(9) +where T is the number of time steps, ˆyt and ˆyt the modelled and observed precipitation +respectively at time step t. +4.5 Evaluating spatial patterns +Quantifying how realistic spatial precipitation fields are is an ongoing research question +in itself, which has become more important with the application of deep learning to weather +forecasting and post-processing. In these applications, neural networks often achieve error +statistics and skill scores competitive with physical models, while the output fields can +at the same time show unphysical characteristics, such as blurring or excessive smoothing. +Ravuri et al. (2021) compare the spatial intermittency, which is characteristic of precipitation +fields, using the power spectral density (PSD) computed from the spatial fields; in the latter +study, the PSD-based quantification was complemented by interviews with a large number +of meteorological experts. We propose the fractal dimension of binary precipitation fields +as an alternative to quantify how realistic the patterns are. +We compute the fractal dimension via the box-counting algorithm (Lovejoy et al., 1987; +Meisel et al., 1992). It quantifies how spatial patterns, for example coastlines (Husain et +al., 2021), change with the scale of measurement. The box-counting algorithm divides the +image into squares and counts the number of squares that cover the binary pattern of +interest, Nsquares. The size of the squares, i.e. the scale of measurement, is then reduced +iteratively by a factor s. The fractal dimension Dfractal can then be determined from the +slope of the resulting log-log scaling, i.e., +–12– + +Dfractal = log(Nsquares) +log(s) +. +(10) +Competing interests +The authors declare no competing interests. +Data availability +The W5E5 data is available for download at https://doi.org/10.48364/ISIMIP.342217. +The GFDL-ESM4 data can be downloaded at https://esgf-node.llnl.gov/projects/ +cmip6/. +Code availability +The Python code for processing and analysing the data, together with the PyTorch +Lightning (Falcon et al., 2019) code is available at https://github.com/p-hss/earth +system model gan bias correction.git. The ISIMIP3BASD code in (Lange, 2022) is +used for this study. +Acknowledgments +NB and PH acknowledge funding by the Volkswagen Foundation, as well as the European +Regional Development Fund (ERDF), the German Federal Ministry of Education and Re- +search and the Land Brandenburg for supporting this project by providing resources on the +high performance computer system at the Potsdam Institute for Climate Impact Research. +N.B. acknowledges funding by the European Union’s Horizon 2020 research and innovation +programme under grant agreement No 820970 and under the Marie Sklodowska-Curie grant +agreement No. 956170, as well as from the Federal Ministry of Education and Research +under grant No. 01LS2001A. SL acknowledges funding from the European Union’s Horizon +2022 research and innovation programme under grant agreement no. 101081193 Optimal +High Resolution Earth System Models for Exploring Future Climate Changes (OptimESM). +References +Beck, H. E., Wood, E. 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In Proceedings of the IEEE international +conference on computer vision (pp. 2223–2232). +–15– + +Supporting Information for ”Deep Learning for +bias-correcting comprehensive high-resolution Earth +system models” +Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3 +1Earth System Modelling, School of Engineering & Design, Technical University of Munich, Munich, Germany +2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany +3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK +Contents of this file +1. Figure S1 to S2 +January 4, 2023, 1:28am +arXiv:2301.01253v1 [physics.ao-ph] 16 Dec 2022 + +X - 2 +: +Figure S1. +Bias maps as in Fig. +2 but with the 95th percentile instead of the mean. +Global mean absolute errors (MAEs) are given in the respective titles. Combining the GAN with +ISIMIP3BASD achieves the lowest error compared to the other methods. +January 4, 2023, 1:28am + +W5E5v2 95th percentile [mm/d] +GFDL-ESM4: MAE = 2.264 +b +a +N.09 +0° +S.09 +0 +25 +50 +120°W +60°W +0° +60°E +120°E +120°W +60°W +0° +60°E +120°E +ISIMIP3BASD: MAE = 1.073 +GAN: MAE = +1.415 +d +N.09 +0° +S.09 +120°W +60°W +.0 +60°E +120°E +60°W +0° +60°E +120°W +120°E +GAN (unconstrained): MAE +1.213 +GAN-ISIMIP3BASD: MAE +0.945 += +e +2 +120°W +60°W +0° +60°E +120°E +120°W +60°W +0° +60°E +120°E +-20 -i5 -i0 -5　0 +5 +10 +15 +20 +Differences in the 95th percentile [mm/d]: +X - 3 +a +50°S +25°S +0° +100°W +75°W +50°W +25°W +W5E5v2 +c +50°S +25°S +0° +100°W +75°W +50°W +25°W +ISIMIP3BASD +b +50°S +25°S +0° +100°W +75°W +50°W +25°W +GFDL-ESM4 +d +50°S +25°S +0° +100°W +75°W +50°W +25°W +GAN-ISIMIP3BASD +5 +10 +15 +20 +25 +30 +35 +Precipitation [mm/d] +Figure S2. +Qualitative comparison of precipitation fields at the same date (December 21st +2014) over the South American continent. The region is used for a comparison of the fractal +dimension in binary precipitation patterns. +January 4, 2023, 1:28am + diff --git a/CNAzT4oBgHgl3EQfTvwq/content/tmp_files/load_file.txt b/CNAzT4oBgHgl3EQfTvwq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eeea002abc440b59378ddb13a80e0cddb71d0d0d --- /dev/null +++ b/CNAzT4oBgHgl3EQfTvwq/content/tmp_files/load_file.txt @@ -0,0 +1,933 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf,len=932 +page_content='Deep Learning for bias-correcting comprehensive high-resolution Earth system models Philipp Hess1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Stefan Lange2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' and Niklas Boers1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='3 1Earth System Modelling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' School of Engineering & Design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Germany 2Potsdam Institute for Climate Impact Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Member of the Leibniz Association,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Germany 3Global Systems Institute and Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' University of Exeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Exeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' UK Key Points: A generative adversarial network is shown to improve daily precipitation fields from a state-of-the-art Earth system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Biases in long-term temporal distributions are strongly reduced by the generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our network-based approach can be complemented with quantile mapping to fur- ther improve precipitation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' –1– arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='01253v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='ao-ph] 16 Dec 2022 Abstract The accurate representation of precipitation in Earth system models (ESMs) is crucial for reliable projections of the ecological and socioeconomic impacts in response to anthropogenic global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The complex cross-scale interactions of processes that produces precipi- tation are challenging to model, however, inducing potentially strong biases in ESM fields, especially regarding extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' State-of-the-art bias correction methods only address errors in the simulated frequency distributions locally, at every individual grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Improving unrealistic spatial patterns of the ESM output, which would require spatial context, has not been possible so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Here, we show that a post-processing method based on physically constrained generative adversarial networks (GANs) can correct biases of a state-of-the-art, CMIP6-class ESM both in local frequency distributions and in the spatial patterns at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' While our method improves local frequency distributions equally well as gold-standard bias- adjustment frameworks it strongly outperforms any existing methods in the correction of spatial patterns, especially in terms of the characteristic spatial intermittency of precipita- tion extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 1 Introduction Precipitation is a crucial climate variable and changing amounts, frequencies, or spatial distributions have potentially severe ecological and socioeconomic impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' With global warming projected to continue in the coming decades, assessing the impacts of changes in precipitation characteristics is an urgent challenge (Wilcox & Donner, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Boyle & Klein, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' IPCC, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Climate impact models are designed to assess the impacts of global warming on, for example, ecosystems, crop yields, vegetation and other land-surface characteristics, infrastructure, water resources, or the economy in general (Kotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022), using the output of climate or Earth system models (ESMs) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Especially for reliable assessments of the ecological and socioeconomic impacts, accurate ESM precipitation fields to feed the impact models are therefore crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' ESMs are integrated on spatial grids with finite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The resolution is limited by the computational resources that are necessary to perform simulations on decadal to centennial time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Current state-of-the-art ESMs have a horizontal resolution on the order of 100km, in exceptional cases going down to 50km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Smaller-scale physical processes that are relevant for the generation of precipitation operate on scales below the size of individual grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' These can therefore not be resolved explicitly in ESMs and have to included as parameterizations of the resolved prognostic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' These include droplet interactions, turbulence, and phase transitions in clouds that play a central role in the generation of precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The limited grid resolution hence introduces errors in the simulated precipitation fields, leading to biases in short-term spatial patterns and long-term summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' These biases need to be addressed prior to passing the ESM precipitation fields to impact mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In particular, climate impact models are often developed and calibrated with input data from reanalysis data rather than ESM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' These reanalyses are created with data assimilation routines and combine various observations with high-resolution weather models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' They hence provide a much more realistic input than the ESM simulations and statistical bias correction methods are necessary to remove biases in the ESM simulations output and to make them more similar to the reanalysis data for which the impact models are calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Quantile mapping (QM) is a standard technique to correct systematic errors in ESM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' QM estimates a mapping between distributions from historical sim- ulations and observations that can thereafter be applied to future simulations in order to provide more accurate simulated precipitation fields to impact models (D´equ´e, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Gudmundsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Cannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' State-of-the-art bias correction methods such as QM are, however, confined to address errors in the simulated frequency distributions locally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', at every grid cell individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' –2– Unrealistic spatial patterns of the ESM output, which would require spatial context, have therefore so far not been addressed by postprocessing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' For precipitation this is particularly important because it has characteristic high intermittency not only in time, but also in its spatial patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Mulitvariate bias correction approaches have recently been developed, aiming to improve spatial dependencies (Vrac, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Cannon, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' However, these approaches are typically only employed in regional studies, as the dimension of the input becomes too large for global high-resolution ESM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Moreover, such meth- ods have been reported to suffer from instabilities and overfitting, while differences in their applicability and assumptions make them challenging to use (Fran¸cois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Here, we employ a recently introduced postprocessing method (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022) based on a cycle-consistent adversarial network (CycleGAN) to consistently improve both local frequency distributions and spatial patterns of state-of-art high-resolution ESM precipita- tion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Artificial neural networks from computer vision and image processing have been successfully applied to various tasks in Earth system science, ranging from weather forecast- ing (Weyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Rasp & Thuerey, 2021) to post-processing (Gr¨onquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Price & Rasp, 2022), by extracting spatial features with convolutional layers (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Generative adversarial networks (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2014) in particular have emerged as a promising architecture that produces sharp images that are necessary to capture the high-frequency variability of precipitation (Ravuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Price & Rasp, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' GANs have been specifically developed to be trained on unpaired image datasets (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This makes them a natural choice for post-processing the output of cli- mate projections, which – unlike weather forecasts – are not nudged to follow the trajectory of observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' due to the chaotic nature of the atmosphere small deviations in the initial conditions or parameters lead to exponentially diverging trajectories (Lorenz, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' As a result, numerical weather forecasts lose their deterministic forecast skill after approximately two weeks at most and century-scale climate simulations do not agree with observed daily weather records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Indeed the task of climate models is rather to produce accurate long-term statistics that to agree with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We apply our CycleGAN approach to correct global high-resolution precipitation simu- lations of the GFDL-ESM4 model (Krasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2018) as a representative ESM from the Climate Model Intercomparison Project phase 6 (CMIP6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' So far, GANs-based approaches have only been applied to postprocess ESM simulations either in a regional context (Fran¸cois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021), or to a very-low-resolution global ESM (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We show here that a suitably designed CycleGAN is capable of improving even the distributions and spatial patterns of precipitation fields from a state-of-the-art comprehensive ESM, namely GFDL- ESM4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In particular, in contrast to rather specific existing methods for postprocessing ESM output for climate impact modelling, we will show that the CycleGAN is general and can readily be applied to different ESMs and observational datasets used as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In order to assure that physical conservation laws are not violated by the GAN-based postprocessing, we include a suitable physical constraint, enforcing that the overall global sum of daily precipitation values is not changed by the GAN-based transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' es- sentially, this assures that precipitation is only spatially redistributed (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' By framing bias correction as an image-to-image translation task, our approach corrects both spatial patterns of daily precipitation fields on short time scales and temporal distributions aggregated over decadal time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We evaluate the skill to improve spatial patterns and temporal distributions against the gold-standard ISIMIP3BASD framework (Lange, 2019), which relies strongly on QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Quantifying the “realisticness” of spatial precipitation patterns is a key problem in current research (Ravuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We use spatial spectral densities and the fractal dimension of spatial patterns as a measure to quantify the similarity of intermittent and un- paired precipitation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We will show that our CycleGAN is indeed spatial context-aware and strongly improves the characteristic intermittency in spatial precipitation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We –3– will also show that our CycleGAN combined with a subseqeunt application of ISIMIP3BASD routine leads to the best overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2 Results We evaluate our CycleGAN method on two different tasks and time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' First, the correction of daily rainfall frequency distributions at each grid cell locally, aggregated from decade-long time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Second, we quantify the ability to improve spatial patterns on daily time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our GAN approach is compared to the raw GFDL-ESM4 model output, as well as to the ISIMIP3BASD methodology applied to the GFDL-ESM4 output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='1 Temporal distributions 10 6 10 5 10 4 10 3 10 2 10 1 100 Histogram a 0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='94 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='993 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='997 W5E5v2 precipitation percentiles W5E5v2 GFDL-ESM4 ISIMIP3BASD GAN GAN (unconstrained) GAN-ISIMIP3BASD 0 25 50 75 100 125 150 Precipitation [mm/d] 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 Absolute error b Figure 1: Histograms of relative precipitation frequencies over the entire globe and test period (2004-2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' (a) The histograms are shown for the W5E5v2 ground truth (black), GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), unconstrained GAN (orange), and the constrained-GAN-ISIMIP3BASD combination (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' (b) Distances of the his- tograms to the W5E5v2 ground truth are shown for the same models as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Percentiles corresponding to the W5E5v2 precipitation values are given on the second x-axis at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Note that GFDL-ESM4 overestimates the frequencies of strong and extreme rainfall events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' All compared methods show similar performance in correcting the local frequency distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' –4– We compute global histograms of relative precipitation frequencies using daily time series (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GFDL-ESM4 model overestimates frequencies in the tail, namely for events above 50 mm/day (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', the 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='7th percentile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our GAN-based method as well as ISIMIP3BASD and the GAN-ISIMIP3BASD combination correct the histogram to match the W5E5v2 ground truth equally well, as can be also seen in the absolute error of the histograms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Comparing the differences in long-term averages of precipitation per grid cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2 and Methods), large biases are apparent in the GFDL-ESM4 model output, especially in the tropics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The double-peaked Intertropical Convergence Zone (ITCZ) bias is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The double-ITCZ bias can also be inferred from the latitudinal profile of the precipitation mean in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Table 1 summarizes the annual biases shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2 as absolute averages, and addi- tionally for the four seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GAN alone reduces the annual bias of the GFDL-ESM4 model by 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The unconstrained GAN performs better than the physically constrained one, with bias reductions of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' As expected, the ISIMIP3BASD gives even better results for correcting the local mean, since it is specifically designed to accurately transform the local frequency distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' It is therefore remarkable that applying the ISIMIP3BASD procedure on the constrained GAN output improves the post-processing further, leading to a local bias reduction of the mean by 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='6%, compared to ISIMIP3BASD with 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' For seasonal time series the order in which the methods perform is the same as for the annual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Besides the error in the mean, we also compute differences in the 95th percentile for each grid cell, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' S1 and as mean absolute errors in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Also in this case of heavy precipitation values we find that ISIMIP3BASD outperforms the GAN, but that combining GAN and ISIMIP3BASD leads to best agreement of the locally computed quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Table 1: The globally averaged absolute value of the grid cell-wise difference in the long- term precipitation average, as well as the 95th percentile, between the W5E5v2 ground truth and GFDL-ESM4, ISIMIP3BASD, GAN, unconstrained GAN, and the GAN-ISIMIP3BASD combination for annual and seasonal time series (in [mm/day]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The relative improvement over the raw GFDL-ESM4 climate model output is shown as percentages for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Season Percentile GFDL- ESM4 ISIMIP3- BASD % GAN % GAN (unconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=') % GAN- ISIMIP3- BASD % Annual 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='217 59.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='592 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='366 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2 –5– Figure 2: Bias in the long-term average precipitation over the entire test set between the W5E5v2 ground truth (a) and GFDL-ESM4 (b), ISIMIP3BASD (c), GAN (d), uncon- strained GAN (e) and the GAN-ISIMIP3BASD combination (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2 Spatial patterns We compare the ability of the GAN to improve spatial patterns based on the W5E5v2 ground truth, against the GFDL-ESM4 simulations and the ISIMIP3BASD method applied to the GFDL-ESM4 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' To model realistic precipitation fields, the characteristic spatial intermittency needs to be captured accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We compute the spatial power spectral density (PSD) of global precipitation fields, averaged over the test set for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' GFDL-ESM4 shows noticeable deviations from W5E5v2 in the PSD (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our GAN can correct these over the entire range of wave- –6– W5E5v2 mean [mm/d] GFDL-ESM4 a b N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0° S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0 ISIMIP3BASD GAN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0° 60°S GAN (unconstrained) GAN-ISIMIP3BASD e f N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0° S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 120°W 60°W 0 60°E 120°E 120°W 60°W 0° 60°E 120°E 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0 Bias [mm/d]80 S 60 S 40 S 20 S 0 20 N 40 N 60 N 80 N Latitude 0 1 2 3 4 5 6 7 Mean precipitation [mm/d] W5E5v2 GFDL-ESM4: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='241 ISIMIP3BASD: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='120 GAN: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='226 GAN (unconstrained): MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='102 GAN-ISIMIP3BASD: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='068 Figure 3: Precipitation averaged over longitudes and the entire test set period from the W5E5v2 ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), unconstrained GAN (orange) and the GAN-ISIMIP3BASD combination (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' To quantify the differences between the shown lines, we show their mean absolute error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='t the W5E5v2 ground truth in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' These values are different from the ones shown in Table 1 as the average is taken here over the longitudes without their absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GAN-ISIMIP3BASD approach shows the lowest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' lengths, closely matching the W5E5v2 ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Improvements over ISIMIP3BASD are especially pronounced in the range of high frequencies (low wavelengths), which are responsible for the intermittent spatial variability of daily precipitation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Adding the physical constraint to the GAN does not affect the ability to produce realistic PSD distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' After applying ISIMIP3BASD to the GAN-processed fields, most of the improvements generated by the GAN are retained, as shown by the GAN-ISIMIP3BASD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' For a second way to quantifying how realistic the simulated and post-processed pre- cipitation fields are, with a focus on high-frequency spatial intermittency, we investigate the fractal dimension (Edgar & Edgar, 2008) of the lines separating grid cells with daily rainfall sums above and below a given quantile threshold (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' For a sample and qualitative comparison of precipitation fields over the South American continent see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The daily spatial precipitation fields are first converted to binary images using a quantile threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The respective quantiles are determined from the precipitation distribution over the entire test set period and globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The mean of the fractal dimension computed with box- counting (see Methods) (Lovejoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Meisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Husain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021) for each time slice is then investigated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Both the GFDL-ESM4 simulations themselves and the results of applying the ISIMIP3BASD post-processing to them exhibit spatial patterns with a lower fractal dimension than the W5E5v2 ground truth, implying too low spatial intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In contrast, the GAN translates spatial fields simulated by GFDL-ESM4 in a way that results in closely matching fractal dimensions over the entire range of quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 3 Discussion Postprocessing climate projections is a fundamentally different task from postprocessing weather forecast simulations (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In the latter case, data-driven postprocess- ing methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' based on deep learning, to minimize differences between paired samples –7– 128 256 512 1024 2048 4096 8192 Wavelength [km] 10 6 10 5 10 4 10 3 10 2 PSD [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='u] W5E5v2 GFDL-ESM4 ISIMIP3BASD GAN GAN (unconstrained) GAN-ISIMIP3BASD Figure 4: The power spectral density (PSD) of the spatial precipitation fields is shown as an average over all samples in the test set for the W5E5v2 ground truth (black) and GFDL- ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan, dashed), unconstrained GAN (orange, dashed-dotted) and the constrained-GAN-ISIMIP3BASD combination (blue, dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GANs and W5E5v2 ground truth agree so closely that they are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In contrast to ISIMIP3BASD, the GAN can correct the intermittent spectrum accurately over the entire range down to the smallest wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' of variables such as spatial precipitation fields (Hess & Boers, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Beyond time scales of a few days, however, the chaotic nature of the atmosphere leads to exponentially diverging trajectories, and for climate or Earth system model output there is no observation-based ground truth to directly compare to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We therefore frame the post-processing of ESM projec- tions, with applications for subsequent 195 impact modelling in mind, as an image-to-image translation task with unpaired samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' To this end we apply a recently developed postprocessing method based on physically constrained CycleGANs to global simulations of a state-of-the-art, high-resolution ESM from the CMIP6 model ensemble, namely the GFDL-ESM4 (Krasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=" O' Neill et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We evaluate our method against the gold-standard bias correction framework ISIMIP3BASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our model can be trained on unpaired samples that are characteristic for climate simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' It is able to correct the ESM simulations in two regards: temporal distributions over long time scales, including extremes in the distrivutions’ tails, as well as spatial patterns of individual global snap shots of the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The latter is not possible with established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our GAN-based approach is designed as a general framework that can be readily applied to different ESMs and observational target datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This is in contrast to existing bias-adjustment methods that are often tailored to specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We chose to correct precipitation because it is arguably one of the hardest variables to represent accurately in ESMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' So far, GANs have only been applied to regional studies or low-resolution global ESMs (Fran¸cois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GFDL-ESM4 model simulations are hence chosen in order to test if our CycleGAN approach would lead –8– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='9 Quantile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='7 Fractal dimension W5E5v2 GFDL-ESM4: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='048 ISIMIP3BASD: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='037 GAN: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='002 GAN (unconstrained): MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='002 GAN-ISIMIP3BASD: MAE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='004 Figure 5: The fractal dimension (see Methods) of binary global precipitation fields is com- pared as averages for different quantile thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Results are shown for the W5E5v2 ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), un- constrained GAN (orange, dashed), and the GAN-ISIMIP3BASD combination (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GAN can accurately reproduce the fractal dimension of the W5E5v2 ground truth spatial precipitation fields over all quantile thresholds, clearly outperforming the ISIMIP3BASD basline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' to improvements even when postprocessing global high-resolution simulations of one of the most complex and sophisticated ESMs to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In the same spirit, we evaluate our ap- proach against a very strong baseline given by the state-of-the-art bias correction framework ISIMIP3BASD, which is based on a trend-preserving QM method (Lange, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Comparing long-term summary statistics, our method yields histograms of relative pre- cipitation frequencies that very closely agree with corresponding histograms from reanalysis data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The means that the extremes in the far end of the tail are accurately cap- tured, with similar skill to the ISIMIP3BASD baseline that is mainly designed for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Differences in the grid cell-wise long-term average show that the GAN skillfully reduces bi- ases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' in particular, the often reported double-peaked ITCZ bias of the GFDL-ESM4 simulations, which is a common feature of most climate models (Tian & Dong, 2020), is strongly reduced (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The ISIMIP3BASD method - being specifically designed for this produces slightly lower biases for grid-cell-wise averages than the GAN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' we show that combining both methods by first applying the GAN and then the ISIMIP3BASD procedure leads to the overall best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Regarding the correction of spatial patterns of the modelled precipitation fields, we compare the spectral density and fractal dimensions of the spatial precipitation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Our results show that indeed only the GAN can capture the characteristic spatial intermittency of precipitation closely (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We believe that the measure of fractal dimension is also relevant for other fields such as nowcasting and medium-range weather forecasting, where blurriness in deep learning-based predictions is often reported (Ravuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021) and needs to be further quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' –9– Post-processing methods for climate projections have to be able to preserve the trends that result from the non-stationary dynamics of the Earth system on long-time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We have therefore introduced the architecture constraint of preserving the global precipitation amount on every day in the climate model output (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We find that this does not affect the quality of the spatial patterns that are produced by our CycleGAN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' However, the skill of correcting mean error biases is slightly reduced by the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This can be expected in part as the constraint is constructed to follow the global mean of the ESM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Hence, biases in the global ESM mean can influence the constrained GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This also motivates our choice to demonstrate the combination of the constrained GAN with the QM- based ISIMIP3BASD procedure, since it can be applied to future climate scenarios, making it more suitable for actual applications than the unconstrained architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' There are several directions to further develop or approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The architecture employed here has been built for equally spaced two-dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Extending the CycleGAN architecture to perform convolutions on the spherical surface, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' using graph neural net- works, might lead to more efficient and accurate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Moreover, GANs are comparably difficult to train, which could make it challenging to identify suitable network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Using large ensembles of climate simulations could provide additional training data that could further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Another straightforward extension of our method would be the inclusion of further input variables or the prediction additional high-impact physical variables, such as near-surface temperatures that are also important for regional impact models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4 Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='1 Training data We use global fields of daily precipitation with a horizontal resolution of 1◦ from the GFDL-ESM4 Earth system model (Krasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2018) and the W5E5v2 reanalysis prod- uct (Cucchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' WFDE5 over land merged with ERA5 over the ocean (W5E5 v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0), 2021) as observation-based ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The W5E5v2 dataset is based on the ERA5 (Hersbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2020) reanalysis and has been bias-adjusted using the Global Precipitation Climatology Centre (GPCC) full data monthly product v2020 (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2011) over land and the Global Precipitation Climatology Project (GPCP) v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='3 dataset (Huffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 1997) over the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Both datasets have been regridded to the same 1◦ horizontal resolution using bilinear interpolation following (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We split the dataset into three periods for training (1950-2000), validation (2001-2003), and testing (2004-2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This corresponds to 8030 samples for training, 1095 for validation, and 4015 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' During pre-processing, the training data is log-transformed with ˜x = log(x+ϵ)−log(ϵ) with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0001, following Rasp and Thuerey (2021), to account for zeros in the transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The data is then normalized to the interval [−1, 1] following (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2 Cycle-consistent generative adversarial networks This section gives a brief overview of the CycleGAN used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We refer to (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022) for a more comprehensive description and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Generative adversarial networks learn to generate images that are nearly indistinguishable from real-world examples through a two-player game (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In this set-up, a first network G, the so-called generator, produces images with the objective to fool a second network D, the discriminator, which has to classify whether a given sample is generated (“fake”) or drawn from a real-world dataset (“real”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Mathematically this can be formalized as G∗ = min G max D LGAN(D, G), (1) –10– with G∗ being the optimal generator network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The loss function LGAN(D, G) can be defined as LGAN(D, G) = Ey∼py(y)[log(D(y))] + Ex∼px(x)[log(1 − D(G(x)))], (2) where py(y) is the distribution of the real-world target data and samples from px(x) are used as inputs by G to produce realistic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2017) consists of two generator-discriminator pairs, where the generators G and F learn inverse mappings between two domains X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This allows to define an additional cycle-consistency loss that constraints the training of the networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Lcycle(G, F) = Ex∼px(x)[||F(G(x)) − x||1] (3) + Ey∼py(y)[||G(F(y)) − y||1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' It measures the error caused by a translation cycle of an image to the other domain and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Further, an additional loss term is introduced to regularize the networks to be close to an identity mapping with, Lident(G, F) = Ex∼px(x)[||G(x) − x||1] (4) + Ey∼py(y)[||F(y) − y||1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In practice, the log-likelihood loss can be replaced by a mean squared error loss to facilitate a more stable training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Further, the generator loss is reformulated to be minimized by inverting the labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' LGenerator = Ex∼px(x)[(DX(G(x)) − 1)2] + Ey∼py(y)[(DY (F(y)) − 1)2] (5) + λLcycle(G, F) + ˜λLident(G, F), where λ and ˜λ are set to 10 and 5 respectively following (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The corresponding loss term for the discriminator networks is given by LDiscriminator = Ey∼py(y)[(DY (y) − 1)2] + Ex∼px(x)[(DX(G(x)))2] (6) + Ex∼px(x)[(DX(x) − 1)2] + Ey∼py(y)[(DY (F(y)))2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' (7) The weights of the generator and discriminator networks are then optimized with the ADAM (Kingma & Ba, 2014) optimizer using a learning rate of 2e−4 and updated in an alternating fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We train the network for 350 epochs and a batch size of 1, saving model checkpoints every other epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We evaluate the checkpoints on the validation dataset to determine the best model instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='3 Network Architectures Both the generator and discriminator have fully convolutional architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The gen- erator uses ReLU activation functions, instance normalization, and reflection padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The discriminator uses leaky ReLU activations with slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2 instead, together with instance normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' For a more detailed description, we refer to our previous study (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The network architectures in this study are the same, only with a change in the number of residual layers in the generator network from 6 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The final layer of the generator can be constrained to preserve the global sum of the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' by rescaling ˜yi = yi �Ngrid i xi �Ngrid i yi , (8) –11– where xi and yi are grid cell values of the generator input and output respectively and Ngrid is the number of grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The generator without this constraint will be referred to as unconstrained in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The global physical constraint enforces that the global daily precipitation sum is not affected by the CycleGAN postprocessing and hence remains identical to the original value from the GFDL-ESM4 simualtions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This is motivated by the observation that large-scale average trends in precipitation follow the Clausius-Clapeyron relation (Traxl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021), which is based on thermodynamic relations and hence can be expected to be modelled well in GFDL-ESM4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='4 Quantile mapping-based bias adjustment We compare the performance of our GAN-based method to the bias adjustment method ISMIP3BASD v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='1 (Lange, 2019, 2022) that has been developed for phase 3 of the Inter- Sectoral Impact Model Intercomparison Project (Warszawski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Frieler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' This state-of-the-art bias-adjustment method is based on a trend-preserving quantile mapping (QM) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' It represents a very strong baseline for comparison as it has been developed prior to this study and used not only in ISIMIP3 but also to prepare many of the climate projections that went into the Interactive Atlas produced as part of the 6th assessment report of working group 1 of the Intergovernmental Panel on Climate Change (IPCC, https://interactive-atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='ipcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='ch/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In QM, a transformation between the cumulative distribution functions (CDFs) of the historical simulation and observations is fitted and then applied to future simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The CDFs can either be empirical or parametric, the latter being a Bernoulli-gamma distribution for the precipitation in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The CFDs are fitted and mapped for each grid cell and day of the year separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' For bias-adjusting the GFDL-ESM4 simulation, parametric QM was found to give the best results, while empirical CDFs are used in combination with the GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' To evaluate the methods in this study we define the grid cell-wise bias as the difference in long-term averages as, Bias(ˆy, y) = 1 T T � t=1 ˆyt − 1 T T � t=1 yt, (9) where T is the number of time steps, ˆyt and ˆyt the modelled and observed precipitation respectively at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='5 Evaluating spatial patterns Quantifying how realistic spatial precipitation fields are is an ongoing research question in itself, which has become more important with the application of deep learning to weather forecasting and post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In these applications, neural networks often achieve error statistics and skill scores competitive with physical models, while the output fields can at the same time show unphysical characteristics, such as blurring or excessive smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Ravuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' (2021) compare the spatial intermittency, which is characteristic of precipitation fields, using the power spectral density (PSD) computed from the spatial fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' in the latter study, the PSD-based quantification was complemented by interviews with a large number of meteorological experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We propose the fractal dimension of binary precipitation fields as an alternative to quantify how realistic the patterns are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' We compute the fractal dimension via the box-counting algorithm (Lovejoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Meisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' It quantifies how spatial patterns, for example coastlines (Husain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2021), change with the scale of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The box-counting algorithm divides the image into squares and counts the number of squares that cover the binary pattern of interest, Nsquares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The size of the squares, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' the scale of measurement, is then reduced iteratively by a factor s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The fractal dimension Dfractal can then be determined from the slope of the resulting log-log scaling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', –12– Dfractal = log(Nsquares) log(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' (10) Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Data availability The W5E5 data is available for download at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='48364/ISIMIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='342217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The GFDL-ESM4 data can be downloaded at https://esgf-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='llnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='gov/projects/ cmip6/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Code availability The Python code for processing and analysing the data, together with the PyTorch Lightning (Falcon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', 2019) code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='com/p-hss/earth system model gan bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The ISIMIP3BASD code in (Lange, 2022) is used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Acknowledgments NB and PH acknowledge funding by the Volkswagen Foundation, as well as the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Re- search and the Land Brandenburg for supporting this project by providing resources on the high performance computer system at the Potsdam Institute for Climate Impact Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' acknowledges funding by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 820970 and under the Marie Sklodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 956170, as well as from the Federal Ministry of Education and Research under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 01LS2001A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' SL acknowledges funding from the European Union’s Horizon 2022 research and innovation programme under grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 101081193 Optimal High Resolution Earth System Models for Exploring Future Climate Changes (OptimESM).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Journal of Climate, 20(1), 53–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', Isola, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=', & Efros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Unpaired image-to-image translation using cycle-consistent adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' In Proceedings of the IEEE international conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2223–2232).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' –15– Supporting Information for ”Deep Learning for bias-correcting comprehensive high-resolution Earth system models” Philipp Hess1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Stefan Lange2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' and Niklas Boers1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='3 1Earth System Modelling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' School of Engineering & Design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Germany 2Potsdam Institute for Climate Impact Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Member of the Leibniz Association,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Germany 3Global Systems Institute and Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' University of Exeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Exeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' UK Contents of this file 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Figure S1 to S2 January 4, 2023, 1:28am arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='01253v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='ao-ph] 16 Dec 2022 X - 2 : Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Bias maps as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' 2 but with the 95th percentile instead of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Global mean absolute errors (MAEs) are given in the respective titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Combining the GAN with ISIMIP3BASD achieves the lowest error compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' January 4, 2023, 1:28am W5E5v2 95th percentile [mm/d] GFDL-ESM4: MAE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='264 b a N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0° S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0 25 50 120°W 60°W 0° 60°E 120°E 120°W 60°W 0° 60°E 120°E ISIMIP3BASD: MAE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='073 GAN: MAE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='415 d N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 0° S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='09 120°W 60°W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='0 60°E 120°E 60°W 0° 60°E 120°W 120°E GAN (unconstrained): MAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='213 GAN-ISIMIP3BASD: MAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content='945 = e 2 120°W 60°W 0° 60°E 120°E 120°W 60°W 0° 60°E 120°E 20 -i5 -i0 -5 0 5 10 15 20 Differences in the 95th percentile [mm/d]: X - 3 a 50°S 25°S 0° 100°W 75°W 50°W 25°W W5E5v2 c 50°S 25°S 0° 100°W 75°W 50°W 25°W ISIMIP3BASD b 50°S 25°S 0° 100°W 75°W 50°W 25°W GFDL-ESM4 d 50°S 25°S 0° 100°W 75°W 50°W 25°W GAN-ISIMIP3BASD 5 10 15 20 25 30 35 Precipitation [mm/d] Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' Qualitative comparison of precipitation fields at the same date (December 21st 2014) over the South American continent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' The region is used for a comparison of the fractal dimension in binary precipitation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} +page_content=' January 4, 2023, 1:28am' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf'} diff --git a/ENAyT4oBgHgl3EQfSPf9/content/tmp_files/2301.00085v1.pdf.txt b/ENAyT4oBgHgl3EQfSPf9/content/tmp_files/2301.00085v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eaa2fb89859157866b4ed6e12a879834a1ccf89e --- /dev/null +++ b/ENAyT4oBgHgl3EQfSPf9/content/tmp_files/2301.00085v1.pdf.txt @@ -0,0 +1,1124 @@ +arXiv:2301.00085v1 [math.CO] 31 Dec 2022 +On the chromatic number of random regular +hypergraphs +Patrick Bennett∗ +Department of Mathematics, +Western Michigan University +Kalamazoo MI 49008 +Alan Frieze† +Department of Mathematical Sciences, +Carnegie Mellon University, +Pittsburgh PA 15213. +Abstract +We estimate the likely values of the chromatic and independence numbers of the +random r-uniform d-regular hypergraph on n vertices for fixed r, large fixed d, and +n → ∞. +1 +Introduction +The study of the chromatic number of random graphs has a long history. It begins with the +work of Bollob´as and Erd˝os [6] and Grimmett and McDiarmid [13] who determined χ(Gn,p), +p constant to within a factor 2, w.h.p. Matula [17] reduced this to a factor of 3/2. Then +we have the discovery of martingale concentration inequalities by Shamir and Spencer [18] +leading to the breakthrough by Bollob´as [5] who determined χ(Gn,p) asymptotically for p +constant. +The case of p → 0 proved a little more tricky, but �Luczak [15] using ideas from Frieze [10] +and [17] determined χ(Gn,p), p = c/n asymptotically for large c. �Luczak [16] showed that +w.h.p. χ(Gn,p), p = c/n took one of two values. It was then that the surprising power of +the second moment method was unleashed by Achlioptas and Naor [3]. Since then there has +been much work tightening our estimates for the k-colorability threshold, k ≥ 3 constant. +See for example Coja-Oghlan [7]. +Random regular graphs of low degree were studied algorithmically by several authors e.g. +Achlioptas and Molloy [2] and by Shi and Wormald [19]. Frieze and �Luczak [12] introduced +∗Research supported in part by Simons Foundation Grant #426894. +†Research supported in part by NSF Grant DMS1661063 +1 + +a way of using our knowledge of χ(Gn,p), p = c/n to tackle χ(Gn,r) where Gn,r denotes a +random r-regular graph and where p = r/n. Subsequently Achlioptas and Moore [2] showed +via the second moment method that w.h.p. χ(Gn,r) was one of 3 values. This was tightened +basically to one value by Coja-Oghlan, Efthymiou and Hetterich [8]. +For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic +number for χ(Hr(n, p) for +�n−1 +r−1 +� +p sufficiently large. Here Hr(n, p) is the binomial r-uniform +hypergraph where each of the +�n +r +� +possible edges is included with probability p. There are +several possibilities of a proper coloring of the vertices of a hypergraph. Here we concentrate +on the case where a vertex coloring is proper if no edge contains vertices of all the same color. +Dyer, Frieze and Greehill [9] and Ayre, Coja-Oghlan and Greehill [1] established showed that +w.h.p. χ(Hr(n, p) took one or two values. When it comes to what ew denote by χ(Hr(n, d), +a random d-regular, r-uniform hypergraph, we are not aware of any results at all. In this +paper we extend the approach of [12] to this case: +Theorem 1. For all fixed r and ε > 0 there exists d0 = d0(r, ε) such that for any fixed +d ≥ d0 we have that w.h.p. +������� +χ(Hr(n, d)) − +� +(r−1)d +r log d +� +1 +r−1 +� +(r−1)d +r log d +� +1 +r−1 +������� +≤ ε, +������� +α(Hr(n, d)) − +� +r log d +(r−1)d +� +1 +r−1 n +� +r log d +(r−1)d +� +1 +r−1 n +������� +≤ ε +(1) +Here α refers to the independence number of a hypergraph. +2 +Preliminaries +2.1 +Tools +We will be using the following forms of Chernoff’s bound (see, e.g., [11]). +Lemma 2 (Chernoff bound). Let X ∼ Bin(n, p). Then for all 0 < λ < np +P(|X − np| ≥ λ) ≤ 2 exp +� +− λ2 +3np +� +. +(2) +Lemma 3 (McDiarmid’s inequality). Let X = f(⃗Z) where ⃗Z = (Z1, . . . Zt) and the Zi are +independent random variables. Assume the function f has the property that whenever ⃗z, ⃗w +differ in only one coordinate we have |f(⃗z) − f(⃗w)| ≤ c. Then for all λ > 0 we have +P(|X − E[X]| ≥ λ) ≤ 2 exp +� +− λ2 +2c2t +� +. +(3) +Bal and the first author [4] showed the following. +2 + +Theorem 4 (Claim 4.2 in [4]). Fix r ≥ 3, d ≥ 2, and 0 < c < r−1 +r . Let z2 be the unique +positive number such that +z2 +� +(z2 + 1)r−1 − zr−1 +2 +� +(z2 + 1)r − zr +2 += c +(4) +and let +z1 = +d +r [(z2 + 1)r − zr +2]. +(5) +Let h(x) = x log x. If it is the case that +h +�d +r +� ++ h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d +r log z1 − dc log z2 < 0 +(6) +then w.h.p. α(Hr(n, d)) < cn. +Krivelevich and Sudakov [14] proved the following. +Theorem 5 (Theorem 5.1 in [14]). For all fixed r and ε > 0 there exists d0 = d0(r, ε) such +that whenever D = D(p) := +�n−1 +r−1 +� +p ≥ d0 we have that +������� +χ(Hr(n, p)) − +� +(r−1)D +r log D +� +1 +r−1 +� +(r−1)D +r log D +� +1 +r−1 +������� +≤ ε, +������� +α(Hr(n, p)) − +� +r log D +(r−1)D +� +1 +r−1 n +� +r log D +(r−1)D +� +1 +r−1 n +������� +≤ ε +with probability at least 1 − o(1/n). +3 +Proof +In this section we prove Theorem 1. First we give an overview. We show in Subsection 3.1 +that the upper bound on α follows from Theorem 4 and some straightforward calculations. +Then the lower bound on χ follows as well. Thus we will be done once we prove the upper +bound on χ (since that proves the lower bound on α). This will be in Subsection 3.2. For +that we follow the methods of Frieze and �Luczak [12]. +We will assume r ≥ 3 since Frieze and �Luczak [12] covered the graph case. We will use +standard asymptotic notation, and we will use big-O notation to suppress any constants +depending on r but not d. Thus, for example we will write r = O(1) and d−1 = O(1) but +not d = O(1). This is convenient for us because even though our theorem is for fixed d, it +requires d to be sufficiently large. +3 + +3.1 +Upper bound on the independence number +We will apply Theorem 4 to show an upper bound on α(Hr(n, d)). Fix ε, r (but not d) and +let c = c(d) := (1 + ε) +� +r log d +(r−1)d +� +1 +r−1. Let z2 be as defined in (4) and z1 be as defined in (5). +We see that +Lemma 6. +z2 = +c +1 − c + O (cr) +Proof. After some algebra, we re-write (4) as +z2 − +zr +2 +(1 + z2)r−1 = +c +1 − c. +and the claim follows. +Now we check (6). +h +�d +r +� ++ h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d +r log z1 − dc log z2 +=d +r log +�d +r +� ++ dc log(dc) + d(1 − c) log(d(1 − c)) − c log c − (1 − c) log(1 − c) +− d log d − d +r log z1 − dc log z2 +=dc log +� +c +(1 − c)z2 +� ++ d +r log [(z2 + 1)r − zr +2] + d log(1 − c) − c log c − (1 − c) log(1 − c). (7) +Now note that the first term of (7) is +dc log +� +c +(1 − c)z2 +� += dc log +� +c +(1 − c) +� +c +1−c + O (cr+1) +� +� += dc log +� +1 +1 + O (cr) +� += O +� +dcr+1� +. +The second term of (7) is +d +r log [(z2 + 1)r − zr +2] = d +r log +�� +1 +1 − c + O +� +cr+1��r +− +� +c +1 − c + O +� +cr+1��r� += d +r log +�� +1 +1 − c +�r � +1 − cr + O +� +cr+1��� += d +r log +� +1 +1 − c +�r ++ d +r log +� +1 − cr + O +� +cr+1�� += −d log(1 − c) − d +r cr + O +� +dcr+1� +. +4 + +The last term of (7) is +(1 − c) log(1 − c) = O(c). +Therefore (7) becomes +− d +r cr − c log c + O +� +c + dcr+1� += − c +�d +rcr−1 + log c +� ++ O +� +c + dcr+1� += − c +� +d +r (1 + ε)r−1 r log d +(r − 1)d + log +� +(1 + ε) +� r log d +(r − 1)d +� +1 +r−1�� ++ O +� +c + dcr+1� += − c +� +(1 + ε)r−1 log d +r − 1 − log d +r − 1 + O(log log d) +� ++ O +� +c + dcr+1� += − Ω (c log d) . +It follows from Theorem 4 that w.h.p. +α(Hr(n, d)) ≤ (1 + ε) +� r log d +(r − 1)d +� +1 +r−1 +. +(8) +3.2 +Upper bound on the chromatic number +Our proof of the upper bound uses the method of Frieze and �Luczak [12]. We will generate +Hr(n, d) in a somewhat complicated way. The way we generate it will allow us to use known +results on Hr(n, p) due to Krivelevich and Sudakov [14]. +Set +m := +�d − d1/2 log d +r +� +n. +(9) +Let H∗ +r(n, m) be an r-uniform multi-hypergraph with m edges, where each multi-edge consists +of r independent uniformly random vertices chosen with replacement. +We will generate +H∗ +r(n, m) as follows. We have n sets (“buckets” ) V1, . . . Vn and a set of rm points P := +{p1, . . . prm}. We put each point pi into a uniform random bucket Vφ(i) independently. We +let R = {R1, . . . , Rm} be a uniform random partition of P into sets of size r. Of course, the +idea here is that the buckets Vi represent vertices and the parts of the partition R represent +edges. Thus Ri defines a hyper-edge {φ(j) : j ∈ Ri} for i = 1, 2, . . . , m. We denote the +hypergraph defined by R by HR. +Note that since r ≥ 3 the expected number of pairs of multi-edges in H∗ +r(n, m) is at most +�n +r +��m +2 +� � +1 +�n +r +� +�2 += O +�m2 +nr +� += O(n−1). +5 + +Thus, w.h.p. there are no multi-edges. Now the expected number of “loops” (edges containing +the same vertex twice) is at most +nm +�r +2 +� �1 +n +�2 += O(1). +Thus w.h.p. there are at most log n loops. We now remove all multi-edges and loops, and +say that M is the (random) number of edges remaining, where m − log n ≤ M ≤ m. The +remaining hypergraph is distributed as H(n, M), the random hypergraph with M edges +chosen uniformly at random without replacement. Next we estimate the chromatic number +of Hr(n, M). +Claim 1. W.h.p. we have +������� +χ(Hr(n, M)) − +� +(r−1)d +r log d +� +1 +r−1 +� +(r−1)d +r log d +� +1 +r−1 +������� +≤ ε +2, +������� +α(Hr(n, M)) − +� +r log D +(r−1)D +� +1 +r−1 n +� +r log D +(r−1)D +� +1 +r−1 n +������� +≤ ε +2. +Proof. We will use Theorem 5 together with a standard argument for comparing Hr(n, p) +with Hr(n, m). Set p := m/ +�n +r +� +and apply Theorem 5 with ε replaced with ε/4 so we get +������� +χ(Hr(n, p)) − +� +(r−1)D +r log D +� +1 +r−1 +� +(r−1)D +r log D +� +1 +r−1 +������� +≤ ε +4 +(10) +with probability at least 1 − o(1/n). Note that here +D = +�n − 1 +r − 1 +� +p = +�n − 1 +r − 1 +� +m/ +�n +r +� += rm/n = d − d1/2 log d. +Now since d, D can be chosen to be arbitrarily large and d = D + O(D1/2 log D) we can +replace D with d in (10) without changing the left hand side by more than ε/4 to obtain +������� +χ(Hr(n, p)) − +� +(r−1)d +r log d +� +1 +r−1 +� +(r−1)d +r log d +� +1 +r−1 +������� +≤ ε +2 +(11) +with probability at least 1−o(1/n). But now note that with probability Ω(n−1/2) the number +of edges in Hr(n, p) is precisely M. Thus we have that +������� +χ(Hr(n, M)) − +� +(r−1)d +r log d +� +1 +r−1 +� +(r−1)d +r log d +� +1 +r−1 +������� +≤ ε +2 +with probability at least 1 − o(n−1/2). This proves the first inequality, and the second one +follows similarly. +6 + +Now we will start to transform Hr(n, m) to the random regular hypergraph Hr(n, d). This +transformation will involve first removing some edges from vertices of degree larger than d, +and then adding some edges to vertices of degree less than d. We define the rank of a point +pi ∈ Vj, to be the number of points pi′ ∈ Vj such that i′ ≤ i. We form a new set of points +P ′ ⊆ P and a partition R′ of P ′ as follows. For any Rk ∈ R containing a point with rank +more than d, we delete Rk from R and delete all points of Rk from P. Note that each bucket +contains at most r points of P ′. Note also that R′ is a uniform random partition of P ′. We +let HR′ be the natural hypergraph associated with R′. +Now we would like to put some more points into the buckets until each bucket has exactly d +points, arriving at some set of points P ′′ ⊇ P ′. We would also like a uniform partition R′′ of +P ′′ into sets of size r, and we would like R′′ to have many of the same parts as R′. We will +accomplish this by constructing a sequence P ′ +1 := P ′ ⊆ P ′ +2 ⊆ . . . ⊆ P ′ +ℓ =: P ′′ of point sets +and a sequence R′ +1 := R′, R′ +2, . . . , R′ +ℓ =: R′′ where R′ +j is a uniform random partition of P ′ +j. +We construct P ′ +j+1, R′ +j+1 from P ′ +j, R′ +j as follows. Suppose |R′ +j| = a (in other words R′ +j has +a parts), so |P ′ +j| = ra. P ′ +j+1 will simply be P ′ +j plus r new points. Now we will choose a +random value K ∈ {1, . . . , r} using the distribution P[K = k] = qk(a), where qk(a) is defined +as follows. +Definition 1. Consider a random partition of ra+r points into a+1 parts of size r, and fix +some set Q of r points. Then for 1 ≤ k ≤ r, the number qk(a) is defined to be the probability +that Q meets exactly k parts of the partition. +We will then remove a uniform random set of K − 1 parts from R′ +j, leaving Kr points in +P ′ +j+1 which are not in any remaining part of R′ +j. We partition those points into K parts of +size r such that each part contains at least one new point (each such partition being equally +likely), arriving at our partition R′ +j+1. +We claim that R′ +j+1 is a uniform random partition of P ′ +j+1 into parts of size r. Indeed, first +consider the r new points that are in P ′ +j+1 which were not in P ′ +j. The probability that a +uniform random partition of P ′ +j+1 would have exactly k parts containing at least one new +point is qk. So we can generate such a random partition as follows: first choose a random +value K with P[K = k] = qk; next we choose a uniform random set of (K − 1)r points from +P ′ +j; next we choose a partition of the set of points consisting of P ′ +j+1 \ P ′ +j together with the +points from P ′ +j we chose in the last step, where the partition we choose is uniformly random +from among all partitions such that each part contains at least one point of P ′ +j+1 \P ′ +j; finally, +we choose a uniform partition of the rest of the points. In our case this partition of the rest +of the points comprises the current partition of the “unused” (a − K + 1)r points. At the +end of this process we have that HR′′ is distributed as Hr(n, d). +7 + +3.2.1 +Bounding the number of low degree vertices in HR′ +We define some sets of buckets. We show that w.h.p. there are few small buckets i.e few +vertices of low degree in the hypregraph HR′. Let S0 be the buckets with at most d−3d1/2 log d +points of P ′, and let S1 be the buckets with at most d−2d1/2 log d points of P. Let S2 be the +set of buckets that, when we remove points from P ′ to get P, have at least d1/2 log d points +removed. Then S0 ⊆ S1 ∪ S2. Our goal is to bound the probability that S0 is too large. +Fix a bucket Vj and let X ∼ Bin +� +rm, 1 +n +� +be the number of points of P in Vj. Then the +probability that Vj is in S1 satisfies +P[Vj ∈ S1] = P +� +X ≤ d − 2d1/2 log d +� += P +� +X − rm +n ≤ −d1/2 log d +� +≤ exp +� +− +d log2 d +3(d − d1/2 log d) +� += exp +� +−Ω +� +log2 d +�� +, +where for our inequality we have used the Chernoff bound (Lemma 2). Therefore E[|S0|] ≤ +exp +� +−Ω +� +log2 d +�� +n. Now we argue that |S1| is concentrated using McDiarmid’s inequality +(Lemma 3). For our application we let X = |S1| which is a function (say f) of the vector +(Z1, . . . Zrm) where Zi tells us which bucket the ith point of P went into. Moving a point +from one bucket to another can only change |S1| by at most 1 so we use c = 1. Thus we get +the bound +P(|X − E[X]| ≥ n2/3) ≤ 2 exp +� +− n4/3 +2rm +� += o(1). +(12) +Now we handle S2. For 1 ≤ j ≤ n let Yj be the number of parts Rk ∈ R such that Rk +contains a point in the bucket Vj as well as a point in some bucket Vj′ where |Vj′| > d. Note +that if Vj ∈ S2 then Yj ≥ d1/2 log d. We view Rk as a set of r points, say {q1, . . . , qr} each +going into a uniform random bucket. Say qi goes to bucket Vji. The probability that Rk is +counted by Yj is at most +rP[j1 = j and |Vj1| > d] + r(r − 1)P[j1 = j and |Vj2| > d] += r +nP[|Vj1| > d +��j1 = j] + r(r − 1) +n +P[|Vj2| > d +��j1 = j] +≤ r2 +n P[|Vj1| > d +��j1 = j] +≤ r2 +n P[Bin(rm − 1, 1/n) ≥ d] = r2 +n exp +� +−Ω +� +log2 d +�� +. +Thus we have +E[Yj] = m · r2 +n exp +� +−Ω +� +log2 d +�� +≤ rd exp +� +−Ω +� +log2 d +�� += rd1/2 exp +� +−Ω +� +log2 d +�� +and so Markov’s inequality gives us +P +� +Yj ≥ d1/2 log d +� +≤ rd exp +� +−Ω +� +log2 d +�� +d1/2 log d += exp +� +−Ω +� +log2 d +�� +8 + +and so E[|S2|] = n exp +� +−Ω +� +log2 d +�� +. We use McDiarmid’s inequality once more, this time +with X = |S2|. +A change in choice of bucket changes |S2| by at most one and so (12) +continues to hold. Thus +|S0| = n exp +� +−Ω +� +log2 d +�� +. +w.h.p. +3.2.2 +A property of independent subsets of Hr(n, m) +Fix 1 ≤ j ≤ r − 1. Set +a := +� +1 + ε +2 +� � r log d +(r − 1)d +� +1 +r−1 +, +κj := 10d +r +�r +j +� +aj, +p := d(r − 1)! +nr−1 +. +The expected number of independent sets A in Hr(n, p) of size at most an such that there +are κjn edges each having j vertices in A is at most +an +� +s=1 +�n +s +� +(1 − p)(s +r) +��s +j +�� n +r−j +� +κjn +� +pκjn +≤ +an +� +s=1 +exp + + +s log +�en +s +� +− +�s +r +� +p + κjn log + +e(an)j +j! +nr−j +(r−j)!p +κjn + + + + + += +an +� +s=1 +exp +� +s log +�en +s +� +− +�s +r +� +p + κjn log +�eaj +10 +�� +≤ an · exp +�� +log +�e +a +� +− 10d +r +�r +j +� +aj−1 log +�10 +e +�� +an +� += o(1/n) +where the last line follows since as d → ∞ we have +log +�e +a +� +∼ +1 +r − 1 log d +and +10d +r +�r +j +� +aj−1 log +�10 +e +� += Ω +� +d +r−j +j−1 log− j−1 +r−1 d +� +≫ log d. +Thus with probability 1 − o(1/n), Hr(n, p) has a coloring using (1 + ε/2) +� +(r−1)d +r log d +� +1 +r−1 colors +such that for each color class A and for each 1 ≤ j ≤ r − 1 there are at most κjn edges with +j vertices in A. The hypergraph Hr(n, m), m = +�n +r +� +p will have this property w.h.p.. +3.2.3 +Transforming HR′ into Hr(n, d) +Now we will complete the transformation to the random regular hypergraph Hr(n, d). We +are open to the possibility that doing so will render our coloring no longer proper, since this +9 + +process will involve changing some edges which might then be contained in a color class. We +will keep track of how many such “bad” edges there are and then repair our coloring at the +end. +We have to add at most (3d1/2 log d + d exp +� +−Ω +� +log2 d +�� +)n < (4d1/2 log d)n points, which +takes at most as many steps. +For each color class A of HR′ define XA,j = XA,j(i) to +be the number of edges with j vertices in A at step i. We have already established that +XA,j(0) ≤ κjn. This follows from Section 3.2.2 and the fact that we have removed edges +from H(n, m) to obtain HR′. Let Ei be the event that at step i we have that for each color +class A and for each 1 ≤ j ≤ r − 1 we have XA,j(i) ≤ 2κjn. Then, assuming Ei holds, the +probability that XA,j increases at step i is at most +� +1≤k≤r, jℓ≥1 +j1+···+jk=j +� +1≤ℓ≤k +2κjℓn +nd/r = +� +1≤k≤r, jℓ≥1 +j1+···+jk=j +� +1≤ℓ≤k +20 +� r +jk +� +ajk ≤ +� +1≤k≤r, jℓ≥1 +j1+···+jk=j +20r2r2aj ≤ 40r2r2aj. +Also, the largest possible increase in XA,j in one step is r. Thus, the final value of XA,j +after at most (4d1/2 log d)n steps is stochastically dominated by κjn + rY where Y +∼ +Bin +� +(4d1/2 log d)n, 40r2r2aj� +. An easy application of the Chernoff bound tells us +P (Y > 2E[Y ]) ≤ exp(−Ω(n)). +(13) +Note that here +2E[Y ] +κjn += 8d1/2 log d · 40r2r2ajn +10d +�r +j +� +ajn/r += O(d−1/2 log d) < 1 +for sufficiently large d. Thus, using (13) and the union bound over all color classes A, we +have w.h.p. the final value of XA,j is at most κjn + 2E[Y ] ≤ 2κjn for all 1 ≤ j ≤ r − 1. +Now we address “bad” edges, i.e. edges contained in a color class. Assuming Ei holds, the ex- +pected number of new edges contained in any color class at step i is at most r(40)r2r2+2rar = +O +�� log d +d +� +r +r−1� +(because it would have to be one of the colors of one of the vertices we are +adding points to). Thus the expected number of bad edges created in (4d1/2 log d)n steps +is stochastically dominated by Z ∼ r · Bin +� +(4d1/2 log d)n, O +��log d +d +� +r +r−1� � +. +Another easy +application of Chernoff shows that w.h.p. Z ≤ 2E[Z] = O(d−1/2n). +We repair the coloring as follows. First we uncolor one vertex from each bad edge, and let +the set of uncolored vertices be U where |U| = u = O +� +d−1/2n +� +. Let +δ := ε +2 +�(r − 1)d +r log d +� +1 +r−1 +. +We claim that for every S ⊆ U, |S| = s, the hypergraph induced on S has at most δs/r +edges. This will complete our proof since it implies that the minimum degree is at most δ +and so U can be recolored using a fresh set of δ colors, yielding a coloring of Hr(n, d) using +10 + +at most +χ(Hr(n, M)) + δ ≤ +� +1 + ε +2 +� �(r − 1)d +r log d +� +1 +r−1 ++ ε +2 +�(r − 1)d +r log d +� +1 +r−1 += (1 + ε) +�(r − 1)d +r log d +� +1 +r−1 +colors. The expected number of sets S with more than δs/r edges is at most +� +1≤s≤u +�n +s +���ds +r +� +δs/r +� +1 +�dn +r +��dn−r +r +� +. . . +�dn−δs+r +r +� +≤ +� +1≤s≤u +�ne +s +�s �(dse/r)re +δs/r +�δs/r +(r!)δs/r +(dn − δs)δs +≤ +� +1≤s≤u +� +ne +s +� +dse +(dn − δs)r +�δ �er · r! +δs +�δ/r�s +. +(14) +Now for 1 ≤ s ≤ √n the term in (14) is at most +� +O(n) · +� +O(n−1/2) +�δ · O(1) +�s += o(1/n) +since δ can be made arbitrarily large by choosing d large. Meanwhile for √n ≤ s ≤ u we +have that the term in (14) is at most +� +O(n1/2) · O(1) · +� +O(n−1/2) +�δ/r�s += o(1/n). +Now since (14) has O(n) terms the whole sum is o(1) and we are done. This completes the +proof of Theorem 1. +4 +Summary +We have asymptotically computed the chromatic number of random r-uniform, d-regular +hypergraphs when proper colorings mean that no edge is mono-chromatic. It would seem +likely that the approach we took would extend to other definitions of proper coloring. We +have not attempted to use second moment calculations to further narrow our estimates. +These would seem to be two natural lines of further research. +References +[1] P. Ayre, A. Coja-Oghlan and C. Greenhill, Hypergraph coloring up to condensation, +Random Structures and Algorithms 54 (2019) 615 - 652. +11 + +[2] D. Achlioptas and C. Moore, The Chromatic Number of Random Regular Graphs, +In Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds) Approximation, Random- +ization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM AP- +PROX 2004 2004. Lecture Notes in Computer Science, vol 3122. Springer, Berlin, Hei- +delberg. Approximation, Randomization, and Combinatorial Optimization. Algorithms +and Techniques (2004) 219–228. +[3] D. Achlioptas and A. Naor, The two possible values of the chromatic number of a +random graph, Annals of Mathematics 162 (2005) 1335-1351. +[4] D. Bal and P. Bennett, The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs. +[5] B. Bollob´as, The chromatic number of random graphs, Combinatorica 8 (1988) 49-55. +[6] B. Bollob´as and P. Erd˝os, Cliques in random graphs, Mathematical Proceedings of the +Cambridge Philosophical Society 80 (1976) 419-427. +[7] A. Coja-Oghlan, Upper-Bounding the k-Colorability Threshold by Counting Covers, +Electronic Journal of Combinatorics 20 (2013). +[8] A. Coja-Oghlan, C. Efthymiou and S. Hetterich, On the chromatic number of random +regular graphs, Journal of Combinatorial Theory B 116 (2016) 367-439. +[9] M. Dyer, A.M. Frieze and C. Greenhill, On the chromatic number of a random hyper- +graph, Journal of Combinatorial Theorey B 113 (2015) 68-122. +[10] A.M. Frieze, On the independence number of random graphs, Discrete Mathematics 81 +(1990) 171-176. +[11] A.M. Frieze and M. Karo´nski, Introduction to Random Graphs, Cambridge University +Press, 2015. +[12] A.M. Frieze and T. �Luczak, On the independence and chromatic numbers of random +regular graphs, Journal of Combinatorial Theory. Series B 54 (1992) 123-132. +[13] G. Grimmett and C. McDiarmid, On colouring random graphs, Mathematical Proceed- +ings of the Cambridge Philosophical Society 77 (1975) 313-324. +[14] M. Krivelevich and B. Sudakov, The chromatic numbers of random hypergraphs, Ran- +dom Structures Algorithms 12 (1998) 381-403. +[15] T. �Luczak, The chromatic number of random graphs, Combinatorica 11 (19990) 45-54. +[16] T. �Luczak, A note on the sharp concentration of the chromatic number of random +graphs, Combinatorica 11 (1991) 295-297. +[17] D. Matula, Expose-and-Merge Exploration and the Chromatic Number of a Random +Graph, Combinatorica 7 (1987) 275-284. +12 + +[18] E. Shamir and J. Spencer, Sharp concentration of the chromatic number od random +graphs Gn,p, Combinatorica 7 (1987) 121-129. +[19] L. Shi and N. Wormald, Coloring random regular graphs, Combinatorics, Probability +and Computing 16 (2007) 459-494. +13 + diff --git a/ENAyT4oBgHgl3EQfSPf9/content/tmp_files/load_file.txt b/ENAyT4oBgHgl3EQfSPf9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7620f583cb41504074a35dec9a85fdf1c69b7a87 --- /dev/null +++ b/ENAyT4oBgHgl3EQfSPf9/content/tmp_files/load_file.txt @@ -0,0 +1,362 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf,len=361 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='00085v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='CO] 31 Dec 2022 On the chromatic number of random regular hypergraphs Patrick Bennett∗ Department of Mathematics, Western Michigan University Kalamazoo MI 49008 Alan Frieze† Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Abstract We estimate the likely values of the chromatic and independence numbers of the random r-uniform d-regular hypergraph on n vertices for fixed r, large fixed d, and n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 1 Introduction The study of the chromatic number of random graphs has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' It begins with the work of Bollob´as and Erd˝os [6] and Grimmett and McDiarmid [13] who determined χ(Gn,p), p constant to within a factor 2, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Matula [17] reduced this to a factor of 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then we have the discovery of martingale concentration inequalities by Shamir and Spencer [18] leading to the breakthrough by Bollob´as [5] who determined χ(Gn,p) asymptotically for p constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The case of p → 0 proved a little more tricky, but �Luczak [15] using ideas from Frieze [10] and [17] determined χ(Gn,p), p = c/n asymptotically for large c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' �Luczak [16] showed that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' χ(Gn,p), p = c/n took one of two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' It was then that the surprising power of the second moment method was unleashed by Achlioptas and Naor [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Since then there has been much work tightening our estimates for the k-colorability threshold, k ≥ 3 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' See for example Coja-Oghlan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Random regular graphs of low degree were studied algorithmically by several authors e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Achlioptas and Molloy [2] and by Shi and Wormald [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Frieze and �Luczak [12] introduced ∗Research supported in part by Simons Foundation Grant #426894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' †Research supported in part by NSF Grant DMS1661063 1 a way of using our knowledge of χ(Gn,p), p = c/n to tackle χ(Gn,r) where Gn,r denotes a random r-regular graph and where p = r/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Subsequently Achlioptas and Moore [2] showed via the second moment method that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' χ(Gn,r) was one of 3 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This was tightened basically to one value by Coja-Oghlan, Efthymiou and Hetterich [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic number for χ(Hr(n, p) for �n−1 r−1 � p sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Here Hr(n, p) is the binomial r-uniform hypergraph where each of the �n r � possible edges is included with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' There are several possibilities of a proper coloring of the vertices of a hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Here we concentrate on the case where a vertex coloring is proper if no edge contains vertices of all the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Dyer, Frieze and Greehill [9] and Ayre, Coja-Oghlan and Greehill [1] established showed that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' χ(Hr(n, p) took one or two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' When it comes to what ew denote by χ(Hr(n, d), a random d-regular, r-uniform hypergraph, we are not aware of any results at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' In this paper we extend the approach of [12] to this case: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For all fixed r and ε > 0 there exists d0 = d0(r, ε) such that for any fixed d ≥ d0 we have that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' ������� χ(Hr(n, d)) − � (r−1)d r log d � 1 r−1 � (r−1)d r log d � 1 r−1 ������� ≤ ε, ������� α(Hr(n, d)) − � r log d (r−1)d � 1 r−1 n � r log d (r−1)d � 1 r−1 n ������� ≤ ε (1) Here α refers to the independence number of a hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='1 Tools We will be using the following forms of Chernoff’s bound (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=', [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Lemma 2 (Chernoff bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let X ∼ Bin(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then for all 0 < λ < np P(|X − np| ≥ λ) ≤ 2 exp � − λ2 3np � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (2) Lemma 3 (McDiarmid’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let X = f(⃗Z) where ⃗Z = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Zt) and the Zi are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Assume the function f has the property that whenever ⃗z, ⃗w differ in only one coordinate we have |f(⃗z) − f(⃗w)| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then for all λ > 0 we have P(|X − E[X]| ≥ λ) ≤ 2 exp � − λ2 2c2t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (3) Bal and the first author [4] showed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 2 Theorem 4 (Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Fix r ≥ 3, d ≥ 2, and 0 < c < r−1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let z2 be the unique positive number such that z2 � (z2 + 1)r−1 − zr−1 2 � (z2 + 1)r − zr 2 = c (4) and let z1 = d r [(z2 + 1)r − zr 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (5) Let h(x) = x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' If it is the case that h �d r � + h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d r log z1 − dc log z2 < 0 (6) then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' α(Hr(n, d)) < cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Krivelevich and Sudakov [14] proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Theorem 5 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='1 in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For all fixed r and ε > 0 there exists d0 = d0(r, ε) such that whenever D = D(p) := �n−1 r−1 � p ≥ d0 we have that ������� χ(Hr(n, p)) − � (r−1)D r log D � 1 r−1 � (r−1)D r log D � 1 r−1 ������� ≤ ε, ������� α(Hr(n, p)) − � r log D (r−1)D � 1 r−1 n � r log D (r−1)D � 1 r−1 n ������� ≤ ε with probability at least 1 − o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 3 Proof In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' First we give an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We show in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='1 that the upper bound on α follows from Theorem 4 and some straightforward calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then the lower bound on χ follows as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus we will be done once we prove the upper bound on χ (since that proves the lower bound on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This will be in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For that we follow the methods of Frieze and �Luczak [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will assume r ≥ 3 since Frieze and �Luczak [12] covered the graph case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will use standard asymptotic notation, and we will use big-O notation to suppress any constants depending on r but not d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus, for example we will write r = O(1) and d−1 = O(1) but not d = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This is convenient for us because even though our theorem is for fixed d, it requires d to be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='1 Upper bound on the independence number We will apply Theorem 4 to show an upper bound on α(Hr(n, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Fix ε, r (but not d) and let c = c(d) := (1 + ε) � r log d (r−1)d � 1 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let z2 be as defined in (4) and z1 be as defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We see that Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' z2 = c 1 − c + O (cr) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' After some algebra, we re-write (4) as z2 − zr 2 (1 + z2)r−1 = c 1 − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now we check (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' h �d r � + h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d r log z1 − dc log z2 =d r log �d r � + dc log(dc) + d(1 − c) log(d(1 − c)) − c log c − (1 − c) log(1 − c) − d log d − d r log z1 − dc log z2 =dc log � c (1 − c)z2 � + d r log [(z2 + 1)r − zr 2] + d log(1 − c) − c log c − (1 − c) log(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (7) Now note that the first term of (7) is dc log � c (1 − c)z2 � = dc log � c (1 − c) � c 1−c + O (cr+1) � � = dc log � 1 1 + O (cr) � = O � dcr+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The second term of (7) is d r log [(z2 + 1)r − zr 2] = d r log �� 1 1 − c + O � cr+1��r − � c 1 − c + O � cr+1��r� = d r log �� 1 1 − c �r � 1 − cr + O � cr+1��� = d r log � 1 1 − c �r + d r log � 1 − cr + O � cr+1�� = −d log(1 − c) − d r cr + O � dcr+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 4 The last term of (7) is (1 − c) log(1 − c) = O(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Therefore (7) becomes − d r cr − c log c + O � c + dcr+1� = − c �d rcr−1 + log c � + O � c + dcr+1� = − c � d r (1 + ε)r−1 r log d (r − 1)d + log � (1 + ε) � r log d (r − 1)d � 1 r−1�� + O � c + dcr+1� = − c � (1 + ε)r−1 log d r − 1 − log d r − 1 + O(log log d) � + O � c + dcr+1� = − Ω (c log d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' It follows from Theorem 4 that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' α(Hr(n, d)) ≤ (1 + ε) � r log d (r − 1)d � 1 r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2 Upper bound on the chromatic number Our proof of the upper bound uses the method of Frieze and �Luczak [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will generate Hr(n, d) in a somewhat complicated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The way we generate it will allow us to use known results on Hr(n, p) due to Krivelevich and Sudakov [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Set m := �d − d1/2 log d r � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (9) Let H∗ r(n, m) be an r-uniform multi-hypergraph with m edges, where each multi-edge consists of r independent uniformly random vertices chosen with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will generate H∗ r(n, m) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We have n sets (“buckets” ) V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Vn and a set of rm points P := {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' prm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We put each point pi into a uniform random bucket Vφ(i) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We let R = {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' , Rm} be a uniform random partition of P into sets of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Of course, the idea here is that the buckets Vi represent vertices and the parts of the partition R represent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus Ri defines a hyper-edge {φ(j) : j ∈ Ri} for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We denote the hypergraph defined by R by HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Note that since r ≥ 3 the expected number of pairs of multi-edges in H∗ r(n, m) is at most �n r ��m 2 � � 1 �n r � �2 = O �m2 nr � = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 5 Thus, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' there are no multi-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now the expected number of “loops” (edges containing the same vertex twice) is at most nm �r 2 � �1 n �2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' there are at most log n loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We now remove all multi-edges and loops, and say that M is the (random) number of edges remaining, where m − log n ≤ M ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The remaining hypergraph is distributed as H(n, M), the random hypergraph with M edges chosen uniformly at random without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Next we estimate the chromatic number of Hr(n, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' we have ������� χ(Hr(n, M)) − � (r−1)d r log d � 1 r−1 � (r−1)d r log d � 1 r−1 ������� ≤ ε 2, ������� α(Hr(n, M)) − � r log D (r−1)D � 1 r−1 n � r log D (r−1)D � 1 r−1 n ������� ≤ ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will use Theorem 5 together with a standard argument for comparing Hr(n, p) with Hr(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Set p := m/ �n r � and apply Theorem 5 with ε replaced with ε/4 so we get ������� χ(Hr(n, p)) − � (r−1)D r log D � 1 r−1 � (r−1)D r log D � 1 r−1 ������� ≤ ε 4 (10) with probability at least 1 − o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Note that here D = �n − 1 r − 1 � p = �n − 1 r − 1 � m/ �n r � = rm/n = d − d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now since d, D can be chosen to be arbitrarily large and d = D + O(D1/2 log D) we can replace D with d in (10) without changing the left hand side by more than ε/4 to obtain ������� χ(Hr(n, p)) − � (r−1)d r log d � 1 r−1 � (r−1)d r log d � 1 r−1 ������� ≤ ε 2 (11) with probability at least 1−o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' But now note that with probability Ω(n−1/2) the number of edges in Hr(n, p) is precisely M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus we have that ������� χ(Hr(n, M)) − � (r−1)d r log d � 1 r−1 � (r−1)d r log d � 1 r−1 ������� ≤ ε 2 with probability at least 1 − o(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This proves the first inequality, and the second one follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 6 Now we will start to transform Hr(n, m) to the random regular hypergraph Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This transformation will involve first removing some edges from vertices of degree larger than d, and then adding some edges to vertices of degree less than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We define the rank of a point pi ∈ Vj, to be the number of points pi′ ∈ Vj such that i′ ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We form a new set of points P ′ ⊆ P and a partition R′ of P ′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For any Rk ∈ R containing a point with rank more than d, we delete Rk from R and delete all points of Rk from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Note that each bucket contains at most r points of P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Note also that R′ is a uniform random partition of P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We let HR′ be the natural hypergraph associated with R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now we would like to put some more points into the buckets until each bucket has exactly d points, arriving at some set of points P ′′ ⊇ P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We would also like a uniform partition R′′ of P ′′ into sets of size r, and we would like R′′ to have many of the same parts as R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will accomplish this by constructing a sequence P ′ 1 := P ′ ⊆ P ′ 2 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' ⊆ P ′ ℓ =: P ′′ of point sets and a sequence R′ 1 := R′, R′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' , R′ ℓ =: R′′ where R′ j is a uniform random partition of P ′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We construct P ′ j+1, R′ j+1 from P ′ j, R′ j as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Suppose |R′ j| = a (in other words R′ j has a parts), so |P ′ j| = ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' P ′ j+1 will simply be P ′ j plus r new points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now we will choose a random value K ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' , r} using the distribution P[K = k] = qk(a), where qk(a) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Consider a random partition of ra+r points into a+1 parts of size r, and fix some set Q of r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then for 1 ≤ k ≤ r, the number qk(a) is defined to be the probability that Q meets exactly k parts of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will then remove a uniform random set of K − 1 parts from R′ j, leaving Kr points in P ′ j+1 which are not in any remaining part of R′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We partition those points into K parts of size r such that each part contains at least one new point (each such partition being equally likely), arriving at our partition R′ j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We claim that R′ j+1 is a uniform random partition of P ′ j+1 into parts of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Indeed, first consider the r new points that are in P ′ j+1 which were not in P ′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The probability that a uniform random partition of P ′ j+1 would have exactly k parts containing at least one new point is qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' So we can generate such a random partition as follows: first choose a random value K with P[K = k] = qk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' next we choose a uniform random set of (K − 1)r points from P ′ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' next we choose a partition of the set of points consisting of P ′ j+1 \\ P ′ j together with the points from P ′ j we chose in the last step, where the partition we choose is uniformly random from among all partitions such that each part contains at least one point of P ′ j+1 \\P ′ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' finally, we choose a uniform partition of the rest of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' In our case this partition of the rest of the points comprises the current partition of the “unused” (a − K + 1)r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' At the end of this process we have that HR′′ is distributed as Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='1 Bounding the number of low degree vertices in HR′ We define some sets of buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We show that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' there are few small buckets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='e few vertices of low degree in the hypregraph HR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let S0 be the buckets with at most d−3d1/2 log d points of P ′, and let S1 be the buckets with at most d−2d1/2 log d points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let S2 be the set of buckets that, when we remove points from P ′ to get P, have at least d1/2 log d points removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then S0 ⊆ S1 ∪ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Our goal is to bound the probability that S0 is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Fix a bucket Vj and let X ∼ Bin � rm, 1 n � be the number of points of P in Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then the probability that Vj is in S1 satisfies P[Vj ∈ S1] = P � X ≤ d − 2d1/2 log d � = P � X − rm n ≤ −d1/2 log d � ≤ exp � − d log2 d 3(d − d1/2 log d) � = exp � −Ω � log2 d �� , where for our inequality we have used the Chernoff bound (Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Therefore E[|S0|] ≤ exp � −Ω � log2 d �� n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now we argue that |S1| is concentrated using McDiarmid’s inequality (Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For our application we let X = |S1| which is a function (say f) of the vector (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Zrm) where Zi tells us which bucket the ith point of P went into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Moving a point from one bucket to another can only change |S1| by at most 1 so we use c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus we get the bound P(|X − E[X]| ≥ n2/3) ≤ 2 exp � − n4/3 2rm � = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (12) Now we handle S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For 1 ≤ j ≤ n let Yj be the number of parts Rk ∈ R such that Rk contains a point in the bucket Vj as well as a point in some bucket Vj′ where |Vj′| > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Note that if Vj ∈ S2 then Yj ≥ d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We view Rk as a set of r points, say {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' , qr} each going into a uniform random bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Say qi goes to bucket Vji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The probability that Rk is counted by Yj is at most rP[j1 = j and |Vj1| > d] + r(r − 1)P[j1 = j and |Vj2| > d] = r nP[|Vj1| > d ��j1 = j] + r(r − 1) n P[|Vj2| > d ��j1 = j] ≤ r2 n P[|Vj1| > d ��j1 = j] ≤ r2 n P[Bin(rm − 1, 1/n) ≥ d] = r2 n exp � −Ω � log2 d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus we have E[Yj] = m · r2 n exp � −Ω � log2 d �� ≤ rd exp � −Ω � log2 d �� = rd1/2 exp � −Ω � log2 d �� and so Markov’s inequality gives us P � Yj ≥ d1/2 log d � ≤ rd exp � −Ω � log2 d �� d1/2 log d = exp � −Ω � log2 d �� 8 and so E[|S2|] = n exp � −Ω � log2 d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We use McDiarmid’s inequality once more, this time with X = |S2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' A change in choice of bucket changes |S2| by at most one and so (12) continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus |S0| = n exp � −Ω � log2 d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2 A property of independent subsets of Hr(n, m) Fix 1 ≤ j ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Set a := � 1 + ε 2 � � r log d (r − 1)d � 1 r−1 , κj := 10d r �r j � aj, p := d(r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' nr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The expected number of independent sets A in Hr(n, p) of size at most an such that there are κjn edges each having j vertices in A is at most an � s=1 �n s � (1 − p)(s r) ��s j �� n r−j � κjn � pκjn ≤ an � s=1 exp \uf8f1 \uf8f2 \uf8f3s log �en s � − �s r � p + κjn log \uf8eb \uf8ede(an)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' nr−j (r−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p κjn \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe = an � s=1 exp � s log �en s � − �s r � p + κjn log �eaj 10 �� ≤ an · exp �� log �e a � − 10d r �r j � aj−1 log �10 e �� an � = o(1/n) where the last line follows since as d → ∞ we have log �e a � ∼ 1 r − 1 log d and 10d r �r j � aj−1 log �10 e � = Ω � d r−j j−1 log− j−1 r−1 d � ≫ log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus with probability 1 − o(1/n), Hr(n, p) has a coloring using (1 + ε/2) � (r−1)d r log d � 1 r−1 colors such that for each color class A and for each 1 ≤ j ≤ r − 1 there are at most κjn edges with j vertices in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The hypergraph Hr(n, m), m = �n r � p will have this property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='. 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='3 Transforming HR′ into Hr(n, d) Now we will complete the transformation to the random regular hypergraph Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We are open to the possibility that doing so will render our coloring no longer proper, since this 9 process will involve changing some edges which might then be contained in a color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We will keep track of how many such “bad” edges there are and then repair our coloring at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We have to add at most (3d1/2 log d + d exp � −Ω � log2 d �� )n < (4d1/2 log d)n points, which takes at most as many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' For each color class A of HR′ define XA,j = XA,j(i) to be the number of edges with j vertices in A at step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We have already established that XA,j(0) ≤ κjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This follows from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='2 and the fact that we have removed edges from H(n, m) to obtain HR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let Ei be the event that at step i we have that for each color class A and for each 1 ≤ j ≤ r − 1 we have XA,j(i) ≤ 2κjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Then, assuming Ei holds, the probability that XA,j increases at step i is at most � 1≤k≤r, jℓ≥1 j1+···+jk=j � 1≤ℓ≤k 2κjℓn nd/r = � 1≤k≤r, jℓ≥1 j1+···+jk=j � 1≤ℓ≤k 20 � r jk � ajk ≤ � 1≤k≤r, jℓ≥1 j1+···+jk=j 20r2r2aj ≤ 40r2r2aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Also, the largest possible increase in XA,j in one step is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus, the final value of XA,j after at most (4d1/2 log d)n steps is stochastically dominated by κjn + rY where Y ∼ Bin � (4d1/2 log d)n, 40r2r2aj� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' An easy application of the Chernoff bound tells us P (Y > 2E[Y ]) ≤ exp(−Ω(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (13) Note that here 2E[Y ] κjn = 8d1/2 log d · 40r2r2ajn 10d �r j � ajn/r = O(d−1/2 log d) < 1 for sufficiently large d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus, using (13) and the union bound over all color classes A, we have w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' the final value of XA,j is at most κjn + 2E[Y ] ≤ 2κjn for all 1 ≤ j ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now we address “bad” edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' edges contained in a color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Assuming Ei holds, the ex- pected number of new edges contained in any color class at step i is at most r(40)r2r2+2rar = O �� log d d � r r−1� (because it would have to be one of the colors of one of the vertices we are adding points to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Thus the expected number of bad edges created in (4d1/2 log d)n steps is stochastically dominated by Z ∼ r · Bin � (4d1/2 log d)n, O ��log d d � r r−1� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Another easy application of Chernoff shows that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Z ≤ 2E[Z] = O(d−1/2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We repair the coloring as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' First we uncolor one vertex from each bad edge, and let the set of uncolored vertices be U where |U| = u = O � d−1/2n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Let δ := ε 2 �(r − 1)d r log d � 1 r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We claim that for every S ⊆ U, |S| = s, the hypergraph induced on S has at most δs/r edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This will complete our proof since it implies that the minimum degree is at most δ and so U can be recolored using a fresh set of δ colors, yielding a coloring of Hr(n, d) using 10 at most χ(Hr(n, M)) + δ ≤ � 1 + ε 2 � �(r − 1)d r log d � 1 r−1 + ε 2 �(r − 1)d r log d � 1 r−1 = (1 + ε) �(r − 1)d r log d � 1 r−1 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' The expected number of sets S with more than δs/r edges is at most � 1≤s≤u �n s ���ds r � δs/r � 1 �dn r ��dn−r r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' �dn−δs+r r � ≤ � 1≤s≤u �ne s �s �(dse/r)re δs/r �δs/r (r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' )δs/r (dn − δs)δs ≤ � 1≤s≤u � ne s � dse (dn − δs)r �δ �er · r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' δs �δ/r�s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' (14) Now for 1 ≤ s ≤ √n the term in (14) is at most � O(n) · � O(n−1/2) �δ · O(1) �s = o(1/n) since δ can be made arbitrarily large by choosing d large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Meanwhile for √n ≤ s ≤ u we have that the term in (14) is at most � O(n1/2) · O(1) · � O(n−1/2) �δ/r�s = o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Now since (14) has O(n) terms the whole sum is o(1) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 4 Summary We have asymptotically computed the chromatic number of random r-uniform, d-regular hypergraphs when proper colorings mean that no edge is mono-chromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' It would seem likely that the approach we took would extend to other definitions of proper coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' We have not attempted to use second moment calculations to further narrow our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' These would seem to be two natural lines of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Ayre, A.' metadata={'source': 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+page_content=' Matula, Expose-and-Merge Exploration and the Chromatic Number of a Random Graph, Combinatorica 7 (1987) 275-284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 12 [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Shamir and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Spencer, Sharp concentration of the chromatic number od random graphs Gn,p, Combinatorica 7 (1987) 121-129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Shi and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' Wormald, Coloring random regular graphs, Combinatorics, Probability and Computing 16 (2007) 459-494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'} diff --git a/GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/2301.01271v1.pdf.txt b/GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/2301.01271v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6d51af6ebbf757bddfa61293682b34cfa4a7572 --- /dev/null +++ b/GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/2301.01271v1.pdf.txt @@ -0,0 +1,1442 @@ +arXiv:2301.01271v1 [econ.GN] 15 Dec 2022 +On the notion of measurable utility on a +connected and separable topological space: +an order isomorphism theorem.∗ +Gianmarco Caldini +7 February 2020 +Abstract +The aim of this article is to define a notion of cardinal utility function called +measurable utility and to define it on a connected and separable subset of a weakly +ordered topological space. The definition is equivalent to the ones given by Frisch +in 1926 and by Shapley in 1975 and postulates axioms on a set of alternatives that +allow both to ordinally rank alternatives and to compare their utility differences. +After a brief review of the philosophy of utilitarianism and the history of utility +theory, the paper introduces the mathematical framework to represent intensity +comparisons of utility and proves a list of topological lemmas that will be used in +the main result. Finally, the article states and proves a representation theorem, +see Theorem 5, for a measurable utility function defined on a connected and sep- +arable subset of a weakly ordered topological space equipped with another weak +order on its cartesian product. Under some assumptions on the order relations, +Theorem 5 proves existence and uniqueness, up to positive affine transformations, +of an order isomorphism with the real line. +∗I am grateful to Professor Massimo Marinacci for letting me know about the open problem. +1 + +Introduction +Together with notions such as value, money, market and economic agents, utility has +been one of the most controversial concepts in the whole history of economic theory. +The most important debate can be considered the one around the question whether it +is possible to define a clear and rigorous concept of utility and an appropriate notion +of unit of measurement for utility, seen as a quantity like the physical ones. In the first +chapter we will give a short introduction to the evolution of the concept of utility from +both a philosophical and a historical point of view. Our treatment is far from being +exhaustive. For an extensive treatment of history of utility and utility measurements, +we refer the interested reader to Stigler [39], Majumdar [26], Adams [1], Luce and +Suppes [24], Fishburn [15], [16], [17] and Moscati [29]. +The second chapter will shift from the descriptive part to more formal concepts +and will be used to introduce the usual mathematical framework of decision theory. +Moreover, we will introduce definitions and axioms that will enable us to represent +comparisons of the intensity that a decision maker feels about the desirability of different +alternatives. For this aim, we will follow the construction of Suppes and Winet [40] +and Shapley [36]. +The third and last chapter will be entirely devoted to the proof of Shapley’s theo- +rem, extending the domain of alternatives X from a convex subset of R to a connected +and separable subset of a topological space, hence providing a generalization of his +theorem. Our intended goal is to define a rigorous notion of a specific kind of cardinal +utility function, not only able to rank alternatives, but also to compare utility differ- +ences. In particular, we define a “twofold” utility function in line with the primordial +axiomatization of Frisch [18], calling it a measurable utility function. In mathematical +terms, we will prove a specific order-isomorphism theorem between a totally ordered, +connected and separable subset of a topological space and the real line. +1 +Philosophy and history of utility theory +Theory of felicity, theory of justice, theory of morality, theory of virtue and theory of +utility are among the most important theories of moral philosophy and, as such, they +are constantly sources of questions that often do not find an immediate answer. When +a human being acts, or when she makes a decision, she is, at the same time, looking for +1 + +justifications, either positive or normative, for the decision she has just made. We, as +human beings, are constantly trying to prove that what we did was the best thing to +do, in some well-defined sense, or, at least, the less harmful. These justifications take +into account the means, the ends and all the possible paths we have to reach our goals. +Moral philosophy is the science that comes into place when we formulate questions +about the ends, the means and the possible ways to achieve them. +Moral philosophy is essentially composed by principles, also called norms, on what +is good and what is bad. They allow to define and to judge human actions, means and +ends. Sometimes, norms take the form of universal laws to which all human beings +are subjected. +Nevertheless, the formulation of moral laws or rules that prescribe +what a single agent should do or not do are intrinsically tied with history. Historical +experiences determine our vision of the world. Our moral philosophy is the result of +different heritages that formed a common culture in which values like human respect, an +idea of equality between human beings and impartiality are among the most important. +Together with this general definition of morality, there exist the similar concepts of +ethics and of role morality - a specific form of professional morality. It was Jeremy Ben- +tham, in an unfinished manuscript which was posthumously published in 1834, to define +the neologism deontology in the title of his book Deontology or the Science of Morality. +The manuscript stated, for the first and only time, the particular aspects of Bentham’s +utilitarian theory as moral philosophy. This passage is clearly mentioned in Sørensen +[37]: +[...] pointing out to each man on each occasion what course of conduct promises to be in +the highest degree conducive to his happiness: to his own happiness, first and last; to the +happiness of others, no farther than in so far as his happiness is promoted by promoting +theirs, than his interest coincides with theirs (p. 5). +In this passage we can see how Bentham considered deontology to be primarily +aimed at one’s own private felicity. Nevertheless, this does not bring any selfish concern. +Bentham’s goal can be identified with the objective study and measurement of passions +and feelings, pleasures and pains, will and action. Among these particular pleasures are +those stemming from sympathy - in Adam Smith’s sense - and they include the genuine +pleasure being happy for the good of others. +In this light, Bentham spent his life in search of the cardinal principle of ethics +and he found it in Epicurean ethics of hedonism. Hedonism comes from Greek ῾ηδον´η, +which means pleasure. Thus, classic utilitarianism, founded on hedonism, started from +2 + +the principle that pleasure is an intrinsic positive value and sorrow is an intrinsic neg- +ative value. It is, for this reason, somehow curious that Bentham conception, founded +on pleasure, had been called utilitarianism, from the simple observation that what is +useful is not necessarily pleasant or providing pleasure. We need always to take into +account that the term utility is intended in a functional sense; what gives utility is what +contributes the most to the individual, or universal, pleasure. +Classical utilitarian philosophers considered utilitarianism well-founded and realistic +thanks to the fact that it is based on pleasure. It is well-founded as its norms are jus- +tified by an intrinsic, absolute value, that does not need any further justification. It is +realistic because they thought human being to ultimately seek the maximum pleasure +and the minimum sorrow. More specifically, human beings try to choose the action +that will provide the maximum excess of pleasure against grief. +For Bentham, what really matters is the total amount of pleasure, intended as +the total excess of pleasure against sorrow: the only reasons for human actions are the +quests for pleasure, avoiding sorrow: they are the sources of our ideas, our judgments +and our determinations. Human moral judgments become statements on happiness; +pleasure (or felicity) is good and sorrow is bad. Utilitarian moral can be considered as +a “calculated hedonism”, that carefully evaluates the characteristics of pleasure. Wise +is the man that is able to restrain from an immediate pleasure for a future good that, +in comparison, will be more beneficial. On the other side, being able to evaluate the +positive or negative consequences of an action without making mistakes is fundamental. +Hence, the correct utilitarian person should reach some kind of “moral arithmetic” that +allows the correct calculations to be carried out. Far from being a unanimously accepted +doctrine, we cannot forget to mention that Alessandro Manzoni wrote an essay [27] in +which he strongly criticized Bentham’s utilitarianism, saying that it is utterly wrong to +think that human beings build their moral values judgment of their actions on utility.1 +From this explanation of utilitarianism, Bentham’s evaluation criterion of actions +follows as an immediate corollary: the maximum happiness for the maximum number +of people. Again, happiness is intended as state of pleasure, or absence of grief. Hence, +individual pleasure becomes no more the ultimate goal: it is the universal pleasure to +be hegemonic.2 +1Manzoni [27] wrote: ”Non ci vuol molto a scoprir qui un falso ragionamento fondato sull’alterazione +d’un fatto. Altro `e che l’utilit`a sia un motivo, cio`e uno de’ motivi per cui gli uomini si determinano +nella scelta dell’azioni, altro `e che sia, per tutti gli uomini, il motivo per eccellenza, l’unico motivo +delle loro determinazioni (p.775). +2This tension between individual pleasure and universal pleasure is one of the principal difficulties +3 + +This view of utilitarianism admits, at least in our minds, the conception of the +existence of a scale of pleasure in which pleasure and sorrow can be added and sub- +tracted. In other words, the idea of a calculus of felicity and grief is not completely +absurd, both in intrapersonal and interpersonal compensations. +1.1 +Brief history of utilitarianism +Although it is possible to find utilitarian reasonings in Aristotele’s works, it is com- +monly agreed that the beginning of the history of utility can be identified with 18th +century moral philosophy. To be even more specific, Bentham’s ideas were not isolated, +since they were already present in works by his illuministic predecessors like Richard +Cumberland, Francis Hutcheson and Cesare Beccaria. Especially Hutcheson [20] had +already defined good as pleasure and good objects as objects that create pleasure. The +novelty of Bentham was to treat pleasure as a measurable quantity, thus making the +utilitarian doctrine directly applicable to issues like tax policies and legislation. In- +deed, not only did Bentham argue that individual pleasure was measurable, but also +that happiness of different people could be compared. Stark [38] cited in his article +Bentham’s writings in the following way: +Fortunes unequal: by a particle of wealth, if added to him who has least, more happiness +will be produced, than if added to the wealth of him who has most (vol. 1, p. 103). +Stark [38] continues: +The quantity of happiness produced by a particle of wealth (each particle being the same +magnitude) will be less and less every particle (vol. 1, p. 113). +It is easy to see how this last concept and the well-known idea of decreasing +marginal utility are related. +In the pioneering work of Jevons [21], utility functions were the primitive mathe- +matical notion to formalize and quantify Bentham’s calculus of pleasure. Utility func- +tions were tools to measure and scale the amount of well-being of human beings. It +seems clear, at this point, how the starting role of utility functions was cardinal,3 in the +sense that utility or, better, pleasure differences were well-founded and realistic notions +with a strong moral philosophy justification. +of utilitarian moral philosophy. +3Note that before the work of Hicks and Allen [19], economists spoke about measurable utility and +not of cardinal utility. +4 + +Summing up, in the beginning, utility functions were designed for the mere purpose +of a calculus of pleasure and sorrow. However, even if the philosophical concept made +sense, the difficulties in the quantification of any experimental measurement of pleasure +led cardinal utility theory to be seen more just like a thought process rather than a +science. +However, utility theory did not rise from philosophy alone, but it was object of +study of other sciences such as statistics, with the so-called St. Petersburg paradox, +and psychophysics, the study of physical stimuli and their relation to sensory reactions. +These two phenomena can be considered the starting point of the law of decreasing +marginal utility. It was Nicolas Bernoulli that, originally, invented what is now called +the St. Petersburg puzzle, which offered the theoretical explanation for the law of de- +creasing marginal utility of wealth. The standard version of the puzzle is the following: +a fair coin is tossed until it lands “head” on the ground. At that point, the player wins +2n dollars, where n is the number of times the coin was flipped. How much should one +be willing to pay for playing this game? In other words, what is the expected value of +the game, given the probability of “head” being 0.5? The mathematical answer is +∞ +� +i=1 +1 +2i · 2i = 1 + 1 + · · · = ∞. +The only rationale for this conundrum is that, if it makes sense to maximize expected +utility and if people are willing to participate to the St. Petersburg game for only a +finite amount of money, then their marginal utility as a function of wealth must be, +somewhere, decreasing. +Neither Bentham nor Bernoulli thought as decreasing marginal utility as a phe- +nomenon in need of scientific justifications. Nevertheless, this came as an immediate +consequence from the psychophysical theories discovered by Weber [42] and generalized +by Fechner [13]. One of the most important questions posed by psychophysics is what is +the functional link between different degrees of a given stimulus and a given sensation. +What Weber did was an in-depth study to try to measure the smallest detectable change +(also called “just noticeable difference” or “minimum perceptible threshold”) in stimuli +like heat, weight and pitch. Moreover, Fechner took this “just noticeable difference” +as a unit of measurement, constructing a scale for subjective sensations. From their +studies we now have the so-called Weber’s law and Fechner’s laws: the former states +that the relative increase of a stimulus needed to produce a just noticeable change is +5 + +constant and, the latter, that the magnitude of sensation is a logarithmic function of +the stimulus. +In conclusion, if wealth is a stimulus, then Benthamian utility must be the cor- +responding sensation. +In this light, St. +Petersburg puzzle can be seen as just one +materialization of these laws. +At the end of 19th century, the marginalist revolution paved the way for an ordinal +approach to the notion of utility. In fact, this was because one of the main economic +problems of late 19th century was the need of a theory of demand. One of the leading +figures that founded neoclassical theory with scientific and analytic rigor was Vilfredo +Pareto. Pareto is considered the father of the so-called ordinal approach. It was a notion +of utility that was purely comparative and it left out from the theory the initial idea for +which utility theory was developed: the existence of psychophysical and physiological +substrates. Pareto’s theory was so successful that was considered a revolution in the +notion of utility. The ordinal approach was extremely successful because it solved the +classic consumer problem based on indifference curves, and the notion of utility had a +central role in its construction. The key aspect was the replacement of marginal utility +- a notion that was meaningless in an ordinal approach - with the trick of marginal rate +of substitutions along indifference curves. +Interesting are the writings of Francis Edgeworth [10] and Pareto [31], starting +from very different assumptions and arriving at different conclusions. +Edgeworth’s +main contribution can be summarized in the synthesis of Bentham’s utilitarianism and +Fechner’s psychophysics: his ideas were based on the unit of utility seen as a just +perceivable increment of pleasure. Moreover, he was interested also in an inter-personal +unit of utility to be able to carry out welfare comparisons among people. Edgeworth +was completely aware of the impossibility of testing these implications, but he was a +strong supporter of the idea of possible comparisons of happiness among people. +Pareto, on the other hand, denied Edgeworth’s intuition of comparisons of utility. +Instead, Pareto [31] reckoned the theoretical possibility of a cardinal notion of utility, +seen as the limit of the purely comparative notion he developed. Nevertheless, he also +argued that such a notion of perfect precision is not attainable and that pleasure is only +imperfectly measurable.4 Summing up, Edgeworth’s and Pareto’s ways of conceiving +measurable utility must be differentiated and utility theory is still today based on the +Paretian notion mainly because of its use in the theory of demand and in the general +4However, Pareto [32] writes: “There is no reason for not accepting it [cardinal utility], with the +reservation that it must be verified by the results deduced from it (p. 73).” +6 + +equilibrium theory. +In 1950, ordinalism was the well-established mainstream ideology in utility theory +and the cardinal notion of utility was almost completely abandoned. Nevertheless, the +purely comparative approach was not convincing everyone, mainly because people’s +introspection suggested the existence something more. One of the main supporters of +cardinalism was Maurice Allais who explicitly wrote in [4]: +The concept of cardinal utility [...] has almost been rejected the literature for half a cen- +tury. This rejection, based on totally unjustified prejudices, deprived economic analysis +of an indispensable tool (p. 1). +Allais [4] admits that the theory of general economic equilibrium can be fully de- +scribed in an ordinal world, but he immediately lists a series of theories that cannot +be adequately developed without a rigorous and well-defined concept of cardinal utility +and interpersonal comparisons. Some examples are the theory of dynamic evolution of +the economy, the theory of fiscal policy, of income transfers, of collective preferences, +social welfare analysis and political choices, of risk, of insurance and the theory of +cooperative games. Then, Allais goes even further in his defense of cardinal utility, +arguing that even the theory of demand could become more intuitive - and with a +simpler exposition - if we could appeal to a notion of intensity of preferences. In any +case, as long as the conclusions of price theory do not change significantly using the +ordinal and the cardinal approach, we should prefer the purely comparative approach +by Occam’s razor. But the problems with group decision making, social choice theory +and cooperative game theory still cannot be solved. Indeed, while classical economists +considered distributional problems as a fundamental part of economic science, the ordi- +nalist approach to utility theory refused completely to deal with questions that involved +interpersonal comparisons of welfare. Economists became more interested in positive +statements, rather than normative ones and the accent was put on efficiency, rather +than equity. This was the case of the optimum allocations in the sense of Pareto. For +a complete overview of the main issues of welfare economics, the main problems with +an ordinal approach and the main literature, we refer the interested reader to Sen [35]. +Hence, one of the main problems, still in the 21st century, is how is it possi- +ble to understand the intuitive tool of introspection to develop a rigorous theory that +economists can apply in their models and explain economic phenomena. The solution +does not exists yet. In the last years, the issue started getting the attention of few deci- +sion theorists mainly because of the powerful developments in the field of neuroscience +7 + +and the new discipline of neuroeconomics. These fields of cognitive sciences are going +into the direction of overcoming the main difficulty of the founders of utilitarianism: +the difficulty of carrying out experiments on pleasure and pain and the construction of +a rigorous and well-defined scale of pleasure. Nevertheless, nothing is clear yet, mainly +because also the theoretical concept of cardinal utility is still vague. Cardinal utility +is still used as a name for a large number of formally distinct concepts and it misses a +precise and well-established definition that can be applied in decision-theoretic models. +During the 20th century a lot of methodologies to try to define a concept of +measurement of human sensation have been defined. +Definition 1. A scale is a rule for the assignment of numbers to aspects5 of objects or +events. +The result was the development of a full taxonomy of scales, with scales that differ +in terms of higher precision of measurement. For an extensive treatment of the theory +of measurement we refer the interested reader to Krantz et alii [22]. +The issue of having a rigorous definition for cardinal utility was not solved by the +theory of measurement. +It was just translated in a different language: what is the +suitable scale for measuring a given aspect? The definition of a unit of measurement +for utility was not an easy task to solve. Even in physics, where experiments can be +carried out with relatively high precision, the way a unit of measurement is defined +is not perfect. One meter was originated as the 1/10-millionth of the distance from +the equator to the north pole along a meridian through Paris. Then, the International +Bureau of Weights and Measures, founded in 1875, defined the meter as the distance +of a particular bar made by platinum and iridium kept in S`evres, near Paris. More +recently, in 1983, the Geneva Conference on Weights and Measures defined the meter +as the distance light travels, in a vacuum, in 1/299,792,458 seconds with time measured +by a cesium-133 atomic clock which emits pulses of radiation at very rapid and regular +intervals. +Increases in science allow the unit of measurement to be duplicated with a better +and better level of precision. The comparison with the unit of measurement of the +quantity utility can be carried out with the philosophical question whether it is, for +some esoteric reason, intrinsically impossible to measure human beings’ pleasure or +whether economic science and neuroeconomics are so underdeveloped that we still have +5For example: hardness, length, volume, density, . . . +8 + +very poor precision in measuring human felicity.6 +The same comparison can be done with light (or heat, color and wave lengths, as +it is mentioned by von Neumann and Morgenstern in [41]). For example, temperature +was, in the original concept, an ordinal quantity as long as the concept warmer was +known. Then, the first transition can be identified with the development of a more pre- +cise science of measurement: thermometry. With thermometry, a scale of temperature +that was unique up to linear transformations was constructed. The main feature was +the association of different temperatures with different classes of systems in thermal +equilibrium. Classes like these were called fixed points for the scale of temperatures. +Then, the second transition can be associated with the development of thermodynam- +ics, where the absolute zero was fixed, defining a reference point for the whole scale. +In physics, these phenomena had to be measured and the individual had to be able to +replicate results of such measurements every time. The same may apply to decision +theory and the notion of utility, someday. At the moment, the issue remains unclear, +even if even Pareto was not completely skeptical about the first transition from an +ordinal purely comparative approach to that of an equality relation for utility differ- +ences. Von Neumann and Morgenstern point out in [41] that the previous concept is +based on the same idea used by Euclid to describe the position on a line: the ordinal +utility concept of preference corresponds to Euclid’s notion of lying to the right of and +the derived concept of equality of utility differences with the geometrical congruence of +intervals. +Hence, the main question becomes whether the derived order relation on utility +differences can be observed and reproduced. Nobody can, at the moment, answer this +question. +1.2 +Axiomatization of utility theories +In 1900, at the International Congress of Mathematicians in Paris, David Hilbert an- +nounced that he was firmly convinced that the foundation of mathematics was almost +complete. Then, he listed 23 problems to be solved and to give full consistency to +mathematics. All the rest was considered, by him, just details. Some of the problems +6Some authors, like Ellingsen [12], are certain, instead, that the philosophical question of whether +utility is intrinsically measurable or not is a spurious one, mainly because they see the issue of “mea- +surement” as a concept that is always invented and never discovered. In this light, our question can +be rephrased as whether it is possible to define a correct notion of measurement that allows some kind +of intrapersonal and interpersonal utility comparisons. +9 + +consisted in the axiomatizations of some fields of mathematics. Indeed, at the begin- +ning of the 20th century, the idea of being able to solve every mathematical problem +led mathematicians to try develop all mathematical theory from a finite set of axioms. +The main advantage of the axiomatic method was to give a clean order and to remove +ambiguity to the theory as a whole. Axioms are the fundamental truths by which it +is possible to start modeling a theory. The careful definition of them is critical in the +development of a theory that does not contain contradictions. +As a result, almost all fields of science started a process of axiomatization, utility +theory as well. +The ordinal Paretian revolution was the fertile environment where +preferences started to be seen as primitive notions. Preference relations began to be +formalized as mathematical order relations on a set of alternatives X and became the +starting point of the whole theory of choice. +As a result, utility functions became +the derived object from the preference relations. The mainstream notion of ordinal +(Paretian) utility reached its maturity with the representation theorems by Eilenberg +[11] and Debreu [8], [9]. Subsequent work in decision theory shifted from decision theory +under certainty to choice problems under uncertainty, with the pioneering article of +Ramsey [33] on the “logic of partial belief.” In short, Ramsey [33] stated the necessity +of the development of a purely psychological method of measuring both probability +and beliefs, in strong contradiction with Keynes’ probability theory. Some years after, +the milestone works of von Neumann and Morgenstern [41] and Savage [34] gave full +authority to decision theory under uncertainty. +One of the first treatments of preference relations as a primitive notion can be +identified with Frisch [18], in his 1926 paper. +Ragnar Frisch was also the first to +formulate an axiomatic notion of utility difference. Hence, two kinds of axioms were +postulated by him: the first ones - called “axioms of the first kind” - regarded the +relation able to rank alternatives in a purely comparative way, while the second axioms +- named “axioms of the second kind” - reflected a notion of intensity of preference and +allowed utility differences to be compared. +So, in parallel to the axiomatization of +ordinal utility, also cardinal utility axiomatizations started to grow. +Frisch’s article did not have the deserved impact in the academic arena, mainly +because his article was written in French and published in a Norwegian mathematical +journal. Hence, the full mathematical formalization of these two notions of preference +axioms resulted almost ten years later from the 1930s debate by Oskar Lange [23] and +Franz Alt [5]. Lange [23] defined an order relation ≻ on the set of alternatives X with +the meaning that, for any two alternatives x, y ∈ X, x ≻ y reads “x is strictly preferred +10 + +to y.” Then, a corresponding relation P on ordering differences is assumed with the +meaning that, for any x, y, z, w ∈ X, xyPzw reads “a change from y to x is strictly +preferred than a change from w to z. +More formally: +x ≻ y ⇐⇒ u(x) > u(y) for all x, y ∈ X +(1) +xyPzw ⇐⇒ u(x) − u(y) > u(z) − u(w) for all x, y, z, w ∈ X +(2) +The main theorem of Lange [23] can be stated as follows: +Theorem 1. If there exists a differentiable utility function u : R → R such that (1) +and (2) hold, then only positive affine transformations of that utility function represent +the given preferences ≻ and P. +It is immediate to see that Lange [23] provides only necessary conditions for a +utility function representation of preference relation. Moreover, it is relatively easy to +see that the assumption of differentiability of u can be largely relaxed. Hence, the issue +becomes whether it is possible to find sufficient conditions on the preference relations +under which Lange’s utility function - a cardinal utility function - exists. This was +done by Franz Alt in his 1936 article [5]. Alt postulated seven axioms that guaranteed +sufficient and necessary conditions for the existence of a continuous utility function +- unique up to positive affine transformations - based on a preference relation and a +utility-difference ordering relation. In his set of axioms, Alt defined a notion that can +be understood as the set of alternatives X to be connected. +With Frisch’s pioneering work of 1926 and 1930s debate by Lange and Alt, the +modern ingredients of cardinal utility axiomatization such as equations (1) and (2) and +connectedness of the domain of alternatives X started to be formalized. In those years, +a lot of different axiomatic models were studied, till the article of the famous philoso- +pher of science Patrick Suppes and his doctoral student Muriel Winet [40]. In their +1955 paper, Suppes and Winet developed an abstract algebraic structure of axioms for +cardinal utility, called a difference structure, in line with old Frisch’s ideas and Lange’s +formalization: not only are individuals able to ordinally rank different alternatives, but +they are also able to compare and rank utility differences of alternatives. Indeed, Sup- +pes and Winet cited the work of Oskar Lange on the notion of utility differences and +stood in favor of the intuitive notion of introspection, elevating it to not just a mere +11 + +intuition, but as a solid base where to build a notion of utility differences. Suppes and +Winet continued their article saying that, up to 1950s, no adequate axiomatization for +intensity comparison had been given. Hence, as Moscati [29] nicely highlights, they +were probably unaware of Alt’s representation theorem and this was probably due to +the fact that Alt [5] was published in German in a German journal. Suppes and Winet +postulated 11 axioms in total, some on the set of alternatives X and others on the +two order relations,7 providing sufficient and necessary condition for a cardinal utility +representation, unique up to positive affine transformations. Another approach to the +field of axiomatization of cardinal utility was taken twenty years later by Lloyd Shap- +ley. While axiomatizations `a la Suppes and Winet started developing a set-theoretic +abstract structure, Shapley substituted the usual long list of postulates with strong +topological conditions both on the domain of alternatives X and on the topology in- +duced by the order relations. Shapley [36] constructed a cardinal utility function u +satisfying some consistency axioms between the orders and assuming the domain of u +to be a convex subset of the real line. We will enter into the details of Shapley [36] in +the next chapters. +In conclusion, the notion of cardinal utility has always suffered a lack of conceptual +precision in its whole history and, for some authors like Ellingsen [12], it can be even +considered the main reason why scientists have disagreed over whether pleasure can be +measured or not.8 What is certain is that the history of cardinal utility, a part from some +sporadic articles, has been a persistent failure, mostly in its applications to economic +theory. While the main reason can be probably identified with the almost total absence +of any rigorous and proven experimental measurement of pleasure, it is fair to observe +that part of its failure must be given to the strong reluctant opinion of the mainstream +ordinal “party.” In fact, a large class of economists classify as “meaningless” even the +mere introspective idea of a comparison of utility difference, and not just the concept +itself, when formalized in a purely comparative environment. This position is shown to +be, with a gentle expression, “epistemological laziness.” We should always remember +that no real progress in economic science can be derived from purely abstract reasoning, +but only from the combined effort of empirical measurements with theoretical analysis, +always under the wise guide of the compass of history and philosophy. +7The conditions these axioms impose are analogous to the conditions defined by Alt [5]: com- +pleteness, transitivity, continuity, and some form of additivity for the two order relations, and an +Archimedean property on the quaternary relation. +8Ellingsen [12] writes about a “fallacy of identity” and “fallacy of unrelatedness.” +12 + +2 +Preliminary results +The aim of this chapter is twofold. On one side, we introduce the mathematical frame- +work that enable us to represent intensity comparisons that a decision maker feels about +the desirability of different alternatives. For this aim, we follow the construction of Sup- +pes and Winet [40] and Shapley [36]. On the other side, we state and prove a list of +lemmas that will be used in Theorem 5 and that allow us to generalize Shapley’s proof +to a connected and separable subset of a topological space. +2.1 +Basic definitions +Definition 2. A relation on a set X is a subset ≿ of the cartesian product X × X, +where x ≿ y means (x, y) ∈ ≿. +In decision theory, ≿ is usually called a preference relation, with the interpretation +that, for any two elements x, y ∈ X, we write x ≿ y if a decision maker either strictly +prefers x to y or is indifferent between the two. +Definition 3. An equivalence relation on a set X is a relation R on X that satisfies +1) Reflexivity: for all x ∈ X, we have xRx. +2) Symmetry: for any two elements x, y ∈ X, if xRy, then yRx. +3) Transitivity: for any three elements x, y and z ∈ X, if xRy and yRz, then xRz. +Definition 4. A relation ≿ on a set X is called a total order relation (or a simple order, +or a linear order) if it has the following properties: +1) Completeness: for any two elements x, y ∈ X, either x ≿ y or y ≿ x or both. +2) Antisymmetry: for any two elements x, y ∈ X, if x ≿ y and y ≿ x, then x = y. +3) Transitivity: for any three elements x, y and z ∈ X, if x ≿ y and y ≿ z, then +x ≿ z. +Note that if ≿ is complete, then it is also reflexive. The relation ≿ induces, in +turns, two other relations. Specifically, for any two elements x, y ∈ X we write: +(i) x ≻ y if x ≿ y but not y ≿ x. +13 + +(ii) x ∼ y if x ≿ y and y ≿ x. +It is easy to see, indeed, that if ≿ is reflexive and transitive, then ∼ is an equivalence +relation. Given an equivalence relation ∼ on a set X and an element x ∈ X, we define +a subset E of X, called the equivalence class determined by x, by the equation +E := {y ∈ X : y ∼ x} +Note that the equivalence class E determined by x contains x, since x ∼ x, hence +E is usually denoted as [x]. +We will denote X/∼ the collection {[x] : x ∈ X} of +all equivalence classes, which is a partition of X: each x ∈ X belongs to one, and +only one, equivalence class. In decision theory, an equivalence class is often called an +indifference curve. +Definition 5. A relation ≿ on a set X is called a weak order if it is complete and +transitive. +The problem of finding a numerical representation for a preference relation ≿, i.e. +an order isomorphism between a generic set X and R, has been widely studied by math- +ematicians and is a familiar and well-understood concept. Such an order isomorphism +is called, in decision theory, a utility function. More formally: +Definition 6. A real-valued function u : X → R is a (Paretian) utility function for ≿ +if for all x, y ∈ X we have +x ≿ y ⇐⇒ u(x) ≥ u(y) +Utility functions “shift” the pairwise comparisons that characterize the order rela- +tion ≿ and its properties in the more analytically convenient space of the real numbers. +Nevertheless, as a result, the only thing that is preserved is the order, and the real +numbers that are images of the utility function cannot be interpreted as a scale where +the decision maker can compare different intensities about the single desirability of any +two alternatives x, y ∈ X. What is important is the ranking given by the real num- +bers, according to the usual order of the ordered field (R, ≥). Indeed, one can easily +prove that every strictly increasing transformation of a utility function is again a utility +function. For this reason, utility functions are called ordinal and their study belong +to what is called ordinal utility theory. The main problem of ordinal utility theory is +to study sufficient and necessary conditions under which a relation ≿ admits a utility +representation. The original reference can be identified with Cantor [7], but the result +has been adapted by Debreu [8]. +14 + +In addition, to be able to solve optimization problems, one of the properties that +is desirable to have is continuity of the utility function. Debreu [8] is the first to state +the theorem in the way we are going to. Nevertheless, he proved it making explicit +reference to Eilenberg [11]. We state here a version of this very well-known theorem. +Definition 7. A weak order ≿ on a set X is said to be continuous if, for every y ∈ X, +the sets {x ∈ X : x ≻ y} and {x ∈ X : x ≺ y} are open.9 +Theorem 2 (Eilenberg). Let ≿ be a complete and transitive relation on a connected +and separable topological space X. The following conditions are equivalent: +(i) ≿ is continuous. +(ii) ≿ admits a continuous utility function u : X → R. +One of the biggest theoretical problems of ordinal utility theory is that the expres- +sion +u(x) − u(y) +is a well-defined real number thanks to the algebraic properties of R, but it is meaning- +less in term of the interpretation of a difference of utility of two alternatives x, y ∈ X. +In other words, a Paretian utility function does not have an intrinsic introspective psy- +chological notion of intensity of the preferences. An immediate corollary of this remark +is that the concept of marginal utility (and what is known under the Gossen’s law of +decreasing marginal utility), based on the notion of different quotient, is meaningless. +More formally, the expression +du(x) +dx += lim +h→0 +u(x + h) − u(x) +h +has no meaning in this setting. Nevertheless, the concept of marginal utility has been a +milestone in economic theory, proving that this notion deserves an adequate theoretical +foundation. +2.2 +An overview on measurable utility theory +Let X be a set of alternatives. Pairs of alternatives (x, y) ∈ X × X are intended to +represent the prospect of replacing alternative y by alternative x, that can be read as +9Note that this is the usual order topology on X. +15 + +“x in lieu of y”. Define the binary relation ≽ on X ×X called intensity preference with +the following interpretation: for any two pairs (x, y) and (z, w) in X × X, +(x, y) ≽ (z, w) +is intended to mean that getting x over y gives at least as much added utility as getting +z over w or (if y ≿ x) at most as much added sadness. As a result, our decision maker +is endowed with a weak order preference relation ≿ on alternatives and an intensity +preference relation ≽ on pairs of alternatives. +Shapley [36] proves his theorem assuming X to be a convex subset of R. As a +result, the proof exploits the full algebraic power of the ordered field and the topological +properties of the linear continuum. Our aim is to generalize the set of alternatives X +to a connected and separable subset of a topological space, ordered with the binary +relations ≿ and ≽ and with the order topology induced by the weak order ≿. +We assume the following axioms for ≿ and ≽, as in Shapley [36]. +Axiom 1. For all x, y, z ∈ X we have (x, z) ≽ (y, z) ⇐⇒ x ≿ y. +Axiom 1 (henceforth A1) is an assumption of consistency between the two order- +ings because it implies that the decision maker prefers to exchange z with x instead of +z with y if and only if she prefers x to y. Together with A1 we can formulate a dual +version of consistency, A1′, that can be derived from the whole set of axioms we are +going to assume later.10 +Axiom 1′. For all x, y, z ∈ X we have (z, x) ≽ (z, y) ⇐⇒ y ≿ x. +We now introduce the main object of this thesis: a joint real-valued representation +for the two orders ≿ and ≽. +Definition 8. A real-valued function u : X → R is a measurable utility function for +(≿, ≽) if for each pair x, y ∈ X +x ≿ y ⇐⇒ u(x) ≥ u(y) +(3) +and if, for each quadruple x, y, z, w ∈ X +(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w). +(4) +The measurable terminology has nothing to do with measure theory, but it refers to +what is known as measurement theory, i.e. the field of science that established the for- +mal foundation of quantitative measurement and the assignment of numbers to objects +10We mention A1′ as a form of axiom only because in this way we can refer to it in the proof of +Theorem 5, but we never assume it formally. A proof of it will be formulated forward with Lemma 13. +16 + +in their structural correspondence. Indeed, not only is a measurable utility function +able to rank pairs of alternatives according to a preference relation, but it also repre- +sents the idea of magnitude and intensity of the preference relation among alternatives. +Therefore, the numerical value u(x) that a measurable utility function assigns to the +alternative x is assuming the role of a particular unit of measurement for pleasure, that +we call util. +Recall that an ordinal utility function u is unique up to strictly monotone trans- +formations f : Im(u) → R. Hence, a measurable utility function is not ordinal. Never- +theless, it is unique up to positive affine transformations. Recall that a positive affine +transformation is a special case of a strictly monotone transformation of the follow- +ing form f(x) = αx + β, with α > 0 and β ∈ R. Positive affine transformations are +order-preserving thanks to α > 0. +Proposition 1. A measurable utility function u : X → R for (≿, ≽) is unique up to +positive affine transformations. +Proof. If u(x) = αu(x) + β then we have +x ≿ y ⇐⇒ u(x) ≥ u(y) ⇐⇒ u(x) = αu(x) + β ≥ αu(y) + β = u(y) +and +(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w) +⇐⇒ u(x) − u(y) = α[u(x) − u(y)] ≥ α[u(z) − u(w)] = u(z) − u(w). +As a result, u and u are two utility representations for (≿, ≽). +The whole class +of utility functions that are unique up to positive affine transformations are called +cardinal. Measurable utility functions are, therefore, cardinal and pertain to the so- +called cardinal utility theory. +Other two axioms (A2, A3) we need to introduce are the following: +Axiom 2. For all x, y, z, w ∈ X we have (x, y) ∼ (z, w) ⇐⇒ (x, z) ∼ (y, w). +Axiom 3. For all x, y, z, w ∈ X the set +{(x, y, z, w) ∈ X × X × X × X : (x, y) ≽ (z, w)} +is closed in the product topology. +Axiom 2 is a “crossover” property that characterizes difference comparisons of util- +ity, while Axiom 3 is a technical assumption defining the order relation ≽ as continuous. +17 + +Shapley [36] proves his theorem on a domain of alternative outcomes that is a +nonempty, convex subset D of the real line where the preference order coincides with +the total order of (R, ≥). Moreover, ≽ is assumed to be a weak order on D × D such +that A1, A2 and A3 are satisfied. +Theorem 3 (Shapley). There exist a utility function u : D ⊆ R → R such that +x ≥ y ⇐⇒ u(x) ≥ u(y) +(5) +and +(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w) +(6) +for all x, y, z, w ∈ D. Moreover, this function is unique up to a positive affine transfor- +mation. +The theorem is stated as a sufficient condition, which is the most difficult part to +prove. The necessary condition of the theorem is easily proved and we state it here as +a proposition. +Proposition 2. If the pair (≥, ≽) has a continuous measurable utility function u : D ⊆ +R → R, then ≥ is complete and transitive, ≽ is complete, transitive, continuous (A3) +and satisfies the crossover axiom (A2), and jointly ≥ and ≽ satisfy the consistency +axiom (A1). +Shapley’s construction of the measurable utility function of Theorem 3 is extremely +elegant, but has the drawback of being too specific as u is defined on a convex subset +of R. On the other side of the spectrum, as mentioned in the first chapter, the field +of utility axiomatization has been prolific in the 20th century and a copious number +of cardinal-utility derivations from preference-intensity axiomatizations were published. +One of the most important papers on this issue was the one published in 1955 by Patrick +Suppes and Muriel Winet. Recalling what described before, Suppes and Winet [40] +advanced an axiomatization of cardinal utility based on the assumption that individuals +are not only able to rank the utility of different alternatives, as is assumed in the ordinal +approach to utility, but are also capable of ranking the differences between the utilities +of commodities. Nevertheless, their 11 axioms on an abstract algebraic structure were +not fully satisfactory in terms of generality: it was too general. Indeed, some of their +axioms can be derived in Shapley [36], thanks to the topological properties of R. +The aim of this research is to settle somewhere in between, finding a representation +theorem for cardinal utility function (in particular, a measurable one) keeping the +18 + +elegance of Shapley’s proof and generalizing the domain of alternatives into the direction +of Suppes and Winet [40]. We will state and prove a representation theorem for a +measurable utility function u : X → R where X is a connected and separable subset of +a topological space, ≿ and ≽ are weak orders and they satisfy (A1), (A2) and (A3). +Before doing this, we need to state and prove some topological preliminary results that +will be used in Theorem 5.11 +2.3 +A few basic lemmas +Definition 9. Let X be a topological space. X is connected if it cannot be separated +into the union of two disjoint nonempty open subsets. Otherwise, such a pair of open +sets is called a separation of X. +Definition 10. Let X be a topological space. X is separable if there exists a countable +dense subset. A dense subset D of a space X is a subset such that its closure equals the +whole space, i.e. D = X. +Definition 11. A totally ordered set (L, ≿) having more than one element is called a +linear continuum if the following hold: +(a) +L has the least upper bound property. +(b) +If x ≻ y, there exists z such that x ≻ z ≻ y +We recall that a ray is a set of the following type (−∞, a) = {x ∈ L : x ≺ a} +and (−∞, a] = {x ∈ L : x ≾ a} in the case L does not have a minimum. In the +case L does have a minimum we write [xm, a) = {x ∈ L : xm ≾ x ≺ a} and [xm, a] = +{x ∈ L : xm ≾ x ≾ a}. Analogously for the sets (a, +∞), [a, +∞), (a, xM] , [a, xM], where +xM is the maximum of L in the case it existed.12 +Given A ⊆ X, an element y ∈ X is an upper bound for a set A if y ≿ x for all +x ∈ A. It is a least upper bound for A if, in addition, it is the minimum of the set of all +upper bounds of A, that is if y′ ≿ x for all x ∈ A then y′ ≿ y. If ≿ is antisymmetric, the +least upper bound is unique and is denoted sup A. The greatest lower bound is defined +analogously and denoted inf A. +11We thank Dr. Hendrik S. Brandsma for providing a feedback and insightful comments. +12Note that in decision theory, rays of a set X equipped with a reflexive and transitive binary +relation ≿ are usually denoted with the following notation L(a, ≿) := (−∞, a] = {x ∈ X : x ≾ a} and +U(a, ≿) := [a, +∞) = {x ∈ X : x ≿ a}, L(a, ≻) := (−∞, a) and U(a, ≻) := (a, +∞). +19 + +Lemma 1. Let ≿ be a total order on a connected set X. Then, X is a linear continuum +in the order topology.13 +Proof. Suppose that a and b are two arbitrary but fixed elements of X such that a ≺ b. +If there is no element c ∈ X such that a ≺ c ≺ b, then X is the union of the open +rays (−∞, b) = {x ∈ X : x ≺ b} and (a, +∞) = {x ∈ X : a ≺ x} both of which are +open sets in the order topology and are also nonempty, as the first contains a, while +the second contains b. But this contradicts the fact that X is connected, so there must +exists an element c ∈ X such that a ≺ c ≺ b. +Now, to show the least upper bound property, let A be a nonempty subset of X +such that A is bounded above in X. Let B be the set of all the upper bounds in X of +set A, i.e. +B := {b ∈ X : b ≿ a for every a ∈ A} +which is nonempty. All we need to show is that B has the least element. If B has a +smallest element (or A has a largest element, which would then be the smallest element +of B), then that element is the least upper bound of A. +Let us assume, instead, that B has no smallest element. Then, for any element +b ∈ B, there exists an element b′ ∈ B such that b′ ≺ b, and so b ∈ (b′, +∞) ⊆ B with +(b′, +∞) being an open set in X. This shows that B is a nonempty open subset of X. +Therefore, B can be closed only in the case when B = X. But we know that B ⊂ X, +since A ⊆ X\B and A ̸= ∅, so it cannot be the case that B = X. Therefore, B has a +limit point b0 that does not belong to B. Then b0 is not an upper bound of set A, which +implies the existence of an element a ∈ A such that b0 ≺ a, we can also conclude that +b0 ∈ (−∞, a) ⊆ X\B, with (−∞, a) being an open set. This contradicts our choice of +b0 as a limit point of set B. Therefore, the set B of all the upper bounds in X of set A +must have a smallest element, and that element is the least upper bound of A. +Given A ⊆ X, we denote A or ClA the topological closure of A, that is defined as +the intersection of all closed sets containing A. +From now on denote X as a subset of a topological space (X, τ), unless otherwise +stated. +Lemma 2. Let ≿ be a complete, transitive and continuous order on a connected set X. +13Note that the converse holds as well: ≿ is a total order on a connected set X if and only if X is a +linear continuum in the order topology. +20 + +Given any x, y ∈ X, with x ≻ y, we have +x ≿ z ≿ y ⇒ z ∈ X +for all z ∈ (X, τ) +Proof. Suppose by contradiction that there exists z ∈ X\X such that x ≻ z ≻ y. +By the continuity of ≿, we can partition X into two nonempty disjoint open sets +{x ∈ X : x ≺ z} and {x ∈ X : x ≻ z}, which contradicts the connectedness of X. +Lemma 3. Suppose that jointly ≿ and ≽ satisfy A1. If ≽ is continuous , then ≿ is +continuous. +Proof. For all arbitrary but fixed y, z ∈ X, by A1 we have {x ∈ X : (x, z) ≽ (y, z)} = +{x : x ≿ y}. By A3, the set {x ∈ X : (x, z) ≽ (y, z)} is closed. Analogous is the case +for {x : y ≿ x}, derived from A1′. +Lemma 4. Fix y ∈ X, the set Iy := {x ∈ X : x ∼ y} is a closed set in X. +Proof. ≿ is continuous, so for every y ∈ X we have that {x ∈ X : x ≿ y} and +{x ∈ X : y ≿ x} are closed. Pick a point x such that x ≿ y and y ≿ x, that is x ∼ y. +So we have {x ∈ X : x ∼ y} = {x ∈ X : x ≿ y} ∩ {x ∈ X : y ≿ x} and the intersection +of two closed sets is closed. +Note that when ≿ is antisymmetric, the set Iy is a singleton and Lemma 4 reduces +to prove that X satisfies the T1 axiom of separation, that is every one-point set is closed. +Clearly, every Hausdorff space satisfies it. +Lemma 5. Let ≿ be a continuous total order on a connected set X. If A ⊆ X is a +nonempty closed set in the order topology and A is bounded above (below), then supA +(infA) belongs to A.14 +Proof. Suppose supA /∈ A. Then supA ∈ X\A, which is open. By definition, there +exists a base element (a, b) such that +supA ∈ (a, b) ⊆ X\A. +A is bounded above so, by Lemma 1, sup A exists and there is an element a⋆ such that +a ≺ a⋆ ≺ sup A, then a⋆ ∈ (a, b) ⊆ X\A, so a⋆ is an upper bound of A smaller that +supA, reaching a contradiction. In the case X had a maximum, then consider the case +where sup A = max X. Let U := (x, sup A] be a basic neighborhood of sup A. Then, x +14The lemma holds even in the case we relaxed connectedness. Nevertheless, we always need to as- +sume sup A exists. If we do not assume the existence of the least upper bound, an easy counterexample +is N ⊂ R that is closed in the order topology, but sup N /∈ N. +21 + +cannot be an upper bound of A as x ≺ sup A. Hence, there exists an element a ∈ A +such that x ≺ a ≾ sup A. Thus, as x was generic, it follows that U ∩ A ̸= ∅. This +means that every neighborhood of sup A intersects A, that is sup A ∈ A. But A is +closed, hence sup A ∈ A and we can conclude sup A = max A. +The case of inf A is specular. +Now we define the notion of convergence in any topological space. +Definition 12. In an arbitrary topological space X, we say that a sequence x1, x2, . . . +of points of the space X converges to the point x of X provided that, corresponding to +each neighborhood U of x, there is a positive integer N such that xn ∈ U for all n ≥ N. +Moreover, let ≿ a total order. We write xn ↑ x if x1 ≾ x2 ≾ · · · ≾ xn ≾ . . . and +supnxn = x where sup is with respect to ≾. The definition xn ↓ x for a ≾-decreasing +sequence is analogous. We say that (xn) converges monotonically to a limit point x +when either xn ↑ x or xn ↓ x. +We now prove one of the fundamental lemmas that allow us to generalize Shapley’s +proof to a connected and separable subset of a topological space. Note that, as long +as Shapley [36] is working on R, sequences as “enough” to characterize the definition +of convergence. +This is due to the fact that there exists a countable collection of +neighborhoods around every point. This is not true in general, but it is for a specific +class of spaces that are said to satisfy the first countability axiom.15 A space X is said +to have a countable basis at the point x if there is a countable collection {Un}n∈N of +neighborhoods of x such that any neighborhood U of x contains at least one of the sets +Un. A space X that has a countable basis at each of its points is said to satisfy the +first countability axiom. +In general, however, sequences are not powerful enough to capture the idea of +convergence we want to capture in a generic topological space. Indeed, there could +be uncountably many neighborhoods around every point, so the countability of the +natural number index of sequences cannot “reach” these points. The ideal solution to +this problem is to define a more general object than a sequence, called a net, and talk +about net-convergence. One can also define a type of object called a filter and show +that filters also provide us a type of convergence which turns out to be equivalent to +net-convergence. With these more powerful tools in place of sequence convergence, one +can fully characterize the notion of convergence in any topological space. +15There are far more general classes of spaces in which convergence can be fully characterized by se- +quences. We refer the interested reader to the notion of Fr´echet-Urysohn spaces and Sequential spaces. +22 + +Nevertheless, we are now going to show that every connected, separable and totally +ordered set X satisfies the first countability axiom. In fact, we are going to prove even +more. We are going to show that X is metrizable, which means there exists a metric d +on the set X that induces the topology of X.16 We give other two definitions that will +be used to prove Lemma 6. +Definition 13. Suppose X is T1. Then X is said to be regular (or T3) if for each pair +consisting of a point x and a closed set B disjoint from x, there exist disjoint open sets +containing x and B, respectively. +Definition 14. If a space X has a countable basis for its topology, then X is said to +satisfy the second countability axiom, or to be second-countable. +Theorem 4 (Urysohn metrization theorem). Every regular space X with a count- +able basis is metrizable. +Lemma 6. Let ≿ be a continuous total order on a connected and separable topological +space X in the order topology and A ⊆ X. We have x ∈ A if and only if there exists a +sequence (xn) ∈ AN that converges monotonically to x. +The steps of the proof are the following: +(i) We show that X is regular17 and second-countable. By the Urysohn metrization +theorem, which provides sufficient (but not necessary) conditions for a space to +be metrizable, there exist a metric d that induces the topology of X. +(ii) Let A ⊆ X with X metrizable, then we have that x ∈ A if and only if there exists +a sequence of points of A converging to x. +(iii) Finally, we use the fact that in every totally ordered topological space X, every +sequence admits a monotone subsequence. Then, if a sequence converges, all of +its subsequences converge to the same limit. Thus, we can extract our monotone +converging sequence. +Lemma 7. A totally ordered topological space X is regular in the order topology. +Proof. It is basic topology to prove that every totally ordered set is Hausdorff, hence +it is T1. Now, suppose x ∈ X and B is a closed set, disjoint from x. So, x ∈ X\B, +16A metrizable space always satisfies the first countability axiom. +17In fact, one could prove that X is also normal. +23 + +which is open. Then, by definition of open set, there exists a basis element (a, b) such +that x ∈ (a, b) and (a, b) ∩ B = ∅. Pick any a0 ∈ (a, x), and let U1 = (−∞, a0) , V1 = +(a0, ∞). If no such a0 exists (in our case it would, by connectedness of X), then let +U1 = (−∞, x), V1 = (a, ∞). In both cases, U1 ∩ V1 = ∅. Similar is the case of the other +side, pick b0 ∈ (x, b), and if that exists, denote U2 = (b0, ∞) , V2 = (−∞, b0) , and if +not, let U2 = (x, ∞), V2 = (−∞, b). Again, in both cases U2 ∩ V2 = ∅. As a result, we +obtained that, in both cases, x ∈ V1 ∩ V2 with V1 ∩ V2 open set and B ⊆ U1 ∪ U2, with +U1 ∪ U2 open set. As V1 ∩ V2 is disjoint from U1 ∪ U2, X is regular. +Lemma 8. A totally ordered, connected and separable topological space X is second- +countable. +Proof. Now we find a countable basis for the order topology of X. As X is separable, +then let D ⊆ X be countable and dense in X, i.e. D = X. Then, define +B := {(a, b) : a, b ∈ D with a ≺ b} +together with, if there exists a minimal element m := min X and a maximal element +M := max X, the set {[m, a), (a, M], a ∈ D}. In both cases, the collection B forms a +countable base for the topology of X. To prove this, we show that for each open set +(a, b) of the order topology of X and for every x ∈ (a, b) there is an element (a′, b′) ∈ B +such that x ∈ (a′, b′) ⊆ (a, b). +Suppose x ∈ (a, b) ⊂ X, then the open intervals (a, x) and (x, b) cannot be empty +by connectedness. Hence, there exist a′ ∈ (a, x) ∩ D and b′ ∈ (x, b) ∩ D. This follows +from the fact that D = X and x ∈ D = X if and only if every open set containing x +intersects D. Then, it follows that x ∈ (a′, b′) ⊆ (a, b). +Now, when m exists, suppose x = m, then x ∈ [m, a) and this set is nonempty +by connectedness. Hence, there exists an element a′′ ∈ [m, a) ∩ D. So, it follows that +x ∈ [m, a′′) ⊆ [m, a). Analogous is the case when M exists. +By Lemma 7 and Lemma 8 , X satisfies all the assumptions of the Urysohn metriza- +tion theorem, hence X is metrizable (and, a fortiori, it is first-countable). +Lemma 9. Let A ⊆ X with X metrizable, then x ∈ A if and only if there exists a +sequence of points of A converging to x. +Proof. Suppose xn → x with xn ∈ A. Then, every neighborhood U of x contains a +point of A, i.e. x ∈ A. Conversely, we use the fact that X is metrizable.18 Let x ∈ A +18Note, once again, that here we do not need the full strength of metrizability. All we really need +24 + +and let d be a metric that induces the order topology. For every n ∈ N, we take the +neighborhood Bd(x, 1 +n), of x of radius 1 +n and we choose xn to be a point such that, for +all n, xn ∈ Bd(x, 1 +n) ∩ A. We show xn → x. Any open set U containing x contains an +ǫ-neighborhood Bd(x, ǫ) centered at x. Choosing N such that +1 +N < ǫ, then U contains +xn for all n ≥ N. +We can finally prove Lemma 6. +Proof. The if part comes trivially by definition. If there exists a sequence that converges +(monotonically) to x, then x ∈ A by Lemma 9. +Conversely, if x ∈ A, then by Lemma 9 we know that there exists a sequence in A +converging to x. Now we show that, in every totally ordered set (X, ≾), every sequence +from N → (X, ≾) has a monotone subsequence. Indeed, this is a property that has +nothing to do with the topology of X. +Let (xi)i∈N be a sequence with values in X. +We say that xk is a peak of the +sequence if h > k ⇒ xh ≾ xk (we admit a slight abuse of notation here, as it would be +better to call peak the index of the sequence, and not its image). We distinguish two +cases: if there are infinitely many peaks, then the subsequence of peaks is an infinite +non-increasing sequence and we are done. If there are only finitely many peaks, then +let i1 be the index such that xi1 is the successor of the last peak. Then, xi1 is not a +peak. Again, we find another index i2 > i1 such that xi2 ≿ xi1. Again, as xi2 is not a +peak, we can find another index i3 > i2 such that xi3 ≿ xi2 ≿ xi1. Keeping defining the +sequence in this way, we get, inductively, a non-decreasing sequence. +In conclusion, as by assumption we have a sequence (xn) ∈ AN converging to x, +this sequence admits a monotone subsequence. But, if a sequence converges to a point +x, then all of its subsequences converge to the same point x. Hence, there exists a +sequence that converges monotonically to x, proving Lemma 6. +Note that Lemma 6 could have been proven just using the notion of first countabil- +ity. Nevertheless, we decided to take the longer path of Urysohn metrization theorem to +is a countable collection of neighborhoods around x. Moreover, both connectedness and separability +are not necessary conditions. We refer the interested reader to the nice two-page paper of Lutzer [25], +that proves a linearly ordered space X is metrizable in the order topology if and only if the diagonal +∆ := {(x, x) : x ∈ X} is a countable intersection of open subsets of X × X, i.e. the diagonal is a Gδ +set. Furthermore, this condition can be shown to be equivalent to have a σ-locally countable basis, +which is a condition more in the spirit of the Nagata-Smirnov metrization theorem which requires a +σ-locally finite basis. +25 + +show how “well-behaved” a totally ordered, connected and separable topological space +can be. +Lemma 10. Let (X, ≿) be a topological space with the order topology. Let ≽ be another +order relation on X × X such that A1 and A3 hold,19 and suppose (xn), (yn) converge +to x and y respectively, and (wn), (zn) converge to w and z respectively. If for every +n ∈ N we have (xn, yn) ≽ (wn, zn) then (x, y) ≽ (w, z). +Proof. Denote the set A := {(x, y, w, z) ∈ X × X × X × X : (x, y) ≽ (w, z)} and pick +a sequence of points with values in A, that is pick (xn, yn, wn, zn) ∈ AN converging to +(x, y, w, z). By assumption, we have that xn → x, yn → y, wn → w, zn → z and this +is equivalent to (xn, yn, wn, zn) → (x, y, w, z). Indeed, a sequence in the product space +X × X × X × X converges to (x, y, w, z) if and only if it converges componentwise, i.e. +xn → x, yn → y, wn → w, zn → z. We now prove this fact. +Assume (xn, yn, wn, zn) → (x, y, w, z) in X ×X ×X ×X. Let U1, U2, U3, U4 be open +sets containing x, y, w, z, respectively. Then U1×U2×U3 ×U4 is a basis element (hence, +open) for the product topology containing (x, y, w, z). By definition of convergence, we +can find n0 such that for all n ≥ n0 we have (xn, yn, wn, zn) ∈ U1 × U2 × U3 × U4. +Thanks to the fact that projections are continuous functions, they preserve convergent +sequences and so for all n ≥ n0 we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4, i.e. +xn → x, yn → y, wn → w, zn → z. +Conversely, if xn → x, yn → y, wn → w, zn → z, let U⋆ be an open subset of +X×X×X×X such that (x, y, w, z) ∈ U⋆. By definition of product topology, we can find +U1 ⊆ X open in X, . . . , U4 ⊆ X open in X such that x ∈ U1, y ∈ U2, w ∈ U3, z ∈ U4. By +convergence, we have that for all i = 1, 2, 3, 4 there exists nki ∈ N such that for all n ≥ +nki we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4. Now pick N := max{nk1, nk2, nk3, nk4} +and for every n ≥ N we have (xn, yn, wn, zn) ∈ U1 × U2 × U3 × U4 ⊆ U⋆. Hence, by +definition of convergence, (xn, yn, wn, zn) → (x, y, w, z). +Now we want to show (x, y, w, z) ∈ A, with A closed in the product topology. +We now prove that every closed set in the product topology is sequentially closed.20 +This means we want to show that if we pick a sequence of points (xn, yn, wn, zn) with +values in A ⊆ X that is converging to a point (x, y, w, z) ∈ X, then (x, y, w, z) ∈ A. +Pick a sequence (xn, yn, wn, zn) with values in A ⊆ X that is converging to a point +19Note that the order topology and A1 are redundant assumptions. The lemma follows immediately +by continuity of ≽ alone. +20Note that when X is metrizable, a set C ⊆ X is closed ⇐⇒ C is sequentially closed. +26 + +(x, y, w, z) ∈ X. Then, let U⋆ be any neighborhood of (x, y, w, z). By convergence, +there exist an n0 ∈ N such that for all n ≥ n0 we have (xn, yn, wn, zn) ∈ U⋆ and, in +particular, (xn, yn, wn, zn) ∈ U⋆ ∩ A. Since U⋆ was an arbitrary but fixed neighborhood +of (x, y, w, z), then (x, y, w, z) is in the closure of A, i.e. (x, y, w, z) ∈ A. But A is +closed, therefore A = A, so (x, y, w, z) ∈ A, hence (x, y) ≽ (w, z). +The proof of Theorem 5 in chapter 3, as in the original version of Shapley [36], +relies on two very interesting lemmas. Similar propositions have been taken as axioms +in environments that lack the topological assumptions on the set of alternatives X. +Lemma 11. Let (w,z) be an element of X × X. If x′, x′′, y ∈ X are such that: +(x′, y) ≽ (w, z) ≽ (x′′, y) +(7) +then there exists a unique, up to indifference, x⋆ ∈ X such that +(x⋆, y) ∼ (w, z) +(8) +and x′ ≿ x⋆ ≿ x′′. +Proof. Define x0 := inf{x ∈ X : (x, y) ≽ (w, z)} and denote A := {x ∈ X : (x, y) ≽ +(w, z)} this set. The set A is nonempty as x′ ∈ A, A is bounded below by x′′ as we +have (w, z) ≽ (x′′, y) and, by transitivity and A1, we reach x ≿ x′′ for every x ∈ A. +Thus, x0 is such that x′ ≿ x0 ≿ x′′ and so x0 ∈ X by Lemma 2. Analogously, we define +x0 := sup{x ∈ X : (w, z) ≽ (x, y)} and denote B := {x ∈ X : (w, z) ≽ (x, y)} this set. +Then, B is nonempty as x′′ ∈ B, B is bounded above by x′ as we have (x′, y) ≽ (w, z) +and, by transitivity and A1, we reach x′ ≿ x for every x ∈ B. Thus, x0 is such that +x′ ≿ x0 ≿ x′′ and so x0 ∈ X by Lemma 2. +By A3, the sets A and B are closed and so, by Lemma 5, we have x0 ∈ A and +x0 ∈ B so that +(x0, y) ≽ (w, z) ≽ (x0, y) +By transitivity and by A1 we have x0 ≿ x0. +Assume now by contradiction that x0 ≻ x0. By Lemma 1 there exists x⋆ ∈ X such +that x0 ≺ x⋆ ≺ x0. But then, comparing x⋆ with (w, z), (x⋆, y) ≽ (w, z) can hold only +if x0 ∼ x⋆ ≻ x0, so x0 ∼ x⋆ and therefore x⋆ should be the infimum of A, reaching a +contradiction. Specular is the contradiction in the other case. Therefore, as there does +not exist any x⋆ ∈ X such that x0 ≺ x⋆ ≺ x0, we must conclude that x0 ∼ x0. By +transitivity and A1 we have +(x0, y) ∼ (w, z) ∼ (x0, y) +27 + +This proves the existence of x⋆ ∈ X for which (8) holds. +Let x ∈ X be any other element of X for which (8) holds. By transitivity, (x⋆, y) ∼ +(x, y). By A1, we have x⋆ ∼ x and this completes the proof. +Lemma 12. Let x, z ∈ X such that x ≻ z. Then, there exists a unique, up to indiffer- +ence, y⋆ ∈ X such that +(x, y⋆) ∼ (y⋆, z) +and x ≻ y⋆ ≻ z. +Proof. Define y0 to be the least upper bound of the set C := {y ∈ X : (x, y) ≽ (y, z)}. +This set is nonempty as if we pick y = z we have (x, z) ≽ (z, z) that by A1 is equivalent +to x ≿ z, that holds by assumption. C is also bounded from above by x as if we pick +y = x we have (x, x) ≽ (x, z) that by A1′ is equivalent to z ≿ x, that, by completeness, +contradicts the assumption of x ≻ z showing that x is an upper bound for C. Since C +is nonempty and bounded above by x, by Lemma 2 we have y0 ∈ X. +Similarly, by defining y0 to be the greatest lower bound of the set D := {y ∈ X : +(y, z) ≽ (x, y)}. This set is nonempty as if we pick y = x we have (x, z) ≽ (x, x) that +by A1′ is if and only if x ≿ z, that holds by assumption. This set is also bounded +from below by z as if we pick y = z we have (z, z) ≽ (x, z) that by A1 is if and only +if z ≿ x, that, by completeness, contradicts the assumption of x ≻ z showing that z is +a lower bound for D. Since D is nonempty and bounded below by z, by Lemma 2 we +have y0 ∈ X. +By A3 the sets C and D are closed, so by Lemma 5 we have y0 ∈ C and y0 ∈ D, +that is +(x, y0) ≽ (y0, z) and (y0, z) ≽ (x, y0) +(9) +We show now that y0 ≿ y0. Suppose, by contradiction, y0 ≻ y0. By Lemma 1 there +exists y⋆ ∈ X such that y0 ≻ y⋆ ≻ y0. Then, by definition of y0 we have (y⋆, z) ≺ (x, y⋆), +while by the definition of y0 we have (x, y⋆) ≺ (y⋆, z). This contradiction shows that +y0 ≿ y0. By A1 this is equivalent to +(y0, z) ≽ (y0, z) for all z ∈ X. +(10) +By A1′ it is also equivalent to +(x, y0) ≽ (x, y0) for all x ∈ X. +(11) +Putting together equation 9 with equations 10 and 11, we reach the loop +(y0, z) ≽ (y0, z) ≽ (x, y0) ≽ (x, y0) ≽ (y0, z). +28 + +By transitivity, we have (y0, z) ∼ (y0, z) and (x, y0) ∼ (x, y0). By A1, we conclude that +y0 ∼ y0. +We conclude proving that from A1, A2 and A3 we can derive A1′. +Lemma 13. Let X be a connected subset of a topological space. +If ≿ is complete +and transitive, ≽ is complete, transitive, satisfies A3 and A2, and jointly ≿ and ≽ +satisfy A1, then A1′ holds, that is, for all x, y, z ∈ X we have x ≿ y if and only if +(z, y) ≽ (z, x). +Proof. By contradiction, suppose A1′ fails. Then, there exist x, y, z ∈ X such that +(z, y) ≽ (z, x) and x ≺ y. We consider two cases: y ≻ z and y ≾ z. +If y ≻ z then, being (z, y) ≽ (z, x) by assumption, we have +(x, x) ∼ (y, y) ≻ (z, y) ≽ (z, x) +by A2 and A1, respectively. We apply Lemma 11 to find a w ∈ X such that +(w, x) ∼ (z, y) and x ≿ w ≿ z. +Being y ≻ x, we have +(z, z) ∼ (y, y) ≻ (x, y) ∼ (w, z) ≿ (z, z) +by A2, A1, A2, A1, respectively. This implies a contradiction in the case y ≻ z. +Assume now y ≾ z. Being y ≾ z and x ≺ y, by transitivity we have x ≺ z. We +can proceed as in the previous case, interchanging the roles of x and y and reversing +all the inequalities. +3 +The theorem +We can now state and prove Shapley’s theorem in our general version. +Theorem 5. Let X be a connected and separable subset of a topological space. If ≿ is +complete and transitive, ≽ is complete, transitive, satisfies A2 and A3, and jointly ≿ +and ≽ satisfy A1, then the pair (≿, ≽) can be represented by a continuous measurable +utility function u: X → R, that is, for each pair x, y ∈ X, +x ≿ y ⇐⇒ u(x) ≥ u(y) +(12) +and for each quadruple x, y, z, w ∈ X, +(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w). +(13) +Moreover, u is unique up to positive affine transformations. +29 + +Proof. We first prove the result when ≿ is antisymmetric. In view of Lemma 1, through- +out the proof we will consider suprema and infima of subsets of X. +Suppose X is not a singleton, otherwise the result is trivially true. Let a0, a1 ∈ X +be two distinct elements of X such that, without loss of generality, a1 ≻ a0. +Assign u(a0) = 0 and u(a1) = 1. Now we want to show that u has a unique +extension on X which is a measurable utility function for (≿, ≽). To ease notation, +denote +1 := (a1, a0) , 0 := (a0, a0) , −1 := (a0, a1). +Clearly, 1, 0, −1 ∈ X × X and, by A1 and A1′, 1 ≻ 0 ≻ −1. Then, by A2 we have +(x, x) ∼ 0 for every x ∈ X. Moreover, for every y ∈ X we have either: +(i) There exists a unique T1(y) ∈ X such that (T1(y), y) ∼ 1 +or +(ii) 1 ≻ (x, y) for all x ∈ X +Indeed, if (ii) fails, there exists x′ ∈ X such that (x′, y) ≽ 1. Since (x′, y) ≽ 1 ≽ 0 ∼ +(y, y), by Lemma 11 there exists an element T1(y) ∈ X such that (T1(y), y) ∼ 1. By +A1 and antisymmetry of ≿ , (T1(y), y) ∼ (y′, y) implies T1(y) = y′, so T1(y) is unique. +In addition, note that y ≺ T1(y). Indeed, (y, y) ∼ 0 ≺ 1 ∼ (T1(y), y), and so A1 +implies y ≺ T1(y). In a similar way as before, for every y ∈ X we have either: +(i.bis) There exists a unique T−1(y) ∈ X such that (T−1(y), y) ∼ −1 +or +(ii.bis) −1 ≺ (x, y) for all x ∈ X +Indeed, if (ii.bis) fails, there exists x′ ∈ X such that (x′, y) ≼ −1. Since (x′, y) ≼ −1 ≼ +0 ∼ (y, y), by Lemma 11 there exists an element T−1(y) ∈ X such that (T1(y), y) ∼ −1. +By A1 and antisymmetry of ≿ , (T−1(y), y) ∼ (y′, y) implies T−1(y) = y′, so T−1(y) is +unique. +In addition, note that T−1(y) ≺ y. Indeed, (T−1(y), y) ∼ −1 ≺ 0 ∼ (y, y), and so +A1 implies T−1(y) ≺ y. +Now define a2 := T1(a1) if (i) holds for y = a1, i.e. if there exists a unique +T1(a1) ∈ X such that (T1(a1), a1) ∼ 1. Similarly, set a3 := T1(a2) if (i) holds for y = a2, +and continue in this way till (if ever) occurs y = an for which (ii) holds, i.e. 1 ≻ (x, an) +for every x ∈ X. Analogously, we define a−1 := T−1(a0) if (i.bis) holds for y = a0, set +a−2 := T−1(a−1) if (i.bis) holds for y = a−1, and continue in this way till (if ever) occurs +y = a−n for which (ii.bis) holds. +30 + +Now define A := {. . . , a−2, a−1, a0, a1, a2, . . . }, with +· · · ≺ a−2 ≺ a−1 ≺ a0 ≺ a1 ≺ a2 ≺ . . . +The set A can be finite or infinite in either direction. If we consider now a sequence that +from an index set Pa ⊆ Z maps to A, we define the following function a : Pa ⊆ Z → A. +Now we start to extend u to A. Define the following: +u(ap) = p +for every p ∈ Pa. +Clearly, we have (12), i.e. x ≿ y if and only if u(x) ≥ u(y) for every x, y that are images +of the sequence a, so (12) holds on A. +Now we show that (13) holds whenever x, y, z, w ∈ A ⊂ X, say x = ap, y = aq, z = +ap−d where p, q, p − d ∈ Pa. Without loss of generality, assume d ≥ 0. We first prove +the “equality” case of (13), that is +(x, y) ∼ (z, w) ⇐⇒ u(x) − u(y) = u(z) − u(w) +(14) +By construction we have +(ap, ap−1) ∼ 1 ∼ (aq, aq−1) +so, by transitivity and A2, we have: +(ap, aq) ∼ (ap−1, aq−1) +Iterating this procedure finitely many times we reach: +(x, y) = (ap, aq) ∼ (ap−d, aq−d) = (z, aq−d) +(15) +By transitivity, (z, aq−d) ∼ (z, w) and so, by A1 aq−d = w, so that u(aq−d) = u(w). +By definition of u we can write +u(x) − u(y) = u(ap) − u(aq) = p − q = u(ap−d) − u(aq−d) = u(z) − u(w) +thus proving (14). Next we prove +(x, y) ≻ (z, w) ⇐⇒ u(x) − u(y) > u(z) − u(w) +(16) +By transitivity, (z, aq−d) ≻ (z, w) and so, by A1′, w ≻ aq−d, so that u(w) > u(aq−d). +By definition of u, from (15) we can write +u(x) − u(y) = u(ap) − u(aq) = p − q = u(ap−d) − u(aq−d) > u(z) − u(w) +thus proving (16). +Summing up, both (12) and (13) hold on the terms of the set A. Using Lemma +12, now we want to extend u to the points of X that lie between terms of the set A. +Set b0 := a0 and since a1 ≻ a0, by Lemma 12 there exists b1 ∈ X, with a1 ≻ b1 ≻ a0, +31 + +such that +(a1, b1) ∼ (b1, a0) +Now build the set B := {. . . , b−2, b−1, b0, b1, b2, . . . }, with +· · · ≺ b−2 ≺ b−1 ≺ b0 ≺ b1 ≺ b2 ≺ . . . +based on b0, b1, in the same way we constructed A from a0, a1. Also here, we can define +a sequence that from an index set Pb ⊆ Z maps to B, that is, we define the following +function b : Pb ⊆ Z → B. +By construction we have +(b2, b1) ∼ (b1, b0) +Together with (a1, b1) ∼ (b1, a0), by transitivity we have (b2, b1) ∼ (a1, b1). By A1, +b2 = a1. Analogously, one can verify that +b2p = ap for every p ∈ Pa +(17) +So, the terms of the set B lie between the terms of the set A, i.e. the set B refines +A and we can write +A ⊆ B +(18) +Denote now c0 := b0 = a0 and we let c1 ∈ X be that element provided by Lemma +12 such that (b1, c1) ∼ (c1, b0). In the same way we constructed B from A, we can +construct, from B, a third set C := {. . . , c−2, c−1, c0, c1, c2, . . . }, with +· · · ≺ c−2 ≺ c−1 ≺ c0 ≺ c1 ≺ c2 ≺ . . . +based on c0, c1. We can see that +c2p = bp for every p ∈ Pc +where Pc ⊆ Z is the collection of indexes of the sequence c : Pc ⊆ Z → C. +The set C refines B +B ⊆ C +(19) +We keep iterating this process, constructing sets that refine one another and, for +ease of notation, we denote them in the following way: +A0 := A +and +a0 +p := ap ∈ A0 +A1 := B +and +a1 +p := bp ∈ A1 +A2 := C +and +a2 +p := cp ∈ A2 +· · · +32 + +These sets generalize the inclusions (18) and (19) as follows: +A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ . . . +(20) +So, in general, an +p for p ̸= 1 is obtained from the construction of (i) and (ii), applied to +the points a0, an +1. The term an +1, for n > 0, is the “midpoint” between an−1 +1 +and a0, that +exists by Lemma 12. By iterating the construction of (17), we have that +p +2n = q +2m =⇒ an +p = am +q +In the spirit of (20), we extend u to all points in A∞ := �∞ +n=1 An by: +u(an +p) = p +2n +for all an +p ∈ An +Relations (12) and (13) hold in this extended domain: given x, y, z, w ∈ �∞ +n=1 An, +just take n large enough so that they become, up to indifference, terms of the set An +and proceed in the same exact way as we did for the set A0. +To complete the construction of u we only remain to show A∞ is dense in X, +that is A∞ = X. We first show that none of the sets An has, for its sequences of +points an, a point of accumulation in X. Indeed, fix n and suppose by contradiction +that an +pk converges monotonically to a⋆ ∈ X, where, without loss of generality, we +assume an +pk ↑ a⋆ with a⋆ ∈ X, i.e. (pk) is an increasing sequence of integers. Denote +1n := (an +1, a0) and we have, for every k ∈ N, +(an +1+pk, an +pk) ≽ 1n ≻ 0 +By Lemma 10, we have (a⋆, a⋆) ≽ 1n. So, by transitivity, we reach (a⋆, a⋆) ≻ 0, a +contradiction. We conclude that, fixed n, none of the sequences an with values in An +has a limit point in X. +To prove A∞ = X, the implication A∞ ⊆ X is trivial by construction. Now we +want to show A∞ ⊇ X, that is all the elements of X belong to the closure of A∞ +as well. Fix x ∈ X such that, without loss of generality, x ≿ a0. For n ≥ 1, define +yn := sup{y ∈ An : x ≿ y}. Note that a0 ∈ {y ∈ An : x ≿ y}, so this set is nonempty +and we can write x ≿ yn ≿ a0. By Lemma 2, yn ∈ X. Note further that, as shown +before, An cannot have accumulation points in X so, as long as yn ∈ X, it follows +yn cannot be an accumulation point of An. So, yn must belong to An and we denote +yn := an +pn. As a result, we have: +an +pn−k ≾ x ≺ an +pn+k for every k > 0 +(21) +We also have that +1n ≻ (x, yn) +(22) +33 + +Indeed, if (22) were not true, then (x, an +pn) ≽ 1n. We consider two cases: an +1+pn ≻ x or +an +1+pn ≾ x. If an +1+pn ≻ x, then, thanks to A1, we reach the following contradiction: +1n ∼ (an +1+pn, an +pn) ≻ (x, an +pn) ≽ 1n +(23) +So an +1+pn ≾ x, but this contradicts (21), that is, it contradicts yn to be the supremum. +Thus, (22) holds. In particular, by A1 and A2, we can write (x, yn) ≽ (yn, yn) ∼ 0, +leading to +1n ≻ (x, yn) ≽ 0 +(24) +Now, when n → ∞, as the sets An+1 ⊇ An ⊇ An−1 . . . are nested one into the +other by (20), we can write, for every n ≥ 1, yn ≾ yn+1 ≾ x. Thus, the points yn form +a non-decreasing sequence that is bounded from above by x. Call y⋆ the limit of this +sequence, that is well-defined by Lemma 1. Since a0 ≾ y⋆ ≾ x, by Lemma 2 it follows +that y⋆ ∈ X. In particular, by Lemma 6 we have y⋆ ∈ A∞, because, for every fixed +n ≥ 1, yn is a term of the sets An, and so (yn) ∈ AN +∞. +As to the 1n terms, for n fixed, we see that +1n ∼ (an +2, an +1) ∼ (an−1 +1 +, an +1) +We also have that, for every n ≥ 1, a0 ≾ an+1 +1 +≾ an +1. +Thus, the points an +1 form, for n → ∞, a non-increasing sequence that is bounded +from below by a0. Call a⋆ the limit of this sequence, that is well-defined by Lemma 1. +Since a0 ≾ a⋆ ≾ a1, by Lemma 2 we have a⋆ ∈ X. +Consider now (an−1 +1 +, an +1) and (x, yn). By Lemma 10 and from (24) it follows that +(a⋆, a⋆) ≽ (x, y⋆) ≽ 0 +Since, by A2, (a⋆, a⋆) ∼ 0, by transitivity (x, y⋆) ∼ 0, so that x ∼ y⋆, i.e. x = y⋆ as ≿ +is antisymmetric. +Since x was arbitrarily chosen in X and y⋆ ∈ A∞, we can conclude x ∈ A∞, so +that A∞ = X. Therefore, we can extend u by continuity to the whole set X by setting +u(x) = lim +n→∞ u(xn) +if (xn) ∈ AN +∞ converges monotonically to x. Note that u : X → R is well-defined. +Indeed, to prove it is well-posed we show that if xn and yn are two sequences that +converge to x, then limn→∞ u(xn) = limn→∞ u(yn). This follows easily by continuity of +u.21 In light of Lemma 6, it is easy to see that u satisfies (12) and (13). +As to uniqueness, observe that any other u that satisfies (12) and (13) can be +21Recall that in every topological space X continuity implies sequential continuity. The converse +holds if X is first-countable. +34 + +normalized so that u(a0) = 0 and u(a1) = 1. So, u must agree on u at each step of +the constructive procedure for u just seen. Indeed, for a given u : X → R, define the +following positive affine transformation f : Im(u) → R such that +f(x) := +x − u(a0) +u(a1) − u(a0) +It is immediate to see that, for the equivalent utility function �u := f ◦ u, we have +�u(a0) = 0 and �u(a1) = 1. +Summing up, we proved Theorem 5 if ≿ is antisymmetric. +Now we drop this +assumption. Let X/∼ be the quotient space with respect to the equivalence relation +∼. The set {x ∈ X : x ∼ y} is a closed set in X by Lemma 4, so (X/∼, ˜≿) is a totally +ordered connected and separable subset of a topological space, where ˜≿ is the total +order induced by the weak order ≿.22 Therefore, the orders ≿ and ≽ induce orders ˜≿ +and ˜≽ on the quotient set X/∼, by setting, for all [x], [y] ∈ X/∼ +[x] ˜≿ [y] ⇐⇒ x ≿ y +and, for all [x], [y], [z], [w] ∈ X/∼ +([x], [y]) ˜≽ ([z], [w]) ⇐⇒ (x, y) ≽ (z, w) +It is routine to show that the orders ˜≿ over X/∼ and ˜≽ over X/∼ × X/∼ inherit the +same properties of ≿ and ≽ used in the theorem. So, by what has been proved so far, +there exists ˜u : X/∼ → R that satisfies (12) and (13) for (˜≿, ˜≽). Let π : X → X/∼ be +the quotient map. Then, the function u : X → R defined as u = ˜u ◦ π is a well-defined +measurable utility function, i.e. it is easily seen to satisfy (12) and (13) for (≿, ≽). +To conclude, we show that u satisfies (12) and (13). If x ∼ y then [x] = [y] and, +by the theorem we have just proved, ˜u([x]) = ˜u([y]), which is (˜u ◦ π)(x) = (˜u ◦ π)(y), +and so u(x) = u(y). If x ≻ y, then [x] ≻ [y], which implies ˜u([x]) > ˜u([y]), which is +(˜u ◦ π)(x) > (˜u ◦ π)(y), and so u(x) > u(y). +Conversely, assume u(x) ≥ u(y) and suppose by contradiction x � y that, by +completeness, is y ≻ x. If u(x) = u(y) then ˜u([x]) = ˜u([y]) ⇐⇒ [x] = [y] ⇐⇒ x ∼ y, +a contradiction. If u(x) > u(y) then ˜u([x]) > ˜u([y]) ⇐⇒ [x] > [y] ⇐⇒ x ≻ y, a +contradiction. Hence, (12) holds for u. +By definition, we have that ([x], [y]) ≽ ([z], [w]) +⇐⇒ +(x, y) ≽ (z, w), for all +[x],[y],[z],[w] ∈ X/∼. So, we can write (x, y) ≽ (z, w) ⇐⇒ ([x], [y]) ≽ ([z], [w]) ⇐⇒ +˜u([x])−˜u([y]) ≥ ˜u([z])−˜u([w]) ⇐⇒ u(x)−u(y) ≥ u(z)−u(w). Hence, also (13) holds +for u. +22That is, ˜≿ := ≿ /∼ ⊆ X/∼ × X/∼. +35 + +This completes the proof of Theorem 5. +Graphically, we can build the following diagram to represent our construction. +X +X/∼ +R +π +u +˜u +References +[1] Adams, E.W. 1960. “Survey of Bernoullian utility theory.” Mathematical Thinking +in the Measurement of Behavior, edited by Solomon, H., 151–268. Glencoe. +[2] Allais, M. 1943. “A la Recherche d’une Discipline Economique. L’Economie Pure.” +Ateliers Industria, Paris. +[3] Allais, M. 1979. “The so-called Allais paradox and rational decisions under uncer- +tainty.” Expected Utility Hypotheses and the Allais Paradox, edited by Allais, M. +and Hagen, O., 437-681. +[4] Allais, M. 1994. “The fundamental cardinalist approach and its prospects.” Cardi- +nalism, edited by Allais, M. and Hagen, O., 289-306. Kluwer Academic Publishers. +[5] Alt, F. 1936. “¨Uber die M¨assbarkeit des Nutzens.” Zeitschrift f¨ur National¨okonomie +7: 161-169. +[6] Banakh, T., Gutik, O., Potiatynyk, O., and Ravsky, A. 2012. “Metrizability of +Clifford topological semigroups.” Semigroup Forum 84: 301–307. +[7] Cantor, G. 1895. “Beitr¨age zur Begr¨undung der transfiniten Mengenlehre.” Mathe- +matische Annalen 46: 481–512. +36 + +[8] Debreu, G. 1954. “Representation of a preference ordering by a numerical function.” +Decision Process, edited by Thrall, R.M., Davis, R. L. and C.H. Coombs, 159-165. +John Wiley and Sons, New York. +[9] Debreu, G. 1964. “Continuity properties of a Paretian utility.” International Eco- +nomic Review 5: 285-293. +[10] Edgeworth, F.Y. 1881. “Mathematical Psychics.” London, Kegan Paul. +[11] Eilenberg, S. 1941. “Ordered Topological Spaces.” American Journal of Mathe- +matics 63: 39-45. +[12] Ellingsen, T. 1994. “Cardinal utility: a history of hedonimetry.” Cardinalism, +edited by Allais, M. and Hagen, O., 105-165. Kluwer Academic Publishers. +[13] Fechner, G.T. 1860. “Elemente der Psychophysik.” Leipzig, Breitkopf und H¨artel. +[14] Feng, Z., and Heath, R. 2008. “Metrizability of topological semigroups on linearly +ordered topological spaces.” Topology Proceedings 32: 83-88. +[15] Fishburn, P.C. 1968. “Utility theory.” Management Science 14: 335-378. +[16] Fishburn, P.C. 1970. “Utility Theory for Decision Making.” John Wiley and Sons, +New York. +[17] Fishburn, P.C. 1976. “Cardinal utility: +An interpretive essay.” Rivista Inter- +nazionale di Scienze Economiche e Commerciale 22: 1102-1114. +[18] Frisch, R. 1926. “Sur un probleme d’economie pure.” Norsk Matematisk Forenings +Skrifter 16: 1-40. +[19] Hicks, J. R., and Allen, R. G. D. 1934. “A Reconsideration of the Theory of Value.” +Economica 1: 52-76. +[20] Hutcheson, P. 1728. “An Essay on the Nature and Conduct of the Passions and +Affections.” edited by Osborn, J., and Longman, T., London. +[21] Jevons, W.S. 1871. “The Theory of Political Economy.” London and New York, +Macmillan. +[22] Krantz, D.H., Luce, R.D., Suppes, P., and Tversky, A. 1971. “Foundations of +Measurement.” New York and London, Academic Press. +37 + +[23] Lange, O. 1934. “The determinateness of the utility function.” Review of Economic +Studies 1: 218-224. +[24] Luce, R.D., and Suppes, P. 1965. “Preference, utility and subjective probability.” +edited by Luce, R.D., Bush, R.R., and Galanter, E. Handbook of Mathematical +Psychology 3. New York: Wiley. +[25] Lutzer, D.J. 1969. “A metrization theorem for linearly orderable spaces.” Proceed- +ings of the American Mathematical Society 22: 557-558. +[26] Majumdar, T. 1958. “Behaviourist cardinalism in utility theory.” Economica 25: +26-33. +[27] Manzoni, A. 1834. “Del sistema che fonda la morale sull’ utilit`a.” Osservazioni +sulla morale cattolica., in Opere Varie, 1845, edited by Giuseppe Redaelli, Milano. +[28] Marinacci, M. 2019. “Utility theory.” Unpublished. +[29] Moscati, I. 2019. “Measuring Utility: From the Marginal Revolution to Behavioral +Economics.” Oxford University Press, Oxford. +[30] Munkres, J.R. 2000. “Topology.” 2nd ed., New York, NY: Pearson. +[31] Pareto, V. 1906. “Manuale di Economia Politica con una Introduzione alla Scienza +Sociali.” Societ`a Editrice Libraria, Milano +[32] Pareto, V. 1911. “Economie mathematique” Encyclopedie des Sciences Mathema- +tiques. Tome 1, vol. 4, Fasc. 4, Paris. English translation: 1955 “Mathematical +Economics.” International Economic Papers 5: 58-102. +[33] Ramsey, F. 1926. “Truth and probability.” In Ramsey, F. 1931. “The Foundations +of Mathematics and Other Logical Essays.” New York, Harcourt Brace and Co. +[34] Savage, L.J. 1954. “The Foundations of Statistics.” New York, John Wiley and +Sons. +[35] Sen, A.K. 1982. “Choice, Welfare and Measurement.” Blackwell, Oxford. +[36] Shapley, L.S. 1975. “Cardinal Utility from Intensity Comparisons.” RAND Report, +R-1683-PR. +38 + +[37] Sørensen, A. 2008. “Deontology - born and kept in servitude by utilitarianism.” +Danish Yearbook of Philosophy 43: 69-96. +[38] Stark, W. 1952. “Jeremy Bentham’s Economic Writings.” 1-3, London, George +Allen and Unwin. +[39] Stigler, G. 1950. “The development of utility theory.” Journal of Political Economy +58: 307-327 and 373-396. +[40] Suppes, P., and Winet, M. 1955. “An axiomatization of utility based on the notion +of utility differences.” Management Science 1: 186-202. +[41] Von Neumann, J., and Morgenstern, O. 1947. “Theory of Games and Economic +Behavior.” 2nd ed., Princeton, Princeton University Press. +[42] Weber, E.H. 1846 “Die Lehre vom Tastsinne und Gemeingef¨uhle auf Versuche +gegr¨undet.” Braunschweig, Wieweg und Sohn. +39 + diff --git a/GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/load_file.txt b/GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5667bee8d4c8f75467230d57de9556d3f9374d58 --- /dev/null +++ b/GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/load_file.txt @@ -0,0 +1,1007 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf,len=1006 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='01271v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='GN] 15 Dec 2022 On the notion of measurable utility on a connected and separable topological space: an order isomorphism theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='∗ Gianmarco Caldini 7 February 2020 Abstract The aim of this article is to define a notion of cardinal utility function called measurable utility and to define it on a connected and separable subset of a weakly ordered topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The definition is equivalent to the ones given by Frisch in 1926 and by Shapley in 1975 and postulates axioms on a set of alternatives that allow both to ordinally rank alternatives and to compare their utility differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' After a brief review of the philosophy of utilitarianism and the history of utility theory, the paper introduces the mathematical framework to represent intensity comparisons of utility and proves a list of topological lemmas that will be used in the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Finally, the article states and proves a representation theorem, see Theorem 5, for a measurable utility function defined on a connected and sep- arable subset of a weakly ordered topological space equipped with another weak order on its cartesian product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Under some assumptions on the order relations, Theorem 5 proves existence and uniqueness, up to positive affine transformations, of an order isomorphism with the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' ∗I am grateful to Professor Massimo Marinacci for letting me know about the open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1 Introduction Together with notions such as value, money, market and economic agents, utility has been one of the most controversial concepts in the whole history of economic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The most important debate can be considered the one around the question whether it is possible to define a clear and rigorous concept of utility and an appropriate notion of unit of measurement for utility, seen as a quantity like the physical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In the first chapter we will give a short introduction to the evolution of the concept of utility from both a philosophical and a historical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Our treatment is far from being exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For an extensive treatment of history of utility and utility measurements, we refer the interested reader to Stigler [39], Majumdar [26], Adams [1], Luce and Suppes [24], Fishburn [15], [16], [17] and Moscati [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The second chapter will shift from the descriptive part to more formal concepts and will be used to introduce the usual mathematical framework of decision theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, we will introduce definitions and axioms that will enable us to represent comparisons of the intensity that a decision maker feels about the desirability of different alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For this aim, we will follow the construction of Suppes and Winet [40] and Shapley [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The third and last chapter will be entirely devoted to the proof of Shapley’s theo- rem, extending the domain of alternatives X from a convex subset of R to a connected and separable subset of a topological space, hence providing a generalization of his theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Our intended goal is to define a rigorous notion of a specific kind of cardinal utility function, not only able to rank alternatives, but also to compare utility differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In particular, we define a “twofold” utility function in line with the primordial axiomatization of Frisch [18], calling it a measurable utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In mathematical terms, we will prove a specific order-isomorphism theorem between a totally ordered, connected and separable subset of a topological space and the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1 Philosophy and history of utility theory Theory of felicity, theory of justice, theory of morality, theory of virtue and theory of utility are among the most important theories of moral philosophy and, as such, they are constantly sources of questions that often do not find an immediate answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' When a human being acts, or when she makes a decision, she is, at the same time, looking for 1 justifications, either positive or normative, for the decision she has just made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We, as human beings, are constantly trying to prove that what we did was the best thing to do, in some well-defined sense, or, at least, the less harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' These justifications take into account the means, the ends and all the possible paths we have to reach our goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moral philosophy is the science that comes into place when we formulate questions about the ends, the means and the possible ways to achieve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moral philosophy is essentially composed by principles, also called norms, on what is good and what is bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' They allow to define and to judge human actions, means and ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Sometimes, norms take the form of universal laws to which all human beings are subjected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, the formulation of moral laws or rules that prescribe what a single agent should do or not do are intrinsically tied with history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Historical experiences determine our vision of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Our moral philosophy is the result of different heritages that formed a common culture in which values like human respect, an idea of equality between human beings and impartiality are among the most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Together with this general definition of morality, there exist the similar concepts of ethics and of role morality - a specific form of professional morality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It was Jeremy Ben- tham, in an unfinished manuscript which was posthumously published in 1834, to define the neologism deontology in the title of his book Deontology or the Science of Morality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The manuscript stated, for the first and only time, the particular aspects of Bentham’s utilitarian theory as moral philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This passage is clearly mentioned in Sørensen [37]: [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='] pointing out to each man on each occasion what course of conduct promises to be in the highest degree conducive to his happiness: to his own happiness, first and last;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' to the happiness of others, no farther than in so far as his happiness is promoted by promoting theirs, than his interest coincides with theirs (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In this passage we can see how Bentham considered deontology to be primarily aimed at one’s own private felicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, this does not bring any selfish concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Bentham’s goal can be identified with the objective study and measurement of passions and feelings, pleasures and pains, will and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Among these particular pleasures are those stemming from sympathy - in Adam Smith’s sense - and they include the genuine pleasure being happy for the good of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In this light, Bentham spent his life in search of the cardinal principle of ethics and he found it in Epicurean ethics of hedonism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hedonism comes from Greek ῾ηδον´η, which means pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, classic utilitarianism, founded on hedonism, started from 2 the principle that pleasure is an intrinsic positive value and sorrow is an intrinsic neg- ative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is, for this reason, somehow curious that Bentham conception, founded on pleasure, had been called utilitarianism, from the simple observation that what is useful is not necessarily pleasant or providing pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We need always to take into account that the term utility is intended in a functional sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' what gives utility is what contributes the most to the individual, or universal, pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Classical utilitarian philosophers considered utilitarianism well-founded and realistic thanks to the fact that it is based on pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is well-founded as its norms are jus- tified by an intrinsic, absolute value, that does not need any further justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is realistic because they thought human being to ultimately seek the maximum pleasure and the minimum sorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' More specifically, human beings try to choose the action that will provide the maximum excess of pleasure against grief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For Bentham, what really matters is the total amount of pleasure, intended as the total excess of pleasure against sorrow: the only reasons for human actions are the quests for pleasure, avoiding sorrow: they are the sources of our ideas, our judgments and our determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Human moral judgments become statements on happiness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' pleasure (or felicity) is good and sorrow is bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Utilitarian moral can be considered as a “calculated hedonism”, that carefully evaluates the characteristics of pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Wise is the man that is able to restrain from an immediate pleasure for a future good that, in comparison, will be more beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' On the other side, being able to evaluate the positive or negative consequences of an action without making mistakes is fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, the correct utilitarian person should reach some kind of “moral arithmetic” that allows the correct calculations to be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Far from being a unanimously accepted doctrine, we cannot forget to mention that Alessandro Manzoni wrote an essay [27] in which he strongly criticized Bentham’s utilitarianism, saying that it is utterly wrong to think that human beings build their moral values judgment of their actions on utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='1 From this explanation of utilitarianism, Bentham’s evaluation criterion of actions follows as an immediate corollary: the maximum happiness for the maximum number of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Again, happiness is intended as state of pleasure, or absence of grief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, individual pleasure becomes no more the ultimate goal: it is the universal pleasure to be hegemonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='2 1Manzoni [27] wrote: ”Non ci vuol molto a scoprir qui un falso ragionamento fondato sull’alterazione d’un fatto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Altro `e che l’utilit`a sia un motivo, cio`e uno de’ motivi per cui gli uomini si determinano nella scelta dell’azioni, altro `e che sia, per tutti gli uomini, il motivo per eccellenza, l’unico motivo delle loro determinazioni (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='775).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 2This tension between individual pleasure and universal pleasure is one of the principal difficulties 3 This view of utilitarianism admits, at least in our minds, the conception of the existence of a scale of pleasure in which pleasure and sorrow can be added and sub- tracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In other words, the idea of a calculus of felicity and grief is not completely absurd, both in intrapersonal and interpersonal compensations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='1 Brief history of utilitarianism Although it is possible to find utilitarian reasonings in Aristotele’s works, it is com- monly agreed that the beginning of the history of utility can be identified with 18th century moral philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' To be even more specific, Bentham’s ideas were not isolated, since they were already present in works by his illuministic predecessors like Richard Cumberland, Francis Hutcheson and Cesare Beccaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Especially Hutcheson [20] had already defined good as pleasure and good objects as objects that create pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The novelty of Bentham was to treat pleasure as a measurable quantity, thus making the utilitarian doctrine directly applicable to issues like tax policies and legislation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In- deed, not only did Bentham argue that individual pleasure was measurable, but also that happiness of different people could be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Stark [38] cited in his article Bentham’s writings in the following way: Fortunes unequal: by a particle of wealth, if added to him who has least, more happiness will be produced, than if added to the wealth of him who has most (vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Stark [38] continues: The quantity of happiness produced by a particle of wealth (each particle being the same magnitude) will be less and less every particle (vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 113).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is easy to see how this last concept and the well-known idea of decreasing marginal utility are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In the pioneering work of Jevons [21], utility functions were the primitive mathe- matical notion to formalize and quantify Bentham’s calculus of pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Utility func- tions were tools to measure and scale the amount of well-being of human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It seems clear, at this point, how the starting role of utility functions was cardinal,3 in the sense that utility or, better, pleasure differences were well-founded and realistic notions with a strong moral philosophy justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' of utilitarian moral philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 3Note that before the work of Hicks and Allen [19], economists spoke about measurable utility and not of cardinal utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 4 Summing up, in the beginning, utility functions were designed for the mere purpose of a calculus of pleasure and sorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' However, even if the philosophical concept made sense, the difficulties in the quantification of any experimental measurement of pleasure led cardinal utility theory to be seen more just like a thought process rather than a science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' However, utility theory did not rise from philosophy alone, but it was object of study of other sciences such as statistics, with the so-called St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Petersburg paradox, and psychophysics, the study of physical stimuli and their relation to sensory reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' These two phenomena can be considered the starting point of the law of decreasing marginal utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It was Nicolas Bernoulli that, originally, invented what is now called the St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Petersburg puzzle, which offered the theoretical explanation for the law of de- creasing marginal utility of wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The standard version of the puzzle is the following: a fair coin is tossed until it lands “head” on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' At that point, the player wins 2n dollars, where n is the number of times the coin was flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' How much should one be willing to pay for playing this game?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In other words, what is the expected value of the game, given the probability of “head” being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='5?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The mathematical answer is ∞ � i=1 1 2i · 2i = 1 + 1 + · · · = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The only rationale for this conundrum is that, if it makes sense to maximize expected utility and if people are willing to participate to the St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Petersburg game for only a finite amount of money, then their marginal utility as a function of wealth must be, somewhere, decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Neither Bentham nor Bernoulli thought as decreasing marginal utility as a phe- nomenon in need of scientific justifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, this came as an immediate consequence from the psychophysical theories discovered by Weber [42] and generalized by Fechner [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One of the most important questions posed by psychophysics is what is the functional link between different degrees of a given stimulus and a given sensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' What Weber did was an in-depth study to try to measure the smallest detectable change (also called “just noticeable difference” or “minimum perceptible threshold”) in stimuli like heat, weight and pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, Fechner took this “just noticeable difference” as a unit of measurement, constructing a scale for subjective sensations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' From their studies we now have the so-called Weber’s law and Fechner’s laws: the former states that the relative increase of a stimulus needed to produce a just noticeable change is 5 constant and, the latter, that the magnitude of sensation is a logarithmic function of the stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In conclusion, if wealth is a stimulus, then Benthamian utility must be the cor- responding sensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In this light, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Petersburg puzzle can be seen as just one materialization of these laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' At the end of 19th century, the marginalist revolution paved the way for an ordinal approach to the notion of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In fact, this was because one of the main economic problems of late 19th century was the need of a theory of demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One of the leading figures that founded neoclassical theory with scientific and analytic rigor was Vilfredo Pareto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pareto is considered the father of the so-called ordinal approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It was a notion of utility that was purely comparative and it left out from the theory the initial idea for which utility theory was developed: the existence of psychophysical and physiological substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pareto’s theory was so successful that was considered a revolution in the notion of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The ordinal approach was extremely successful because it solved the classic consumer problem based on indifference curves, and the notion of utility had a central role in its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The key aspect was the replacement of marginal utility a notion that was meaningless in an ordinal approach - with the trick of marginal rate of substitutions along indifference curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Interesting are the writings of Francis Edgeworth [10] and Pareto [31], starting from very different assumptions and arriving at different conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Edgeworth’s main contribution can be summarized in the synthesis of Bentham’s utilitarianism and Fechner’s psychophysics: his ideas were based on the unit of utility seen as a just perceivable increment of pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, he was interested also in an inter-personal unit of utility to be able to carry out welfare comparisons among people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Edgeworth was completely aware of the impossibility of testing these implications, but he was a strong supporter of the idea of possible comparisons of happiness among people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pareto, on the other hand, denied Edgeworth’s intuition of comparisons of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Instead, Pareto [31] reckoned the theoretical possibility of a cardinal notion of utility, seen as the limit of the purely comparative notion he developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, he also argued that such a notion of perfect precision is not attainable and that pleasure is only imperfectly measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='4 Summing up, Edgeworth’s and Pareto’s ways of conceiving measurable utility must be differentiated and utility theory is still today based on the Paretian notion mainly because of its use in the theory of demand and in the general 4However, Pareto [32] writes: “There is no reason for not accepting it [cardinal utility], with the reservation that it must be verified by the results deduced from it (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 73).” 6 equilibrium theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In 1950, ordinalism was the well-established mainstream ideology in utility theory and the cardinal notion of utility was almost completely abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, the purely comparative approach was not convincing everyone, mainly because people’s introspection suggested the existence something more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One of the main supporters of cardinalism was Maurice Allais who explicitly wrote in [4]: The concept of cardinal utility [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='] has almost been rejected the literature for half a cen- tury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This rejection, based on totally unjustified prejudices, deprived economic analysis of an indispensable tool (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Allais [4] admits that the theory of general economic equilibrium can be fully de- scribed in an ordinal world, but he immediately lists a series of theories that cannot be adequately developed without a rigorous and well-defined concept of cardinal utility and interpersonal comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Some examples are the theory of dynamic evolution of the economy, the theory of fiscal policy, of income transfers, of collective preferences, social welfare analysis and political choices, of risk, of insurance and the theory of cooperative games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, Allais goes even further in his defense of cardinal utility, arguing that even the theory of demand could become more intuitive - and with a simpler exposition - if we could appeal to a notion of intensity of preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In any case, as long as the conclusions of price theory do not change significantly using the ordinal and the cardinal approach, we should prefer the purely comparative approach by Occam’s razor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But the problems with group decision making, social choice theory and cooperative game theory still cannot be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, while classical economists considered distributional problems as a fundamental part of economic science, the ordi- nalist approach to utility theory refused completely to deal with questions that involved interpersonal comparisons of welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Economists became more interested in positive statements, rather than normative ones and the accent was put on efficiency, rather than equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This was the case of the optimum allocations in the sense of Pareto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For a complete overview of the main issues of welfare economics, the main problems with an ordinal approach and the main literature, we refer the interested reader to Sen [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, one of the main problems, still in the 21st century, is how is it possi- ble to understand the intuitive tool of introspection to develop a rigorous theory that economists can apply in their models and explain economic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The solution does not exists yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In the last years, the issue started getting the attention of few deci- sion theorists mainly because of the powerful developments in the field of neuroscience 7 and the new discipline of neuroeconomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' These fields of cognitive sciences are going into the direction of overcoming the main difficulty of the founders of utilitarianism: the difficulty of carrying out experiments on pleasure and pain and the construction of a rigorous and well-defined scale of pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, nothing is clear yet, mainly because also the theoretical concept of cardinal utility is still vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Cardinal utility is still used as a name for a large number of formally distinct concepts and it misses a precise and well-established definition that can be applied in decision-theoretic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' During the 20th century a lot of methodologies to try to define a concept of measurement of human sensation have been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A scale is a rule for the assignment of numbers to aspects5 of objects or events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The result was the development of a full taxonomy of scales, with scales that differ in terms of higher precision of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For an extensive treatment of the theory of measurement we refer the interested reader to Krantz et alii [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The issue of having a rigorous definition for cardinal utility was not solved by the theory of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It was just translated in a different language: what is the suitable scale for measuring a given aspect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The definition of a unit of measurement for utility was not an easy task to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Even in physics, where experiments can be carried out with relatively high precision, the way a unit of measurement is defined is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One meter was originated as the 1/10-millionth of the distance from the equator to the north pole along a meridian through Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, the International Bureau of Weights and Measures, founded in 1875, defined the meter as the distance of a particular bar made by platinum and iridium kept in S`evres, near Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' More recently, in 1983, the Geneva Conference on Weights and Measures defined the meter as the distance light travels, in a vacuum, in 1/299,792,458 seconds with time measured by a cesium-133 atomic clock which emits pulses of radiation at very rapid and regular intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Increases in science allow the unit of measurement to be duplicated with a better and better level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The comparison with the unit of measurement of the quantity utility can be carried out with the philosophical question whether it is, for some esoteric reason, intrinsically impossible to measure human beings’ pleasure or whether economic science and neuroeconomics are so underdeveloped that we still have 5For example: hardness, length, volume, density, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 8 very poor precision in measuring human felicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='6 The same comparison can be done with light (or heat, color and wave lengths, as it is mentioned by von Neumann and Morgenstern in [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For example, temperature was, in the original concept, an ordinal quantity as long as the concept warmer was known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, the first transition can be identified with the development of a more pre- cise science of measurement: thermometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' With thermometry, a scale of temperature that was unique up to linear transformations was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The main feature was the association of different temperatures with different classes of systems in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Classes like these were called fixed points for the scale of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, the second transition can be associated with the development of thermodynam- ics, where the absolute zero was fixed, defining a reference point for the whole scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In physics, these phenomena had to be measured and the individual had to be able to replicate results of such measurements every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The same may apply to decision theory and the notion of utility, someday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' At the moment, the issue remains unclear, even if even Pareto was not completely skeptical about the first transition from an ordinal purely comparative approach to that of an equality relation for utility differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Von Neumann and Morgenstern point out in [41] that the previous concept is based on the same idea used by Euclid to describe the position on a line: the ordinal utility concept of preference corresponds to Euclid’s notion of lying to the right of and the derived concept of equality of utility differences with the geometrical congruence of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, the main question becomes whether the derived order relation on utility differences can be observed and reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nobody can, at the moment, answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='2 Axiomatization of utility theories In 1900, at the International Congress of Mathematicians in Paris, David Hilbert an- nounced that he was firmly convinced that the foundation of mathematics was almost complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, he listed 23 problems to be solved and to give full consistency to mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' All the rest was considered, by him, just details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Some of the problems 6Some authors, like Ellingsen [12], are certain, instead, that the philosophical question of whether utility is intrinsically measurable or not is a spurious one, mainly because they see the issue of “mea- surement” as a concept that is always invented and never discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In this light, our question can be rephrased as whether it is possible to define a correct notion of measurement that allows some kind of intrapersonal and interpersonal utility comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 9 consisted in the axiomatizations of some fields of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, at the begin- ning of the 20th century, the idea of being able to solve every mathematical problem led mathematicians to try develop all mathematical theory from a finite set of axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The main advantage of the axiomatic method was to give a clean order and to remove ambiguity to the theory as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Axioms are the fundamental truths by which it is possible to start modeling a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The careful definition of them is critical in the development of a theory that does not contain contradictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, almost all fields of science started a process of axiomatization, utility theory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The ordinal Paretian revolution was the fertile environment where preferences started to be seen as primitive notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Preference relations began to be formalized as mathematical order relations on a set of alternatives X and became the starting point of the whole theory of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, utility functions became the derived object from the preference relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The mainstream notion of ordinal (Paretian) utility reached its maturity with the representation theorems by Eilenberg [11] and Debreu [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Subsequent work in decision theory shifted from decision theory under certainty to choice problems under uncertainty, with the pioneering article of Ramsey [33] on the “logic of partial belief.” In short, Ramsey [33] stated the necessity of the development of a purely psychological method of measuring both probability and beliefs, in strong contradiction with Keynes’ probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Some years after, the milestone works of von Neumann and Morgenstern [41] and Savage [34] gave full authority to decision theory under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One of the first treatments of preference relations as a primitive notion can be identified with Frisch [18], in his 1926 paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Ragnar Frisch was also the first to formulate an axiomatic notion of utility difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, two kinds of axioms were postulated by him: the first ones - called “axioms of the first kind” - regarded the relation able to rank alternatives in a purely comparative way, while the second axioms named “axioms of the second kind” - reflected a notion of intensity of preference and allowed utility differences to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, in parallel to the axiomatization of ordinal utility, also cardinal utility axiomatizations started to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Frisch’s article did not have the deserved impact in the academic arena, mainly because his article was written in French and published in a Norwegian mathematical journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, the full mathematical formalization of these two notions of preference axioms resulted almost ten years later from the 1930s debate by Oskar Lange [23] and Franz Alt [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lange [23] defined an order relation ≻ on the set of alternatives X with the meaning that, for any two alternatives x, y ∈ X, x ≻ y reads “x is strictly preferred 10 to y.” Then, a corresponding relation P on ordering differences is assumed with the meaning that, for any x, y, z, w ∈ X, xyPzw reads “a change from y to x is strictly preferred than a change from w to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' More formally: x ≻ y ⇐⇒ u(x) > u(y) for all x, y ∈ X (1) xyPzw ⇐⇒ u(x) − u(y) > u(z) − u(w) for all x, y, z, w ∈ X (2) The main theorem of Lange [23] can be stated as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If there exists a differentiable utility function u : R → R such that (1) and (2) hold, then only positive affine transformations of that utility function represent the given preferences ≻ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is immediate to see that Lange [23] provides only necessary conditions for a utility function representation of preference relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, it is relatively easy to see that the assumption of differentiability of u can be largely relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, the issue becomes whether it is possible to find sufficient conditions on the preference relations under which Lange’s utility function - a cardinal utility function - exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This was done by Franz Alt in his 1936 article [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Alt postulated seven axioms that guaranteed sufficient and necessary conditions for the existence of a continuous utility function unique up to positive affine transformations - based on a preference relation and a utility-difference ordering relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In his set of axioms, Alt defined a notion that can be understood as the set of alternatives X to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' With Frisch’s pioneering work of 1926 and 1930s debate by Lange and Alt, the modern ingredients of cardinal utility axiomatization such as equations (1) and (2) and connectedness of the domain of alternatives X started to be formalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In those years, a lot of different axiomatic models were studied, till the article of the famous philoso- pher of science Patrick Suppes and his doctoral student Muriel Winet [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In their 1955 paper, Suppes and Winet developed an abstract algebraic structure of axioms for cardinal utility, called a difference structure, in line with old Frisch’s ideas and Lange’s formalization: not only are individuals able to ordinally rank different alternatives, but they are also able to compare and rank utility differences of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, Sup- pes and Winet cited the work of Oskar Lange on the notion of utility differences and stood in favor of the intuitive notion of introspection, elevating it to not just a mere 11 intuition, but as a solid base where to build a notion of utility differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppes and Winet continued their article saying that, up to 1950s, no adequate axiomatization for intensity comparison had been given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, as Moscati [29] nicely highlights, they were probably unaware of Alt’s representation theorem and this was probably due to the fact that Alt [5] was published in German in a German journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppes and Winet postulated 11 axioms in total, some on the set of alternatives X and others on the two order relations,7 providing sufficient and necessary condition for a cardinal utility representation, unique up to positive affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Another approach to the field of axiomatization of cardinal utility was taken twenty years later by Lloyd Shap- ley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' While axiomatizations `a la Suppes and Winet started developing a set-theoretic abstract structure, Shapley substituted the usual long list of postulates with strong topological conditions both on the domain of alternatives X and on the topology in- duced by the order relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Shapley [36] constructed a cardinal utility function u satisfying some consistency axioms between the orders and assuming the domain of u to be a convex subset of the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We will enter into the details of Shapley [36] in the next chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In conclusion, the notion of cardinal utility has always suffered a lack of conceptual precision in its whole history and, for some authors like Ellingsen [12], it can be even considered the main reason why scientists have disagreed over whether pleasure can be measured or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='8 What is certain is that the history of cardinal utility, a part from some sporadic articles, has been a persistent failure, mostly in its applications to economic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' While the main reason can be probably identified with the almost total absence of any rigorous and proven experimental measurement of pleasure, it is fair to observe that part of its failure must be given to the strong reluctant opinion of the mainstream ordinal “party.” In fact, a large class of economists classify as “meaningless” even the mere introspective idea of a comparison of utility difference, and not just the concept itself, when formalized in a purely comparative environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This position is shown to be, with a gentle expression, “epistemological laziness.” We should always remember that no real progress in economic science can be derived from purely abstract reasoning, but only from the combined effort of empirical measurements with theoretical analysis, always under the wise guide of the compass of history and philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 7The conditions these axioms impose are analogous to the conditions defined by Alt [5]: com- pleteness, transitivity, continuity, and some form of additivity for the two order relations, and an Archimedean property on the quaternary relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 8Ellingsen [12] writes about a “fallacy of identity” and “fallacy of unrelatedness.” 12 2 Preliminary results The aim of this chapter is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' On one side, we introduce the mathematical frame- work that enable us to represent intensity comparisons that a decision maker feels about the desirability of different alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For this aim, we follow the construction of Sup- pes and Winet [40] and Shapley [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' On the other side, we state and prove a list of lemmas that will be used in Theorem 5 and that allow us to generalize Shapley’s proof to a connected and separable subset of a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='1 Basic definitions Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A relation on a set X is a subset ≿ of the cartesian product X × X, where x ≿ y means (x, y) ∈ ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In decision theory, ≿ is usually called a preference relation, with the interpretation that, for any two elements x, y ∈ X, we write x ≿ y if a decision maker either strictly prefers x to y or is indifferent between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' An equivalence relation on a set X is a relation R on X that satisfies 1) Reflexivity: for all x ∈ X, we have xRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 2) Symmetry: for any two elements x, y ∈ X, if xRy, then yRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 3) Transitivity: for any three elements x, y and z ∈ X, if xRy and yRz, then xRz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A relation ≿ on a set X is called a total order relation (or a simple order, or a linear order) if it has the following properties: 1) Completeness: for any two elements x, y ∈ X, either x ≿ y or y ≿ x or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 2) Antisymmetry: for any two elements x, y ∈ X, if x ≿ y and y ≿ x, then x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 3) Transitivity: for any three elements x, y and z ∈ X, if x ≿ y and y ≿ z, then x ≿ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note that if ≿ is complete, then it is also reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The relation ≿ induces, in turns, two other relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Specifically, for any two elements x, y ∈ X we write: (i) x ≻ y if x ≿ y but not y ≿ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 13 (ii) x ∼ y if x ≿ y and y ≿ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is easy to see, indeed, that if ≿ is reflexive and transitive, then ∼ is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Given an equivalence relation ∼ on a set X and an element x ∈ X, we define a subset E of X, called the equivalence class determined by x, by the equation E := {y ∈ X : y ∼ x} Note that the equivalence class E determined by x contains x, since x ∼ x, hence E is usually denoted as [x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We will denote X/∼ the collection {[x] : x ∈ X} of all equivalence classes, which is a partition of X: each x ∈ X belongs to one, and only one, equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In decision theory, an equivalence class is often called an indifference curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A relation ≿ on a set X is called a weak order if it is complete and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The problem of finding a numerical representation for a preference relation ≿, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' an order isomorphism between a generic set X and R, has been widely studied by math- ematicians and is a familiar and well-understood concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Such an order isomorphism is called, in decision theory, a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' More formally: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A real-valued function u : X → R is a (Paretian) utility function for ≿ if for all x, y ∈ X we have x ≿ y ⇐⇒ u(x) ≥ u(y) Utility functions “shift” the pairwise comparisons that characterize the order rela- tion ≿ and its properties in the more analytically convenient space of the real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, as a result, the only thing that is preserved is the order, and the real numbers that are images of the utility function cannot be interpreted as a scale where the decision maker can compare different intensities about the single desirability of any two alternatives x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' What is important is the ranking given by the real num- bers, according to the usual order of the ordered field (R, ≥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, one can easily prove that every strictly increasing transformation of a utility function is again a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For this reason, utility functions are called ordinal and their study belong to what is called ordinal utility theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The main problem of ordinal utility theory is to study sufficient and necessary conditions under which a relation ≿ admits a utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The original reference can be identified with Cantor [7], but the result has been adapted by Debreu [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 14 In addition, to be able to solve optimization problems, one of the properties that is desirable to have is continuity of the utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Debreu [8] is the first to state the theorem in the way we are going to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, he proved it making explicit reference to Eilenberg [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We state here a version of this very well-known theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A weak order ≿ on a set X is said to be continuous if, for every y ∈ X, the sets {x ∈ X : x ≻ y} and {x ∈ X : x ≺ y} are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='9 Theorem 2 (Eilenberg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let ≿ be a complete and transitive relation on a connected and separable topological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The following conditions are equivalent: (i) ≿ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (ii) ≿ admits a continuous utility function u : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One of the biggest theoretical problems of ordinal utility theory is that the expres- sion u(x) − u(y) is a well-defined real number thanks to the algebraic properties of R, but it is meaning- less in term of the interpretation of a difference of utility of two alternatives x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In other words, a Paretian utility function does not have an intrinsic introspective psy- chological notion of intensity of the preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' An immediate corollary of this remark is that the concept of marginal utility (and what is known under the Gossen’s law of decreasing marginal utility), based on the notion of different quotient, is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' More formally, the expression du(x) dx = lim h→0 u(x + h) − u(x) h has no meaning in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, the concept of marginal utility has been a milestone in economic theory, proving that this notion deserves an adequate theoretical foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='2 An overview on measurable utility theory Let X be a set of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pairs of alternatives (x, y) ∈ X × X are intended to represent the prospect of replacing alternative y by alternative x, that can be read as 9Note that this is the usual order topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 15 “x in lieu of y”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Define the binary relation ≽ on X ×X called intensity preference with the following interpretation: for any two pairs (x, y) and (z, w) in X × X, (x, y) ≽ (z, w) is intended to mean that getting x over y gives at least as much added utility as getting z over w or (if y ≿ x) at most as much added sadness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, our decision maker is endowed with a weak order preference relation ≿ on alternatives and an intensity preference relation ≽ on pairs of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Shapley [36] proves his theorem assuming X to be a convex subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, the proof exploits the full algebraic power of the ordered field and the topological properties of the linear continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Our aim is to generalize the set of alternatives X to a connected and separable subset of a topological space, ordered with the binary relations ≿ and ≽ and with the order topology induced by the weak order ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We assume the following axioms for ≿ and ≽, as in Shapley [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Axiom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For all x, y, z ∈ X we have (x, z) ≽ (y, z) ⇐⇒ x ≿ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Axiom 1 (henceforth A1) is an assumption of consistency between the two order- ings because it implies that the decision maker prefers to exchange z with x instead of z with y if and only if she prefers x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Together with A1 we can formulate a dual version of consistency, A1′, that can be derived from the whole set of axioms we are going to assume later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='10 Axiom 1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For all x, y, z ∈ X we have (z, x) ≽ (z, y) ⇐⇒ y ≿ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We now introduce the main object of this thesis: a joint real-valued representation for the two orders ≿ and ≽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A real-valued function u : X → R is a measurable utility function for (≿, ≽) if for each pair x, y ∈ X x ≿ y ⇐⇒ u(x) ≥ u(y) (3) and if, for each quadruple x, y, z, w ∈ X (x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (4) The measurable terminology has nothing to do with measure theory, but it refers to what is known as measurement theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' the field of science that established the for- mal foundation of quantitative measurement and the assignment of numbers to objects 10We mention A1′ as a form of axiom only because in this way we can refer to it in the proof of Theorem 5, but we never assume it formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A proof of it will be formulated forward with Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 16 in their structural correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, not only is a measurable utility function able to rank pairs of alternatives according to a preference relation, but it also repre- sents the idea of magnitude and intensity of the preference relation among alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Therefore, the numerical value u(x) that a measurable utility function assigns to the alternative x is assuming the role of a particular unit of measurement for pleasure, that we call util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Recall that an ordinal utility function u is unique up to strictly monotone trans- formations f : Im(u) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, a measurable utility function is not ordinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Never- theless, it is unique up to positive affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Recall that a positive affine transformation is a special case of a strictly monotone transformation of the follow- ing form f(x) = αx + β, with α > 0 and β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Positive affine transformations are order-preserving thanks to α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A measurable utility function u : X → R for (≿, ≽) is unique up to positive affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If u(x) = αu(x) + β then we have x ≿ y ⇐⇒ u(x) ≥ u(y) ⇐⇒ u(x) = αu(x) + β ≥ αu(y) + β = u(y) and (x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w) ⇐⇒ u(x) − u(y) = α[u(x) − u(y)] ≥ α[u(z) − u(w)] = u(z) − u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, u and u are two utility representations for (≿, ≽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The whole class of utility functions that are unique up to positive affine transformations are called cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Measurable utility functions are, therefore, cardinal and pertain to the so- called cardinal utility theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Other two axioms (A2, A3) we need to introduce are the following: Axiom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For all x, y, z, w ∈ X we have (x, y) ∼ (z, w) ⇐⇒ (x, z) ∼ (y, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Axiom 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For all x, y, z, w ∈ X the set {(x, y, z, w) ∈ X × X × X × X : (x, y) ≽ (z, w)} is closed in the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Axiom 2 is a “crossover” property that characterizes difference comparisons of util- ity, while Axiom 3 is a technical assumption defining the order relation ≽ as continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 17 Shapley [36] proves his theorem on a domain of alternative outcomes that is a nonempty, convex subset D of the real line where the preference order coincides with the total order of (R, ≥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, ≽ is assumed to be a weak order on D × D such that A1, A2 and A3 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Theorem 3 (Shapley).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' There exist a utility function u : D ⊆ R → R such that x ≥ y ⇐⇒ u(x) ≥ u(y) (5) and (x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w) (6) for all x, y, z, w ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, this function is unique up to a positive affine transfor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The theorem is stated as a sufficient condition, which is the most difficult part to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The necessary condition of the theorem is easily proved and we state it here as a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If the pair (≥, ≽) has a continuous measurable utility function u : D ⊆ R → R, then ≥ is complete and transitive, ≽ is complete, transitive, continuous (A3) and satisfies the crossover axiom (A2), and jointly ≥ and ≽ satisfy the consistency axiom (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Shapley’s construction of the measurable utility function of Theorem 3 is extremely elegant, but has the drawback of being too specific as u is defined on a convex subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' On the other side of the spectrum, as mentioned in the first chapter, the field of utility axiomatization has been prolific in the 20th century and a copious number of cardinal-utility derivations from preference-intensity axiomatizations were published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One of the most important papers on this issue was the one published in 1955 by Patrick Suppes and Muriel Winet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Recalling what described before, Suppes and Winet [40] advanced an axiomatization of cardinal utility based on the assumption that individuals are not only able to rank the utility of different alternatives, as is assumed in the ordinal approach to utility, but are also capable of ranking the differences between the utilities of commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, their 11 axioms on an abstract algebraic structure were not fully satisfactory in terms of generality: it was too general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, some of their axioms can be derived in Shapley [36], thanks to the topological properties of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The aim of this research is to settle somewhere in between, finding a representation theorem for cardinal utility function (in particular, a measurable one) keeping the 18 elegance of Shapley’s proof and generalizing the domain of alternatives into the direction of Suppes and Winet [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We will state and prove a representation theorem for a measurable utility function u : X → R where X is a connected and separable subset of a topological space, ≿ and ≽ are weak orders and they satisfy (A1), (A2) and (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Before doing this, we need to state and prove some topological preliminary results that will be used in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='3 A few basic lemmas Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let X be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' X is connected if it cannot be separated into the union of two disjoint nonempty open subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Otherwise, such a pair of open sets is called a separation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let X be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' X is separable if there exists a countable dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A dense subset D of a space X is a subset such that its closure equals the whole space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' D = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A totally ordered set (L, ≿) having more than one element is called a linear continuum if the following hold: (a) L has the least upper bound property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (b) If x ≻ y, there exists z such that x ≻ z ≻ y We recall that a ray is a set of the following type (−∞, a) = {x ∈ L : x ≺ a} and (−∞, a] = {x ∈ L : x ≾ a} in the case L does not have a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In the case L does have a minimum we write [xm, a) = {x ∈ L : xm ≾ x ≺ a} and [xm, a] = {x ∈ L : xm ≾ x ≾ a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Analogously for the sets (a, +∞), [a, +∞), (a, xM] , [a, xM], where xM is the maximum of L in the case it existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='12 Given A ⊆ X, an element y ∈ X is an upper bound for a set A if y ≿ x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is a least upper bound for A if, in addition, it is the minimum of the set of all upper bounds of A, that is if y′ ≿ x for all x ∈ A then y′ ≿ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If ≿ is antisymmetric, the least upper bound is unique and is denoted sup A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The greatest lower bound is defined analogously and denoted inf A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 11We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hendrik S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Brandsma for providing a feedback and insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 12Note that in decision theory, rays of a set X equipped with a reflexive and transitive binary relation ≿ are usually denoted with the following notation L(a, ≿) := (−∞, a] = {x ∈ X : x ≾ a} and U(a, ≿) := [a, +∞) = {x ∈ X : x ≿ a}, L(a, ≻) := (−∞, a) and U(a, ≻) := (a, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 19 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let ≿ be a total order on a connected set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, X is a linear continuum in the order topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose that a and b are two arbitrary but fixed elements of X such that a ≺ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If there is no element c ∈ X such that a ≺ c ≺ b, then X is the union of the open rays (−∞, b) = {x ∈ X : x ≺ b} and (a, +∞) = {x ∈ X : a ≺ x} both of which are open sets in the order topology and are also nonempty, as the first contains a, while the second contains b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But this contradicts the fact that X is connected, so there must exists an element c ∈ X such that a ≺ c ≺ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now, to show the least upper bound property, let A be a nonempty subset of X such that A is bounded above in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let B be the set of all the upper bounds in X of set A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' B := {b ∈ X : b ≿ a for every a ∈ A} which is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' All we need to show is that B has the least element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If B has a smallest element (or A has a largest element, which would then be the smallest element of B), then that element is the least upper bound of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let us assume, instead, that B has no smallest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, for any element b ∈ B, there exists an element b′ ∈ B such that b′ ≺ b, and so b ∈ (b′, +∞) ⊆ B with (b′, +∞) being an open set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This shows that B is a nonempty open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Therefore, B can be closed only in the case when B = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But we know that B ⊂ X, since A ⊆ X\\B and A ̸= ∅, so it cannot be the case that B = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Therefore, B has a limit point b0 that does not belong to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then b0 is not an upper bound of set A, which implies the existence of an element a ∈ A such that b0 ≺ a, we can also conclude that b0 ∈ (−∞, a) ⊆ X\\B, with (−∞, a) being an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This contradicts our choice of b0 as a limit point of set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Therefore, the set B of all the upper bounds in X of set A must have a smallest element, and that element is the least upper bound of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Given A ⊆ X, we denote A or ClA the topological closure of A, that is defined as the intersection of all closed sets containing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' From now on denote X as a subset of a topological space (X, τ), unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let ≿ be a complete, transitive and continuous order on a connected set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 13Note that the converse holds as well: ≿ is a total order on a connected set X if and only if X is a linear continuum in the order topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 20 Given any x, y ∈ X, with x ≻ y, we have x ≿ z ≿ y ⇒ z ∈ X for all z ∈ (X, τ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose by contradiction that there exists z ∈ X\\X such that x ≻ z ≻ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By the continuity of ≿, we can partition X into two nonempty disjoint open sets {x ∈ X : x ≺ z} and {x ∈ X : x ≻ z}, which contradicts the connectedness of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose that jointly ≿ and ≽ satisfy A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If ≽ is continuous , then ≿ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For all arbitrary but fixed y, z ∈ X, by A1 we have {x ∈ X : (x, z) ≽ (y, z)} = {x : x ≿ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A3, the set {x ∈ X : (x, z) ≽ (y, z)} is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Analogous is the case for {x : y ≿ x}, derived from A1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Fix y ∈ X, the set Iy := {x ∈ X : x ∼ y} is a closed set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' ≿ is continuous, so for every y ∈ X we have that {x ∈ X : x ≿ y} and {x ∈ X : y ≿ x} are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pick a point x such that x ≿ y and y ≿ x, that is x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So we have {x ∈ X : x ∼ y} = {x ∈ X : x ≿ y} ∩ {x ∈ X : y ≿ x} and the intersection of two closed sets is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note that when ≿ is antisymmetric, the set Iy is a singleton and Lemma 4 reduces to prove that X satisfies the T1 axiom of separation, that is every one-point set is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Clearly, every Hausdorff space satisfies it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let ≿ be a continuous total order on a connected set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If A ⊆ X is a nonempty closed set in the order topology and A is bounded above (below), then supA (infA) belongs to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose supA /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then supA ∈ X\\A, which is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By definition, there exists a base element (a, b) such that supA ∈ (a, b) ⊆ X\\A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A is bounded above so, by Lemma 1, sup A exists and there is an element a⋆ such that a ≺ a⋆ ≺ sup A, then a⋆ ∈ (a, b) ⊆ X\\A, so a⋆ is an upper bound of A smaller that supA, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In the case X had a maximum, then consider the case where sup A = max X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let U := (x, sup A] be a basic neighborhood of sup A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, x 14The lemma holds even in the case we relaxed connectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, we always need to as- sume sup A exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If we do not assume the existence of the least upper bound, an easy counterexample is N ⊂ R that is closed in the order topology, but sup N /∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 21 cannot be an upper bound of A as x ≺ sup A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, there exists an element a ∈ A such that x ≺ a ≾ sup A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, as x was generic, it follows that U ∩ A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This means that every neighborhood of sup A intersects A, that is sup A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But A is closed, hence sup A ∈ A and we can conclude sup A = max A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The case of inf A is specular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we define the notion of convergence in any topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In an arbitrary topological space X, we say that a sequence x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' of points of the space X converges to the point x of X provided that, corresponding to each neighborhood U of x, there is a positive integer N such that xn ∈ U for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, let ≿ a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We write xn ↑ x if x1 ≾ x2 ≾ · · · ≾ xn ≾ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' and supnxn = x where sup is with respect to ≾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The definition xn ↓ x for a ≾-decreasing sequence is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We say that (xn) converges monotonically to a limit point x when either xn ↑ x or xn ↓ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We now prove one of the fundamental lemmas that allow us to generalize Shapley’s proof to a connected and separable subset of a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note that, as long as Shapley [36] is working on R, sequences as “enough” to characterize the definition of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This is due to the fact that there exists a countable collection of neighborhoods around every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This is not true in general, but it is for a specific class of spaces that are said to satisfy the first countability axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='15 A space X is said to have a countable basis at the point x if there is a countable collection {Un}n∈N of neighborhoods of x such that any neighborhood U of x contains at least one of the sets Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A space X that has a countable basis at each of its points is said to satisfy the first countability axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In general, however, sequences are not powerful enough to capture the idea of convergence we want to capture in a generic topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, there could be uncountably many neighborhoods around every point, so the countability of the natural number index of sequences cannot “reach” these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The ideal solution to this problem is to define a more general object than a sequence, called a net, and talk about net-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' One can also define a type of object called a filter and show that filters also provide us a type of convergence which turns out to be equivalent to net-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' With these more powerful tools in place of sequence convergence, one can fully characterize the notion of convergence in any topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 15There are far more general classes of spaces in which convergence can be fully characterized by se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We refer the interested reader to the notion of Fr´echet-Urysohn spaces and Sequential spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 22 Nevertheless, we are now going to show that every connected, separable and totally ordered set X satisfies the first countability axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In fact, we are going to prove even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We are going to show that X is metrizable, which means there exists a metric d on the set X that induces the topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='16 We give other two definitions that will be used to prove Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose X is T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then X is said to be regular (or T3) if for each pair consisting of a point x and a closed set B disjoint from x, there exist disjoint open sets containing x and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If a space X has a countable basis for its topology, then X is said to satisfy the second countability axiom, or to be second-countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Theorem 4 (Urysohn metrization theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Every regular space X with a count- able basis is metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let ≿ be a continuous total order on a connected and separable topological space X in the order topology and A ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We have x ∈ A if and only if there exists a sequence (xn) ∈ AN that converges monotonically to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The steps of the proof are the following: (i) We show that X is regular17 and second-countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By the Urysohn metrization theorem, which provides sufficient (but not necessary) conditions for a space to be metrizable, there exist a metric d that induces the topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (ii) Let A ⊆ X with X metrizable, then we have that x ∈ A if and only if there exists a sequence of points of A converging to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (iii) Finally, we use the fact that in every totally ordered topological space X, every sequence admits a monotone subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, if a sequence converges, all of its subsequences converge to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, we can extract our monotone converging sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A totally ordered topological space X is regular in the order topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' It is basic topology to prove that every totally ordered set is Hausdorff, hence it is T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now, suppose x ∈ X and B is a closed set, disjoint from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, x ∈ X\\B, 16A metrizable space always satisfies the first countability axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 17In fact, one could prove that X is also normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 23 which is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, by definition of open set, there exists a basis element (a, b) such that x ∈ (a, b) and (a, b) ∩ B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pick any a0 ∈ (a, x), and let U1 = (−∞, a0) , V1 = (a0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If no such a0 exists (in our case it would, by connectedness of X), then let U1 = (−∞, x), V1 = (a, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In both cases, U1 ∩ V1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Similar is the case of the other side, pick b0 ∈ (x, b), and if that exists, denote U2 = (b0, ∞) , V2 = (−∞, b0) , and if not, let U2 = (x, ∞), V2 = (−∞, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Again, in both cases U2 ∩ V2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, we obtained that, in both cases, x ∈ V1 ∩ V2 with V1 ∩ V2 open set and B ⊆ U1 ∪ U2, with U1 ∪ U2 open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As V1 ∩ V2 is disjoint from U1 ∪ U2, X is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' A totally ordered, connected and separable topological space X is second- countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we find a countable basis for the order topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As X is separable, then let D ⊆ X be countable and dense in X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' D = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, define B := {(a, b) : a, b ∈ D with a ≺ b} together with, if there exists a minimal element m := min X and a maximal element M := max X, the set {[m, a), (a, M], a ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In both cases, the collection B forms a countable base for the topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' To prove this, we show that for each open set (a, b) of the order topology of X and for every x ∈ (a, b) there is an element (a′, b′) ∈ B such that x ∈ (a′, b′) ⊆ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose x ∈ (a, b) ⊂ X, then the open intervals (a, x) and (x, b) cannot be empty by connectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, there exist a′ ∈ (a, x) ∩ D and b′ ∈ (x, b) ∩ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This follows from the fact that D = X and x ∈ D = X if and only if every open set containing x intersects D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, it follows that x ∈ (a′, b′) ⊆ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now, when m exists, suppose x = m, then x ∈ [m, a) and this set is nonempty by connectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, there exists an element a′′ ∈ [m, a) ∩ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, it follows that x ∈ [m, a′′) ⊆ [m, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Analogous is the case when M exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By Lemma 7 and Lemma 8 , X satisfies all the assumptions of the Urysohn metriza- tion theorem, hence X is metrizable (and, a fortiori, it is first-countable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let A ⊆ X with X metrizable, then x ∈ A if and only if there exists a sequence of points of A converging to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose xn → x with xn ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, every neighborhood U of x contains a point of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Conversely, we use the fact that X is metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='18 Let x ∈ A 18Note, once again, that here we do not need the full strength of metrizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' All we really need 24 and let d be a metric that induces the order topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For every n ∈ N, we take the neighborhood Bd(x, 1 n), of x of radius 1 n and we choose xn to be a point such that, for all n, xn ∈ Bd(x, 1 n) ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We show xn → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Any open set U containing x contains an ǫ-neighborhood Bd(x, ǫ) centered at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Choosing N such that 1 N < ǫ, then U contains xn for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We can finally prove Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The if part comes trivially by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If there exists a sequence that converges (monotonically) to x, then x ∈ A by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Conversely, if x ∈ A, then by Lemma 9 we know that there exists a sequence in A converging to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we show that, in every totally ordered set (X, ≾), every sequence from N → (X, ≾) has a monotone subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, this is a property that has nothing to do with the topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let (xi)i∈N be a sequence with values in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We say that xk is a peak of the sequence if h > k ⇒ xh ≾ xk (we admit a slight abuse of notation here, as it would be better to call peak the index of the sequence, and not its image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We distinguish two cases: if there are infinitely many peaks, then the subsequence of peaks is an infinite non-increasing sequence and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If there are only finitely many peaks, then let i1 be the index such that xi1 is the successor of the last peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, xi1 is not a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Again, we find another index i2 > i1 such that xi2 ≿ xi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Again, as xi2 is not a peak, we can find another index i3 > i2 such that xi3 ≿ xi2 ≿ xi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Keeping defining the sequence in this way, we get, inductively, a non-decreasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In conclusion, as by assumption we have a sequence (xn) ∈ AN converging to x, this sequence admits a monotone subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But, if a sequence converges to a point x, then all of its subsequences converge to the same point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, there exists a sequence that converges monotonically to x, proving Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note that Lemma 6 could have been proven just using the notion of first countabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Nevertheless, we decided to take the longer path of Urysohn metrization theorem to is a countable collection of neighborhoods around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, both connectedness and separability are not necessary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We refer the interested reader to the nice two-page paper of Lutzer [25], that proves a linearly ordered space X is metrizable in the order topology if and only if the diagonal ∆ := {(x, x) : x ∈ X} is a countable intersection of open subsets of X × X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' the diagonal is a Gδ set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Furthermore, this condition can be shown to be equivalent to have a σ-locally countable basis, which is a condition more in the spirit of the Nagata-Smirnov metrization theorem which requires a σ-locally finite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 25 show how “well-behaved” a totally ordered, connected and separable topological space can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let (X, ≿) be a topological space with the order topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let ≽ be another order relation on X × X such that A1 and A3 hold,19 and suppose (xn), (yn) converge to x and y respectively, and (wn), (zn) converge to w and z respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If for every n ∈ N we have (xn, yn) ≽ (wn, zn) then (x, y) ≽ (w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Denote the set A := {(x, y, w, z) ∈ X × X × X × X : (x, y) ≽ (w, z)} and pick a sequence of points with values in A, that is pick (xn, yn, wn, zn) ∈ AN converging to (x, y, w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By assumption, we have that xn → x, yn → y, wn → w, zn → z and this is equivalent to (xn, yn, wn, zn) → (x, y, w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, a sequence in the product space X × X × X × X converges to (x, y, w, z) if and only if it converges componentwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' xn → x, yn → y, wn → w, zn → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We now prove this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Assume (xn, yn, wn, zn) → (x, y, w, z) in X ×X ×X ×X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let U1, U2, U3, U4 be open sets containing x, y, w, z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then U1×U2×U3 ×U4 is a basis element (hence, open) for the product topology containing (x, y, w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By definition of convergence, we can find n0 such that for all n ≥ n0 we have (xn, yn, wn, zn) ∈ U1 × U2 × U3 × U4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thanks to the fact that projections are continuous functions, they preserve convergent sequences and so for all n ≥ n0 we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' xn → x, yn → y, wn → w, zn → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Conversely, if xn → x, yn → y, wn → w, zn → z, let U⋆ be an open subset of X×X×X×X such that (x, y, w, z) ∈ U⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By definition of product topology, we can find U1 ⊆ X open in X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' , U4 ⊆ X open in X such that x ∈ U1, y ∈ U2, w ∈ U3, z ∈ U4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By convergence, we have that for all i = 1, 2, 3, 4 there exists nki ∈ N such that for all n ≥ nki we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now pick N := max{nk1, nk2, nk3, nk4} and for every n ≥ N we have (xn, yn, wn, zn) ∈ U1 × U2 × U3 × U4 ⊆ U⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, by definition of convergence, (xn, yn, wn, zn) → (x, y, w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we want to show (x, y, w, z) ∈ A, with A closed in the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We now prove that every closed set in the product topology is sequentially closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='20 This means we want to show that if we pick a sequence of points (xn, yn, wn, zn) with values in A ⊆ X that is converging to a point (x, y, w, z) ∈ X, then (x, y, w, z) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Pick a sequence (xn, yn, wn, zn) with values in A ⊆ X that is converging to a point 19Note that the order topology and A1 are redundant assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The lemma follows immediately by continuity of ≽ alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 20Note that when X is metrizable, a set C ⊆ X is closed ⇐⇒ C is sequentially closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 26 (x, y, w, z) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, let U⋆ be any neighborhood of (x, y, w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By convergence, there exist an n0 ∈ N such that for all n ≥ n0 we have (xn, yn, wn, zn) ∈ U⋆ and, in particular, (xn, yn, wn, zn) ∈ U⋆ ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since U⋆ was an arbitrary but fixed neighborhood of (x, y, w, z), then (x, y, w, z) is in the closure of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (x, y, w, z) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But A is closed, therefore A = A, so (x, y, w, z) ∈ A, hence (x, y) ≽ (w, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The proof of Theorem 5 in chapter 3, as in the original version of Shapley [36], relies on two very interesting lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Similar propositions have been taken as axioms in environments that lack the topological assumptions on the set of alternatives X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let (w,z) be an element of X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If x′, x′′, y ∈ X are such that: (x′, y) ≽ (w, z) ≽ (x′′, y) (7) then there exists a unique, up to indifference, x⋆ ∈ X such that (x⋆, y) ∼ (w, z) (8) and x′ ≿ x⋆ ≿ x′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Define x0 := inf{x ∈ X : (x, y) ≽ (w, z)} and denote A := {x ∈ X : (x, y) ≽ (w, z)} this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The set A is nonempty as x′ ∈ A, A is bounded below by x′′ as we have (w, z) ≽ (x′′, y) and, by transitivity and A1, we reach x ≿ x′′ for every x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, x0 is such that x′ ≿ x0 ≿ x′′ and so x0 ∈ X by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Analogously, we define x0 := sup{x ∈ X : (w, z) ≽ (x, y)} and denote B := {x ∈ X : (w, z) ≽ (x, y)} this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, B is nonempty as x′′ ∈ B, B is bounded above by x′ as we have (x′, y) ≽ (w, z) and, by transitivity and A1, we reach x′ ≿ x for every x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, x0 is such that x′ ≿ x0 ≿ x′′ and so x0 ∈ X by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A3, the sets A and B are closed and so, by Lemma 5, we have x0 ∈ A and x0 ∈ B so that (x0, y) ≽ (w, z) ≽ (x0, y) By transitivity and by A1 we have x0 ≿ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Assume now by contradiction that x0 ≻ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By Lemma 1 there exists x⋆ ∈ X such that x0 ≺ x⋆ ≺ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' But then, comparing x⋆ with (w, z), (x⋆, y) ≽ (w, z) can hold only if x0 ∼ x⋆ ≻ x0, so x0 ∼ x⋆ and therefore x⋆ should be the infimum of A, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Specular is the contradiction in the other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Therefore, as there does not exist any x⋆ ∈ X such that x0 ≺ x⋆ ≺ x0, we must conclude that x0 ∼ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By transitivity and A1 we have (x0, y) ∼ (w, z) ∼ (x0, y) 27 This proves the existence of x⋆ ∈ X for which (8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let x ∈ X be any other element of X for which (8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By transitivity, (x⋆, y) ∼ (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A1, we have x⋆ ∼ x and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let x, z ∈ X such that x ≻ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, there exists a unique, up to indiffer- ence, y⋆ ∈ X such that (x, y⋆) ∼ (y⋆, z) and x ≻ y⋆ ≻ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Define y0 to be the least upper bound of the set C := {y ∈ X : (x, y) ≽ (y, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This set is nonempty as if we pick y = z we have (x, z) ≽ (z, z) that by A1 is equivalent to x ≿ z, that holds by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' C is also bounded from above by x as if we pick y = x we have (x, x) ≽ (x, z) that by A1′ is equivalent to z ≿ x, that, by completeness, contradicts the assumption of x ≻ z showing that x is an upper bound for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since C is nonempty and bounded above by x, by Lemma 2 we have y0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Similarly, by defining y0 to be the greatest lower bound of the set D := {y ∈ X : (y, z) ≽ (x, y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This set is nonempty as if we pick y = x we have (x, z) ≽ (x, x) that by A1′ is if and only if x ≿ z, that holds by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This set is also bounded from below by z as if we pick y = z we have (z, z) ≽ (x, z) that by A1 is if and only if z ≿ x, that, by completeness, contradicts the assumption of x ≻ z showing that z is a lower bound for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since D is nonempty and bounded below by z, by Lemma 2 we have y0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A3 the sets C and D are closed, so by Lemma 5 we have y0 ∈ C and y0 ∈ D, that is (x, y0) ≽ (y0, z) and (y0, z) ≽ (x, y0) (9) We show now that y0 ≿ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose, by contradiction, y0 ≻ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By Lemma 1 there exists y⋆ ∈ X such that y0 ≻ y⋆ ≻ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, by definition of y0 we have (y⋆, z) ≺ (x, y⋆), while by the definition of y0 we have (x, y⋆) ≺ (y⋆, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This contradiction shows that y0 ≿ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A1 this is equivalent to (y0, z) ≽ (y0, z) for all z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (10) By A1′ it is also equivalent to (x, y0) ≽ (x, y0) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (11) Putting together equation 9 with equations 10 and 11, we reach the loop (y0, z) ≽ (y0, z) ≽ (x, y0) ≽ (x, y0) ≽ (y0, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 28 By transitivity, we have (y0, z) ∼ (y0, z) and (x, y0) ∼ (x, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A1, we conclude that y0 ∼ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We conclude proving that from A1, A2 and A3 we can derive A1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let X be a connected subset of a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If ≿ is complete and transitive, ≽ is complete, transitive, satisfies A3 and A2, and jointly ≿ and ≽ satisfy A1, then A1′ holds, that is, for all x, y, z ∈ X we have x ≿ y if and only if (z, y) ≽ (z, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By contradiction, suppose A1′ fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, there exist x, y, z ∈ X such that (z, y) ≽ (z, x) and x ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We consider two cases: y ≻ z and y ≾ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If y ≻ z then, being (z, y) ≽ (z, x) by assumption, we have (x, x) ∼ (y, y) ≻ (z, y) ≽ (z, x) by A2 and A1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We apply Lemma 11 to find a w ∈ X such that (w, x) ∼ (z, y) and x ≿ w ≿ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Being y ≻ x, we have (z, z) ∼ (y, y) ≻ (x, y) ∼ (w, z) ≿ (z, z) by A2, A1, A2, A1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This implies a contradiction in the case y ≻ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Assume now y ≾ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Being y ≾ z and x ≺ y, by transitivity we have x ≺ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We can proceed as in the previous case, interchanging the roles of x and y and reversing all the inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 3 The theorem We can now state and prove Shapley’s theorem in our general version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let X be a connected and separable subset of a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If ≿ is complete and transitive, ≽ is complete, transitive, satisfies A2 and A3, and jointly ≿ and ≽ satisfy A1, then the pair (≿, ≽) can be represented by a continuous measurable utility function u: X → R, that is, for each pair x, y ∈ X, x ≿ y ⇐⇒ u(x) ≥ u(y) (12) and for each quadruple x, y, z, w ∈ X, (x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (13) Moreover, u is unique up to positive affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We first prove the result when ≿ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In view of Lemma 1, through- out the proof we will consider suprema and infima of subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Suppose X is not a singleton, otherwise the result is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let a0, a1 ∈ X be two distinct elements of X such that, without loss of generality, a1 ≻ a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Assign u(a0) = 0 and u(a1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we want to show that u has a unique extension on X which is a measurable utility function for (≿, ≽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' To ease notation, denote 1 := (a1, a0) , 0 := (a0, a0) , −1 := (a0, a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Clearly, 1, 0, −1 ∈ X × X and, by A1 and A1′, 1 ≻ 0 ≻ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, by A2 we have (x, x) ∼ 0 for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Moreover, for every y ∈ X we have either: (i) There exists a unique T1(y) ∈ X such that (T1(y), y) ∼ 1 or (ii) 1 ≻ (x, y) for all x ∈ X Indeed, if (ii) fails, there exists x′ ∈ X such that (x′, y) ≽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since (x′, y) ≽ 1 ≽ 0 ∼ (y, y), by Lemma 11 there exists an element T1(y) ∈ X such that (T1(y), y) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A1 and antisymmetry of ≿ , (T1(y), y) ∼ (y′, y) implies T1(y) = y′, so T1(y) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In addition, note that y ≺ T1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, (y, y) ∼ 0 ≺ 1 ∼ (T1(y), y), and so A1 implies y ≺ T1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In a similar way as before, for every y ∈ X we have either: (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='bis) There exists a unique T−1(y) ∈ X such that (T−1(y), y) ∼ −1 or (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='bis) −1 ≺ (x, y) for all x ∈ X Indeed, if (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='bis) fails, there exists x′ ∈ X such that (x′, y) ≼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since (x′, y) ≼ −1 ≼ 0 ∼ (y, y), by Lemma 11 there exists an element T−1(y) ∈ X such that (T1(y), y) ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A1 and antisymmetry of ≿ , (T−1(y), y) ∼ (y′, y) implies T−1(y) = y′, so T−1(y) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In addition, note that T−1(y) ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, (T−1(y), y) ∼ −1 ≺ 0 ∼ (y, y), and so A1 implies T−1(y) ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now define a2 := T1(a1) if (i) holds for y = a1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' if there exists a unique T1(a1) ∈ X such that (T1(a1), a1) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Similarly, set a3 := T1(a2) if (i) holds for y = a2, and continue in this way till (if ever) occurs y = an for which (ii) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1 ≻ (x, an) for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Analogously, we define a−1 := T−1(a0) if (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='bis) holds for y = a0, set a−2 := T−1(a−1) if (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='bis) holds for y = a−1, and continue in this way till (if ever) occurs y = a−n for which (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='bis) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 30 Now define A := {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' , a−2, a−1, a0, a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' }, with · · ≺ a−2 ≺ a−1 ≺ a0 ≺ a1 ≺ a2 ≺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The set A can be finite or infinite in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If we consider now a sequence that from an index set Pa ⊆ Z maps to A, we define the following function a : Pa ⊆ Z → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we start to extend u to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Define the following: u(ap) = p for every p ∈ Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Clearly, we have (12), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' x ≿ y if and only if u(x) ≥ u(y) for every x, y that are images of the sequence a, so (12) holds on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we show that (13) holds whenever x, y, z, w ∈ A ⊂ X, say x = ap, y = aq, z = ap−d where p, q, p − d ∈ Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Without loss of generality, assume d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We first prove the “equality” case of (13), that is (x, y) ∼ (z, w) ⇐⇒ u(x) − u(y) = u(z) − u(w) (14) By construction we have (ap, ap−1) ∼ 1 ∼ (aq, aq−1) so, by transitivity and A2, we have: (ap, aq) ∼ (ap−1, aq−1) Iterating this procedure finitely many times we reach: (x, y) = (ap, aq) ∼ (ap−d, aq−d) = (z, aq−d) (15) By transitivity, (z, aq−d) ∼ (z, w) and so, by A1 aq−d = w, so that u(aq−d) = u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By definition of u we can write u(x) − u(y) = u(ap) − u(aq) = p − q = u(ap−d) − u(aq−d) = u(z) − u(w) thus proving (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Next we prove (x, y) ≻ (z, w) ⇐⇒ u(x) − u(y) > u(z) − u(w) (16) By transitivity, (z, aq−d) ≻ (z, w) and so, by A1′, w ≻ aq−d, so that u(w) > u(aq−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By definition of u, from (15) we can write u(x) − u(y) = u(ap) − u(aq) = p − q = u(ap−d) − u(aq−d) > u(z) − u(w) thus proving (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Summing up, both (12) and (13) hold on the terms of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Using Lemma 12, now we want to extend u to the points of X that lie between terms of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Set b0 := a0 and since a1 ≻ a0, by Lemma 12 there exists b1 ∈ X, with a1 ≻ b1 ≻ a0, 31 such that (a1, b1) ∼ (b1, a0) Now build the set B := {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' , b−2, b−1, b0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' }, with · · ≺ b−2 ≺ b−1 ≺ b0 ≺ b1 ≺ b2 ≺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' based on b0, b1, in the same way we constructed A from a0, a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Also here, we can define a sequence that from an index set Pb ⊆ Z maps to B, that is, we define the following function b : Pb ⊆ Z → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By construction we have (b2, b1) ∼ (b1, b0) Together with (a1, b1) ∼ (b1, a0), by transitivity we have (b2, b1) ∼ (a1, b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By A1, b2 = a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Analogously, one can verify that b2p = ap for every p ∈ Pa (17) So, the terms of the set B lie between the terms of the set A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' the set B refines A and we can write A ⊆ B (18) Denote now c0 := b0 = a0 and we let c1 ∈ X be that element provided by Lemma 12 such that (b1, c1) ∼ (c1, b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In the same way we constructed B from A, we can construct, from B, a third set C := {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' , c−2, c−1, c0, c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' }, with · · ≺ c−2 ≺ c−1 ≺ c0 ≺ c1 ≺ c2 ≺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' based on c0, c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We can see that c2p = bp for every p ∈ Pc where Pc ⊆ Z is the collection of indexes of the sequence c : Pc ⊆ Z → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The set C refines B B ⊆ C (19) We keep iterating this process, constructing sets that refine one another and, for ease of notation, we denote them in the following way: A0 := A and a0 p := ap ∈ A0 A1 := B and a1 p := bp ∈ A1 A2 := C and a2 p := cp ∈ A2 · · 32 These sets generalize the inclusions (18) and (19) as follows: A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (20) So, in general, an p for p ̸= 1 is obtained from the construction of (i) and (ii), applied to the points a0, an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The term an 1, for n > 0, is the “midpoint” between an−1 1 and a0, that exists by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By iterating the construction of (17), we have that p 2n = q 2m =⇒ an p = am q In the spirit of (20), we extend u to all points in A∞ := �∞ n=1 An by: u(an p) = p 2n for all an p ∈ An Relations (12) and (13) hold in this extended domain: given x, y, z, w ∈ �∞ n=1 An, just take n large enough so that they become, up to indifference, terms of the set An and proceed in the same exact way as we did for the set A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' To complete the construction of u we only remain to show A∞ is dense in X, that is A∞ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We first show that none of the sets An has, for its sequences of points an, a point of accumulation in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, fix n and suppose by contradiction that an pk converges monotonically to a⋆ ∈ X, where, without loss of generality, we assume an pk ↑ a⋆ with a⋆ ∈ X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' (pk) is an increasing sequence of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Denote 1n := (an 1, a0) and we have, for every k ∈ N, (an 1+pk, an pk) ≽ 1n ≻ 0 By Lemma 10, we have (a⋆, a⋆) ≽ 1n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, by transitivity, we reach (a⋆, a⋆) ≻ 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We conclude that, fixed n, none of the sequences an with values in An has a limit point in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' To prove A∞ = X, the implication A∞ ⊆ X is trivial by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we want to show A∞ ⊇ X, that is all the elements of X belong to the closure of A∞ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Fix x ∈ X such that, without loss of generality, x ≿ a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' For n ≥ 1, define yn := sup{y ∈ An : x ≿ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note that a0 ∈ {y ∈ An : x ≿ y}, so this set is nonempty and we can write x ≿ yn ≿ a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By Lemma 2, yn ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note further that, as shown before, An cannot have accumulation points in X so, as long as yn ∈ X, it follows yn cannot be an accumulation point of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, yn must belong to An and we denote yn := an pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As a result, we have: an pn−k ≾ x ≺ an pn+k for every k > 0 (21) We also have that 1n ≻ (x, yn) (22) 33 Indeed, if (22) were not true, then (x, an pn) ≽ 1n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' We consider two cases: an 1+pn ≻ x or an 1+pn ≾ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If an 1+pn ≻ x, then, thanks to A1, we reach the following contradiction: 1n ∼ (an 1+pn, an pn) ≻ (x, an pn) ≽ 1n (23) So an 1+pn ≾ x, but this contradicts (21), that is, it contradicts yn to be the supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In particular, by A1 and A2, we can write (x, yn) ≽ (yn, yn) ∼ 0, leading to 1n ≻ (x, yn) ≽ 0 (24) Now, when n → ∞, as the sets An+1 ⊇ An ⊇ An−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' are nested one into the other by (20), we can write, for every n ≥ 1, yn ≾ yn+1 ≾ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, the points yn form a non-decreasing sequence that is bounded from above by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Call y⋆ the limit of this sequence, that is well-defined by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since a0 ≾ y⋆ ≾ x, by Lemma 2 it follows that y⋆ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' In particular, by Lemma 6 we have y⋆ ∈ A∞, because, for every fixed n ≥ 1, yn is a term of the sets An, and so (yn) ∈ AN ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As to the 1n terms, for n fixed, we see that 1n ∼ (an 2, an 1) ∼ (an−1 1 , an 1) We also have that, for every n ≥ 1, a0 ≾ an+1 1 ≾ an 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Thus, the points an 1 form, for n → ∞, a non-increasing sequence that is bounded from below by a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Call a⋆ the limit of this sequence, that is well-defined by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since a0 ≾ a⋆ ≾ a1, by Lemma 2 we have a⋆ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Consider now (an−1 1 , an 1) and (x, yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By Lemma 10 and from (24) it follows that (a⋆, a⋆) ≽ (x, y⋆) ≽ 0 Since, by A2, (a⋆, a⋆) ∼ 0, by transitivity (x, y⋆) ∼ 0, so that x ∼ y⋆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' x = y⋆ as ≿ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Since x was arbitrarily chosen in X and y⋆ ∈ A∞, we can conclude x ∈ A∞, so that A∞ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Therefore, we can extend u by continuity to the whole set X by setting u(x) = lim n→∞ u(xn) if (xn) ∈ AN ∞ converges monotonically to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Note that u : X → R is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, to prove it is well-posed we show that if xn and yn are two sequences that converge to x, then limn→∞ u(xn) = limn→∞ u(yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' This follows easily by continuity of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='21 In light of Lemma 6, it is easy to see that u satisfies (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' As to uniqueness, observe that any other u that satisfies (12) and (13) can be 21Recall that in every topological space X continuity implies sequential continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The converse holds if X is first-countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 34 normalized so that u(a0) = 0 and u(a1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, u must agree on u at each step of the constructive procedure for u just seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Indeed, for a given u : X → R, define the following positive affine transformation f : Im(u) → R such that f(x) := x − u(a0) u(a1) − u(a0) It is immediate to see that, for the equivalent utility function �u := f ◦ u, we have �u(a0) = 0 and �u(a1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Summing up, we proved Theorem 5 if ≿ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Now we drop this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let X/∼ be the quotient space with respect to the equivalence relation ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' The set {x ∈ X : x ∼ y} is a closed set in X by Lemma 4, so (X/∼, ˜≿) is a totally ordered connected and separable subset of a topological space, where ˜≿ is the total order induced by the weak order ≿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='22 Therefore, the orders ≿ and ≽ induce orders ˜≿ and ˜≽ on the quotient set X/∼, by setting, for all [x], [y] ∈ X/∼ [x] ˜≿ [y] ⇐⇒ x ≿ y and, for all [x], [y], [z], [w] ∈ X/∼ ([x], [y]) ˜≽ ([z], [w]) ⇐⇒ (x, y) ≽ (z, w) It is routine to show that the orders ˜≿ over X/∼ and ˜≽ over X/∼ × X/∼ inherit the same properties of ≿ and ≽ used in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, by what has been proved so far, there exists ˜u : X/∼ → R that satisfies (12) and (13) for (˜≿, ˜≽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Let π : X → X/∼ be the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Then, the function u : X → R defined as u = ˜u ◦ π is a well-defined measurable utility function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' it is easily seen to satisfy (12) and (13) for (≿, ≽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' To conclude, we show that u satisfies (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If x ∼ y then [x] = [y] and, by the theorem we have just proved, ˜u([x]) = ˜u([y]), which is (˜u ◦ π)(x) = (˜u ◦ π)(y), and so u(x) = u(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If x ≻ y, then [x] ≻ [y], which implies ˜u([x]) > ˜u([y]), which is (˜u ◦ π)(x) > (˜u ◦ π)(y), and so u(x) > u(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Conversely, assume u(x) ≥ u(y) and suppose by contradiction x � y that, by completeness, is y ≻ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If u(x) = u(y) then ˜u([x]) = ˜u([y]) ⇐⇒ [x] = [y] ⇐⇒ x ∼ y, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' If u(x) > u(y) then ˜u([x]) > ˜u([y]) ⇐⇒ [x] > [y] ⇐⇒ x ≻ y, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, (12) holds for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' By definition, we have that ([x], [y]) ≽ ([z], [w]) ⇐⇒ (x, y) ≽ (z, w), for all [x],[y],[z],[w] ∈ X/∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' So, we can write (x, y) ≽ (z, w) ⇐⇒ ([x], [y]) ≽ ([z], [w]) ⇐⇒ ˜u([x])−˜u([y]) ≥ ˜u([z])−˜u([w]) ⇐⇒ u(x)−u(y) ≥ u(z)−u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' Hence, also (13) holds for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 22That is, ˜≿ := ≿ /∼ ⊆ X/∼ × X/∼.' metadata={'source': 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+page_content=' 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' “Theory of Games and Economic Behavior.” 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=', Princeton, Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' [42] Weber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 1846 “Die Lehre vom Tastsinne und Gemeingef¨uhle auf Versuche gegr¨undet.” Braunschweig, Wieweg und Sohn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} +page_content=' 39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfUvxQ/content/2301.01271v1.pdf'} diff --git a/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf b/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6172efbcd942a89600fc40e250d12fdcfb791fb9 --- /dev/null +++ b/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fcfc187e6e20dc7f03a8982da6e1ac5120c9469737d16148883308bec63f21c6 +size 408232 diff --git a/H9E4T4oBgHgl3EQfIAys/vector_store/index.faiss b/H9E4T4oBgHgl3EQfIAys/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7f394d5bfa1923eaf858ddfde57a246409441b8c --- /dev/null +++ b/H9E4T4oBgHgl3EQfIAys/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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elliptic surface +Renato Dias Costa +Abstract +Given a rational elliptic surface X over an algebraically closed field, we investigate whether a +given natural number k can be the intersection number of two sections of X. If not, we say that +k a gap number. We try to answer when gap numbers exist, how they are distributed and how to +identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation +to the classical problem of representing integers by positive-definite quadratic forms. +Contents +1 +Introduction +2 +2 +Preliminaries +4 +2.1 +The Mordell-Weil Lattice +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Gap numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.3 +Bounds cmax, cmin for the contribution term . . . . . . . . . . . . . . . . . . . . . . . +6 +2.4 +The difference ∆ = cmax − cmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.5 +The quadratic form QX +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3 +Intersection with a torsion section +10 +4 +Existence of a pair of sections with a given intersection number +11 +4.1 +Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +4.2 +Sufficient conditions when ∆ ≤ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +4.2.1 +The case ∆ < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +4.2.2 +The case ∆ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +4.3 +Necessary and sufficient conditions for ∆ ≤ 2 +. . . . . . . . . . . . . . . . . . . . . . +14 +4.4 +Summary of sufficient conditions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5 +Main Results +15 +5.1 +No gap numbers in rank r ≥ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5.2 +Gaps with probability 1 in rank r ≤ 2 +. . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5.3 +Identification of gaps when E(K) is torsion-free with rank r = 1 +. . . . . . . . . . . +18 +5.4 +Surfaces with a 1-gap +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +6 +Appendix +23 +1 + +1 +Introduction +Description of the problem. Let X be a rational elliptic surface over an algebraically closed +field, i.e. a smooth, rational projective surface with a fibration π : X → P1 whose general fiber +is a smooth curve of genus 1. Assume also that π is relatively minimal, i.e. no fiber contains an +exceptional curve in its support. We use E/K to denote the generic fiber of π, which is an elliptic +curve over the function field K := k(P1). By the Mordell-Weil theorem, the set E(K) of K-points +is a finitely generated Abelian group, whose rank we denote by r. The points on E(K) are in +bijective correspondence with the sections of π, as well as with the exceptional curves on X, so +we use these terms interchangeably. This paper addresses the following question: given sections +P1, P2 ∈ E(K), what values can the intersection number P1 · P2 possibly attain? +Original motivation. +The problem originates from a previous investigation of conic bundles +on X, i.e. morphisms ϕ : X → P1 whose general fiber is a smooth curve of genus zero [Cos]. More +specifically, one of the ways to produce a conic bundle is by finding a pair of sections P1, P2 ∈ E(K) +with P1 · P2 = 1, so that the linear system |P1 + P2| induces a conic bundle ϕ|P1+P2| : X → P1 +having P1 + P2 as a reducible fiber. We may ask under which conditions such a pair exists. An +immediate necessary condition is that r ≥ 1, for if r = 0 any two distinct sections must be disjoint +[SS19, Cor. 8.30]. Conversely, given that r ≥ 1, does X admit such a pair? The first observation +is that r ≥ 1 implies an infinite number of sections, so we should expect infinitely many values for +P1·P2 as P1, P2 run through E(K). Then the question is ultimately: what values may P1·P2 assume? +Mordell-Weil lattices. The computation of intersection numbers on a surface is a difficult prob- +lem in general. However, as we are concerned with sections on an elliptic surface, the information +we need is considerably more accessible. The reason for this lies in the Mordell-Weil lattice, a +concept first established in [Elk90], [Shi89], [Shi90]. It involves the definition of a Q-valued pair- +ing ⟨·, ·⟩ on E(K), called the height pairing [SS19, Section 6.5], inducing a positive-definite lattice +(E(K)/E(K)tor, ⟨·, ·⟩), named the Mordell-Weil lattice. +A key aspect of its construction is the +connection with the Néron-Severi lattice, so that the height pairing and the intersection pairing +of sections are strongly intertwined. In the case of rational elliptic surfaces, the possibilities for +the Mordell-Weil lattice have already been classified in [OS91], which gives us a good starting point. +Representation of integers. +The use of Mordell-Weil lattices in our investigation leads to +a classical problem in number theory, which is the representation of integers by positive-definite +quadratic forms. Indeed, the free part of E(K) is generated by r terms, so the height h(P) := ⟨P, P⟩ +induces a positive-definite quadratic form on r variables with coefficients in Q. If O ∈ E(K) is the +neutral section and R is the set of reducible fibers of π, then by the height formula (2) +h(P) = 2 + 2(P · O) − +� +v∈R +contrv(P), +where the sum over v is a rational number which can be estimated. By clearing denominators, +we see that the possible values of P · O depend on a certain range of integers represented by a +positive-definite quadratic form with coefficients in Z. This point of view is explored in some parts +of the paper, where we apply results such as the classical Lagrange’s four-square theorem [HW79, +§20.5], the counting of integers represented by a binary quadratic form [Ber12, p. 91] and the more +recent Bhargava-Hanke’s 290-theorem on universal quadratic forms [BH, Thm. 1]. +2 + +Statement of results. Given k ∈ Z≥0 we investigate whether there is a pair of sections P1, P2 ∈ +E(K) such that P1 · P2 = k. If such a pair does not exist, we say that X has a k-gap, or that k is +a gap number. Our first result is a complete identification of gap numbers in some cases: +Theorem 5.7. If E(K) is torsion-free with rank r = 1, we have the following characterization of +gap numbers on X according to the lattice T associated to the reducible fibers of π. +T +k is a gap number ⇔ none of +the following are perfect squares +E7 +k + 1, 4k + 1 +A7 +k+1 +4 , 16k, ..., 16k + 9 +D7 +k+1 +2 , 8k + 1, ..., 8k + 4 +A6 ⊕ A1 +k+1 +7 , 28k − 3, ..., 28k + 21 +E6 ⊕ A1 +k+1 +3 , 12k + 1, ..., 12k + 9 +D5 ⊕ A2 +k+1 +6 , 24k + 1, ..., 24k + 16 +A4 ⊕ A3 +k+1 +10 , 40k − 4, ..., 40k + 25 +A4 ⊕ A2 ⊕ A1 +k+1 +15 , 60k − 11, ..., 60k + 45 +We also explore the possibility of X having no gap numbers. We prove that, in fact, this is +always the case if the Mordell-Weil rank is big enough. +Theorem 5.2. If r ≥ 5, then X has no gap numbers. +On the other hand, for r ≤ 2 we show that gap numbers occur with probability 1. +Theorem 5.4. If r ≤ 2, then the set of gap numbers of X, i.e. G := {k ∈ N | k is a gap number of X} +has density 1 in N, i.e. +lim +n→∞ +#G ∩ {1, ..., n} +n += 1. +At last we answer the question from the original motivation, which consists in classifying the +rational elliptic surfaces with a 1-gap: +Theorem 5.8. X has a 1-gap if and only if r = 0 or r = 1 and π has a III∗ fiber. +3 + +Structure of the paper. The text is organized as follows. Section 2 introduces the main objects, +namely the Mordell-Weil lattice, the bounds cmax, cmin for the contribution term, the difference +∆ = cmax −cmin and the quadratic form QX induced by the height pairing. In Section 3 we explain +the role of torsion sections in the investigation. The key technical results are gathered in Section 4, +where we state necessary and sufficient conditions for having P1 · P2 = k for a given k. Section 5 +contains the main results of the paper, namely: the description of gap numbers when E(K) is +torsion-free with r = 1 (Subsection 5.3), the absence of gap numbers for r ≥ 5 (Subsection 5.1), +density of gap numbers when r ≤ 2 (Subsection 5.2) and the classification of surfaces with a 1-gap +(Subsection 5.4). Section 6 is an appendix containing Table 8, which stores the relevant information +about the Mordell-Weil lattices of rational elliptic surfaces with r ≥ 1. +2 +Preliminaries +Throughout the paper X denotes a rational elliptic surface over an algebraically closed field +k of any characteristic. More precisely, X is a smooth rational projective surface with a fibration +π : X → P1, with a section, whose general fiber is a smooth curve of genus 1. We assume moreover +that π is relatively minimal (i.e. each fiber has no exceptional curve in its support) [SS19, Def. +5.2]. The generic fiber of π is an elliptic curve E/K over K := k(P1). The set E(K) of K-points is +called the Mordell-Weil group of X, whose rank is called the Mordell-Weil rank of X, denoted by +r := rank E(K). +In what follows we introduce the main objects of our investigation and stablish some notation. +2.1 +The Mordell-Weil Lattice +We give a brief description of the Mordell-Weil lattice, which is the central tool used in the +paper. Although it can be defined on elliptic surfaces in general, we restrict ourselves to rational +elliptic surfaces. For more information on Mordell-Weil lattices, we refer the reader to the com- +prehensive introduction by Schuett and Shioda [SS19] in addition to the original sources, namely +[Elk90], [Shi89], [Shi90]. +We begin by noting that points in E(K) can be regarded as curves on X and by defining the +lattice T and the trivial lattice Triv(X), which are needed to define the Mordell-Weil lattice. +Sections, points on E(K) and exceptional curves. The sections of π are in bijective cor- +respondence with points on E(K). Moreover, since X is rational and relatively minimal, points on +E(K) also correspond to exceptional curves on X [SS10, Section 8.2]. For this reason we identify +sections of π, points on E(K) and exceptional curves on X. +The lattice T and the trivial lattice Triv(X). Let O ∈ E(K) be the neutral section and +R := {v ∈ P1 | π−1(v) is reducible} the set of reducible fibers of π. The components of a fiber +π−1(v) are denoted by Θv,i, where Θv,0 is the only component intersected by O. The Néron-Severi +group NS(X) together with the intersection pairing is called the Néron-Severi lattice. +4 + +We define the following sublattices of NS(X), which encode the reducible fibers of π: +Tv := Z⟨Θv,i | i ̸= 0⟩ for v ∈ R, +T := +� +v∈R +Tv. +By Kodaira’s classification [SS19, Thm. 5.12], each Tv with v ∈ R is represented by a Dynkin +diagram Am, Dm or Em for some m. We also define the trivial lattice of X, namely +Triv(X) := Z⟨O, Θv,i | i ≥ 0, v ∈ R⟩. +Next we define the Mordell-Weil lattice and present the height formula. +The Mordell-Weil lattice. In order to give E(K) a lattice structure, we cannot use the inter- +section pairing directly, which only defines a lattice on NS(X) but not on E(K). This is achieved +by defining a Q-valued pairing, called the height pairing, given by +⟨·, ·⟩ : E(K) × E(K) → Q +P, Q �→ −ϕ(P) · ϕ(Q), +where ϕ : E(K) → NS(X) ⊗Z Q is defined from the orthogonal projection with respect to Triv(X) +(for a detailed exposition, see [SS19, Section 6.5]). Moreover, dividing by torsion elements we get +a positive-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩) [SS19, Thm. 6.20], called the Mordell-Weil lattice. +The height formula. The height pairing can be explicitly computed by the height formula [SS19, +Thm. 6.24]. For rational elliptic surfaces, it is given by +⟨P, Q⟩ = 1 + (P · O) + (Q · O) − (P · Q) − +� +v∈R +contrv(P, Q), +(1) +h(P) := ⟨P, P⟩ = 2 + 2(P · O) − +� +v∈R +contrv(P), +(2) +where contrv(P) := contrv(P, P) and contrv(P, Q) are given by Table 1 [SS19, Table 6.1] assuming +P, Q meet π−1(v) at Θv,i, Θv,j resp. with 0 < i < j. If P or Q meets Θv,0, then contrv(P, Q) := 0. +The minimal norm. Since E(K) is finitely generated, there is a minimal positive value for h(P) +as P runs through E(K) with h(P) > 0. It is called the minimal norm, denoted by +µ := min{h(P) > 0 | P ∈ E(K)}. +The narrow Mordell-Weil lattice. An important sublattice of E(K) is the narrow Mordell-Weil +lattice E(K)0, defined as +E(K)0 := {P ∈ E(K) | P intersects Θv,0 for all v ∈ R} += {P ∈ E(K) | contrv(P) = 0 for all v ∈ R}. +As a subgroup, E(K)0 is torsion-free; as a sublattice, it is a positive-definite even integral lattice +with finite index in E(K) [SS19, Thm. 6.44]. The importance of the narrow lattice can be explained +by its considerable size as a sublattice and by the easiness to compute the height pairing on it, +since all contribution terms vanish. A complete classification of the lattices E(K) and E(K)0 on +rational elliptic surfaces is found in [OS91, Main Thm.]. +5 + +Tv +A1 +E7 +A2 +E6 +An−1 +Dn+4 +Type of π−1(v) +III +III∗ +IV +IV∗ +In +I∗ +n +contrv(P) +1 +2 +3 +2 +2 +3 +4 +3 +i(n−i) +n +� +1 +(i = 1) +1 + n +4 +(i > 1) +contrv(P, Q) +- +- +1 +3 +2 +3 +i(n−j) +n +� 1 +2 +(i = 1) +1 +2 + n +4 +(i > 1) +Table 1: Local contributions from reducible fibers to the height pairing. +2.2 +Gap numbers +We introduce some convenient terminology to express the possibility of finding a pair of sections +with a given intersection number. +Definition 2.1. If there are no sections P1, P2 ∈ E(K) such that P1 · P2 = k, we say that X has +a k-gap or that k is a gap number of X. +Definition 2.2. We say that X is gap-free if for every k ∈ Z≥0 there are sections P1, P2 ∈ E(K) +such that P1 · P2 = k. +Remark 2.3. In case the Mordell-Weil rank is r = 0, we have E(K) = E(K)tor. In particular, +any two distinct sections are disjoint [SS19, Cor. 8.30], hence every k ≥ 1 is a gap number of X. +For positive rank, the description of gap numbers is less trivial, thus our focus on r ≥ 1. +2.3 +Bounds cmax, cmin for the contribution term +We define the estimates cmax, cmin for the contribution term � +v contrv(P) and state some +simple facts about them. We also provide an example to illustrate how they are computed. +The need for these estimates comes from the following. Suppose we are given a section P ∈ E(K) +whose height h(P) is known and we want to determine P · O. In case P ∈ E(K)0 we have a direct +answer, namely P · O = h(P)/2 − 1 by the height formula (2). +However if P /∈ E(K)0, the +computation of P · O depends on the contribution term cP := � +v∈R contrv(P), which by Table 1 +depends on how P intersects the reducible fibers of π. Usually we do not have this intersection +data at hand, which is why we need estimates for cP not depending on P. +Definition 2.4. If the set R of reducible fibers of π is not empty, we define +cmax := +� +v∈R +max{contrv(P) | P ∈ E(K)}, +cmin := min {contrv(P) > 0 | P ∈ E(K), v ∈ R} . +Remark 2.5. The case R = ∅ only occurs when X has Mordell-Weil rank r = 8 (No. 1 in Table 8). +In this case E(K)0 = E(K) and � +v∈R contrv(P) = 0 ∀P ∈ E(K), hence we adopt the convention +cmax = cmin = 0. +6 + +Remark 2.6. We use cmax, cmin as bounds for cP := � +v contrv(P). For our purposes it is not +necessary to know whether cP actually attains one of these bounds for some P, so that cmax, cmin +should be understood as hypothetical values. +We state some facts about cmax, cmin. +Lemma 2.7. Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1. If π admits a +reducible fiber, then: +i) cmin > 0. +ii) cmax < 4. +iii) cmin ≤ � +v∈R contrv(P) ≤ cmax ∀P /∈ E(K)0. For P ∈ E(K)0, only the second inequality holds. +iv) If � +v∈R contrv(P) = cmin, then contrv′(P) = cmin for some v′ and contrv(P) = 0 for v ̸= v′. +Proof. Item i) is immediate from the definition of cmin. For ii) it is enough to check the values +of cmax directly in Table 8. For iii), the second inequality follows from the definition of cmax and +clearly holds for any P ∈ E(K). If P /∈ E(K)0, then cP := � +v contrv(P) > 0, so contrv0(P) > 0 +for some v0. Therefore cP ≥ contrv0(P) ≥ cmin. +For iv), let � +v contrv(P) = cmin. Assume by contradiction that there are distinct v1, v2 such +that contrvi(P) > 0 for i = 1, 2. By definition of cmin we have cmin ≤ contrvi(P) for i = 1, 2 so +cmin = +� +v +contrv(P) ≥ contrv1(P) + contrv2(P) ≥ 2cmin, +which is absurd because cmin > 0 by i). Therefore there is only one v′ with contrv′(P) > 0, while +contrv(P) = 0 for all v ̸= v′. In particular, contrv′(P) = cmin. ■ +Explicit computation. Once we know the lattice T associated with the reducible fibers of π +(Section 2.1), the computation of cmax, cmin is simple. For a fixed v ∈ R, the extreme values of the +local contribution contrv(P) are given in Table 2, which is derived from Table 1. We provide an +example to illustrate this computation. +Tv +max{contrv(P) | P ∈ E(K)} +min{contrv(P) > 0 | P ∈ E(K)} +An−1 +ℓ(n−ℓ) +n +, where ℓ := +�n +2 +� +n−1 +n +Dn+4 +1 + n +4 +1 +E6 +4 +3 +4 +3 +E7 +3 +2 +3 +2 +Table 2: Extreme values of contrv(P). +7 + +Example: Let π with fiber configuration (I4, IV, III, I1). The reducible fibers are I4, IV, III, so +T = A3 ⊕ A2 ⊕ A1. +By Table 2, the maximal contributions for A3, A2, A1 are 2·2 +4 += 1, +2 +3, +1 +2 +respectively. The minimal positive contributions are 1·3 +4 = 3 +4, 2 +3, 1 +2 respectively. Then +cmax = 1 + 2 +3 + 1 +2 = 13 +6 , +cmin = min +�3 +4, 2 +3, 1 +2 +� += 1 +2. +2.4 +The difference ∆ = cmax − cmin +In this section we explain why the value of ∆ := cmax − cmin is relevant to our discussion, +specially in Subsection 4.2. We also verify that ∆ < 2 in most cases and identify the exceptional +ones in Table 3 and Table 4. +As noted in Subsection 2.3, in case P /∈ E(K)0 and h(P) is known, the difficulty of determining +P ·O lies in the contribution term cP := � +v∈R contrv(P). In particular, the range of possible values +for cP determines the possibilities for P · O. This range is measured by the difference +∆ := cmax − cmin. +Hence a smaller ∆ means a better control over the intersection number P · O, which is why ∆ +plays an important role in determining possible intersection numbers. In Subsection 4.3 we assume +∆ ≤ 2 and state necessary and sufficient conditions for having a pair P1, P2 such that P1 · P2 = k +for a given k ≥ 0. If however ∆ > 2, the existence of such a pair is not guaranteed a priori, so a +case-by-case treatment is needed. Fortunately by Lemma 2.8 the case ∆ > 2 is rare. +Lemma 2.8. Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1. The only cases +with ∆ = 2 and ∆ > 2 are in Table 3 and 4 respectively. In particular we have ∆ < 2 whenever +E(K) is torsion-free. +No. +T +E(K) +cmax +cmin +24 +A⊕5 +1 +A∗ +1 +⊕3 ⊕ Z/2Z +5 +2 +1 +2 +38 +A3 ⊕ A⊕3 +1 +A∗ +1 ⊕ ⟨1/4⟩ ⊕ Z/2Z +5 +2 +1 +2 +53 +A5 ⊕ A⊕2 +1 +⟨1/6⟩ ⊕ Z/2Z +5 +2 +1 +2 +57 +D4 ⊕ A⊕3 +1 +A∗ +1 ⊕ (Z/2Z)⊕2 +5 +2 +1 +2 +58 +A⊕2 +3 +⊕ A1 +A∗ +1 ⊕ Z/4Z +5 +2 +1 +2 +61 +A⊕3 +2 +⊕ A1 +⟨1/6⟩ ⊕ Z/3Z +5 +2 +1 +2 +Table 3: Cases with ∆ = 2 +8 + +No. +T +E(K) +cmax +cmin +∆ +41 +A2 ⊕ A⊕4 +1 +1 +6 +� +2 +1 +1 +2 +� +⊕ Z/2Z +8 +3 +1 +2 +13 +6 +42 +A⊕6 +1 +A∗ +1 +⊕2 ⊕ (Z/2Z)⊕2 +3 +1 +2 +5 +2 +59 +A3 ⊕ A2 ⊕ A⊕2 +1 +⟨1/12⟩ ⊕ Z/2Z +8 +3 +1 +2 +13 +6 +60 +A3 ⊕ A⊕4 +1 +⟨1/4⟩ ⊕ (Z/2Z)⊕2 +3 +1 +2 +5 +2 +Table 4: Cases with ∆ > 2 +Proof. By searching Table 8 for all cases with ∆ = 2 and ∆ > 2, we obtain Table 3 and Table 4 +respectively. Notice in particular that in both tables the torsion part of E(K) is always nontrivial. +Consequently, if E(K) is torsion-free, then ∆ < 2. ■ +2.5 +The quadratic form QX +We define the positive-definite quadratic form with integer coefficients QX derived from the +height pairing. The relevance of QX is due to the fact that some conditions for having P1 · P2 = k +for some P1, P2 ∈ E(K) can be stated in terms of what integers can be represented by QX (see +Corollary 4.2 and Proposition 4.12). +The definition of QX consists in clearing denominators of the rational quadratic form induced +by the height pairing; the only question is how to find a scale factor that works in every case. More +precisely, if E(K) has rank r ≥ 1 and P1, ..., Pr are generators of its free part, then q(x1, ..., xr) := +h(x1P1 + ... + xrPr) is a quadratic form with coefficients in Q; we define QX by multiplying q by +some integer d > 0 so as to produce coefficients in Z. We show that d may always be chosen as the +determinant of the narrow lattice E(K)0. +Definition 2.9. Let X with r ≥ 1. Let P1, ..., Pr be generators of the free part of E(K). Define +QX(x1, ..., xr) := (det E(K)0) · h(x1P1 + ... + xrPr). +We check that the matrix representing QX has entries in Z, therefore QX has coefficients in Z. +Lemma 2.10. Let A be the matrix representing the quadratic form QX, i.e. Q(x1, ..., xr) = xtAx, +where x := (x1, ..., xr)t. Then A has integer entries. In particular, QX has integer coefficients. +Proof. Let P1, ..., Pr be generators of the free part of E(K) and let L := E(K)0. The free part of +E(K) is isomorphic to the dual lattice L∗ [OS91, Main Thm.], so we may find generators P 0 +1 , ..., P 0 +r +of L such that the Gram matrix B0 := (⟨P 0 +i , P 0 +j ⟩)i,j of L is the inverse of the Gram matrix +B := (⟨Pi, Pj⟩)i,j of L∗. +9 + +We claim that QX is represented by the adjugate matrix of B0, i.e. the matrix adj(B0) such +that B0 · adj(B0) = (det B0) · Ir, where Ir is the r × r identity matrix. Indeed, by construction B +represents the quadratic form h(x1P1 + ... + xrPr), therefore +QX(x1, ..., xr) = (det E(K)0) · h(x1P1 + ... + xrPr) += (det B0) · xtBx += (det B0) · xt(B0)−1x += xtadj(B0)x, +as claimed. To prove that A := adj(B0) has integer coefficients, notice that the Gram matrix +B0 of L = E(K)0 has integer coefficients (as E(K)0 is an even lattice), then so does A. ■ +We close this subsection with a simple consequence of the definition of QX. +Lemma 2.11. If h(P) = m for some P ∈ E(K), then QX represents d · m, where d := det E(K)0. +Proof. Let P1, ..., Pr be generators for the free part of E(K). Let P = a1P1 + ... + arPr + Q, where +ai ∈ Z and Q is a torsion element (possibly zero). Since torsion sections do not contribute to the +height pairing, then h(P − Q) = h(P) = m. Hence +QX(a1, ..., ar) = d · h(a1P1 + ... + arPr) += d · h(P − Q) += d · m. ■ +3 +Intersection with a torsion section +Before dealing with more technical details in Section 4, we explain how torsion sections can be +of help in our investigation, specially in Subsection 4.2. +We first note some general properties of torsion sections. As the height pairing is positive- +definite on E(K)/E(K)tor, torsion sections are inert in the sense that for each Q ∈ E(K)tor we +have ⟨Q, P⟩ = 0 for all P ∈ E(K). +Moreover, in the case of rational elliptic surfaces, torsion +sections also happen to be mutually disjoint: +Theorem 3.1. [MP89, Lemma 1.1] On a rational elliptic surface, Q1 · Q2 = 0 for any distinct +Q1, Q2 ∈ E(K)tor. In particular, if O is the neutral section, then Q·O = 0 for all Q ∈ E(K)tor\{O}. +Remark 3.2. As stated in [MP89, Lemma 1.1], Theorem 3.1 holds for elliptic surfaces over C even +without assuming X is rational. However, for an arbitrary algebraically closed field the rationality +hypothesis is needed, and a proof can be found in [SS19, Cor. 8.30]. +By taking advantage of the properties above, we use torsion sections to help us find P1, P2 ∈ +E(K) such that P1 · P2 = k for a given k ∈ Z≥0. This is particularly useful when ∆ ≥ 2, in which +case E(K)tor is not trivial by Lemma 2.8. +The idea is as follows. Given k ∈ Z≥0, suppose we can find P ∈ E(K)0 with height h(P) = 2k. +By the height formula (2), P · O = k − 1 < k, which is not yet what we need. In the next lemma +we show that replacing O with a torsion section Q ̸= O gives P · Q = k, as desired. +10 + +Lemma 3.3. Let P ∈ E(K)0 such that h(P) = 2k. Then P · Q = k for all Q ∈ E(K)tor \ {O}. +Proof. Assume there is some Q ∈ E(K)tor \ {O}. By Theorem 3.1, Q · O = 0 and by the height +formula (2), 2k = 2 + 2(P · O) − 0, hence P · O = k − 1. We use the height formula (1) for ⟨P, Q⟩ +in order to conclude that P · Q = k. Since P ∈ E(K)0, it intersects the neutral component Θv,0 of +every reducible fiber π−1(v), so contrv(P, Q) = 0 for all v ∈ R. Hence +0 = ⟨P, Q⟩ += 1 + P · O + Q · O − P · Q − +� +v∈R +contrv(P, Q) += 1 + (k − 1) + 0 − P · Q − 0 += k − P · Q. ■ +4 +Existence of a pair of sections with a given intersection number +Given k ∈ Z≥0, we state necessary and (in most cases) sufficient conditions for having +P1 ·P2 = k for some P1, P2 ∈ E(K). Necessary conditions are stated in generality in Subsection 4.1, +while sufficient ones depend on the value of ∆ and are treated separately in Subsection 4.2. In +Subsection 4.4, we collect all sufficient conditions proven in this section. +4.1 +Necessary Conditions +If k ∈ Z≥0, we state necessary conditions for having P1·P2 = k for some sections P1, P2 ∈ E(K). +We note that the value of ∆ is not relevant in this subsection, although it plays a decisive role for +sufficient conditions in Subsection 4.2. +Lemma 4.1. Let k ∈ Z≥0. If P1 · P2 = k for some P1, P2 ∈ E(K), then one of the following holds: +i) h(P) = 2 + 2k for some P ∈ E(K)0. +ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some P /∈ E(K)0. +Proof. Without loss of generality we may assume P2 is the neutral section, so that P1 · O = k. By +the height formula (2), h(P1) = 2 + 2k − c, where c := � +v contrv(P1). If P1 ∈ E(K)0, then c = 0 +and h(P1) = 2 + 2k, hence i) holds. If P1 /∈ E(K)0, then cmin ≤ c ≤ cmax by Lemma 2.7. But +h(P1) = 2 + 2k − c, therefore 2 + 2k − cmax ≤ h(P1) ≤ 2 + 2k − cmin, i.e. ii) holds. ■ +Corollary 4.2. Let k ∈ Z≥0. If P1 · P2 = k for some P1, P2 ∈ E(K), then QX represents some +integer in [d · (2 + 2k − cmax), d · (2 + 2k)], where d := det E(K)0. +Proof. +We apply Lemma 4.1 and rephrase it in terms of QX. If i) holds, then QX represents +d · (2 + 2k) by Lemma 2.11. But if ii) holds, then h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] and by +Lemma 2.11, QX represents d · h(P) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)]. In both i) and ii), +QX represents some integer in [d · (2 + 2k − cmax), d · (2 + 2k)]. ■ +11 + +4.2 +Sufficient conditions when ∆ ≤ 2 +In this subsection we state sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) +under the assumption that ∆ ≤ 2. By Lemma 2.8, this covers almost all cases (more precisely, all +but No. 41, 42, 59, 60 in Table 8). We treat ∆ < 2 and ∆ = 2 separately, as the latter needs more +attention. +4.2.1 +The case ∆ < 2 +We first prove Lemma 4.3, which gives sufficient conditions assuming ∆ < 2, then Corollary 4.5, +which states sufficient conditions in terms of integers represented by QX. +This is followed by +Corollary 4.6, which is a simplified version of Corollary 4.5. +Lemma 4.3. Assume ∆ < 2 and let k ∈ Z≥0. If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some +P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K). +Proof. Let O ∈ E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P · O) − c, +where c := � +v contrv(P). Since h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin], then +2 + 2k − cmax ≤ 2 + 2(P · O) − c ≤ 2 + 2k − cmin +⇒ c − cmax +2 +≤ P · O − k ≤ c − cmin +2 +. +Therefore P · O − k is an integer in I := +� c−cmax +2 +, c−cmin +2 +�. We prove that 0 is the only integer in +I, so that P · O − k = 0, i.e. P · O = k. First notice that c ̸= 0, as P /∈ E(K)0. By Lemma 2.7 iii), +cmin ≤ c ≤ cmax, consequently c−cmax +2 +≤ 0 ≤ c−cmin +2 +, i.e. 0 ∈ I. Moreover ∆ < 2 implies that I has +length cmax−cmin +2 += ∆ +2 < 1, so I contains no integer except 0 as desired. ■ +Remark 4.4. Lemma 4.3 also applies when cmax = cmin, in which case the closed interval degen- +erates into a point. +The following corollary of Lemma 4.3 states a sufficient condition in terms of integers represented +by the quadratic form QX (Section 2.5). +Corollary 4.5. Assume ∆ < 2 and let d := det E(K)0. If QX represents an integer not divisible +by d in the interval [d · (2+ 2k − cmax), d · (2+ 2k − cmin)], then P1 · P2 = k for some P1, P2 ∈ E(K). +Proof. Let a1, ..., ar ∈ Z such that QX(a1, ..., ar) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)] with +d ∤ QX(a1, ..., ar). Let P := a1P1 + ... + arPr, where P1, ..., Pr are generators of the free part of +E(K). Then d ∤ QX(a1, ..., ar) = d · h(P), which implies that h(P) /∈ Z. In particular P /∈ E(K)0 +since E(K)0 is an integer lattice. Moreover h(P) = 1 +dQX(a1, ..., ar) ∈ [2 + 2k − cmax, 2 + 2k − cmin] +and we are done by Lemma 4.3. ■ +12 + +The next corollary, although weaker than Corollary 4.5, is more practical for concrete examples +and is frequently used in Subsection 5.4. It does not involve finding integers represented by QX, +but only finding perfect squares in an interval depending on the minimal norm µ (Subsection 2.1). +Corollary 4.6. Assume ∆ < 2. If there is a perfect square n2 ∈ +� +2+2k−cmax +µ +, 2+2k−cmin +µ +� +such that +n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K). +Proof. Take P ∈ E(K) such that h(P) = µ. Since h(nP) = n2µ /∈ Z, we must have nP /∈ E(K)0 +as E(K)0 is an integer lattice. Moreover h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin] and we are +done by Lemma 4.3. ■ +4.2.2 +The case ∆ = 2 +The statement of sufficient conditions for ∆ = 2 is almost identical to the one for ∆ < 2: the +only difference is that the closed interval Lemma 4.3 is substituted by a right half-open interval +in Lemma 4.8. This change, however, is associated with a technical difficulty in the case when a +section has minimal contribution term, thus the separate treatment for ∆ = 2. +The results are presented in the following order. First we prove Lemma 4.7, which is a statement +about sections whose contribution term is minimal. +Next we prove Lemma 4.8, which states +sufficient conditions for ∆ = 2, then Corollaries 4.9 and 4.10. +Lemma 4.7. Assume ∆ = 2. +If there is P ∈ E(K) such that � +v∈R contrv(P) = cmin, then +P · Q = P · O + 1 for every Q ∈ E(K)tor \ {O}. +Proof. If Q ∈ E(K)tor \ {O}, then Q · O = 0 by Theorem 3.1. Moreover, by the height formula (1), +0 = ⟨P, Q⟩ = 1 + P · O + 0 − P · Q − +� +v∈R +contrv(P, Q). (∗) +Hence it suffices to show that contrv(P, Q) = 0 ∀v ∈ R. By Lemma 2.7 iv), contrv′(P) = cmin +for some v′ and contrv(P) = 0 for all v ̸= v′. In particular P meets Θv,0, hence contrv(P, Q) = 0 +for all v ̸= v′. Thus from (∗) we see that contrv′(P, Q) is an integer, which we prove is 0. +We claim that Tv′ = A1, so that contrv′(P, Q) = 0 or 1 +2 by Table 1. In this case, as contrv′(P, Q) +is an integer, it must be 0, and we are done. To see that Tv′ = A1 we analyse contrv′(P). Since +∆ = 2, then cmin = 1 +2 by Table 3 and contrv′(P) = cmin = 1 +2. By Table 1, this only happens if +Tv′ = An−1 and 1 +2 = i(n−i) +n +for some 0 ≤ i < n. The only possibility is i = 1, n = 2 and Tv′ = A1. ■ +With the aid of Lemma 4.7 we are able to state sufficient conditions for ∆ = 2. +13 + +Lemma 4.8. Assume ∆ = 2 and let k ∈ Z≥0. If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some +P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K). +Proof. Let O ∈ E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P · O) − c, +where c := � +v contrv(P). We repeat the arguments from Lemma 4.3, in this case with the right +half-open interval, so that the hypothesis that h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin), implies that +P · O − k is an integer in I′ := +� c−cmax +2 +, c−cmin +2 +�. +Since I′ is half-open with length cmax−cmin +2 += ∆ +2 = 1, then I′ contains exactly one integer. If +0 ∈ I′, then P · O − k = 0, i.e. P · O = k and we are done. Hence we assume 0 /∈ I′. +We claim that P ·O = k −1. First, notice that if c > cmin, then the inequalities cmin < c ≤ cmax +give c−cmax +2 +≤ 0 < c−cmin +2 +, i.e. 0 ∈ I′, which is a contradiction. Hence c = cmin. Since ∆ = 2, then +I′ = [−1, 0), whose only integer is −1. Thus P · O − k = −1, i.e. P · O = k − 1, as claimed. +Finally, let Q ∈ E(K)tor \ {O}, so that P · Q = P · O + 1 = k by Lemma 4.7 and we are done. +We remark that E(K)tor is not trivial by Table 3, therefore such Q exists. ■ +The following corollaries are analogues to Corollary 4.5 and Corollary 4.6 adapted to ∆ = 2. +Similarly to the case ∆ < 2, Corollary 4.9 is stronger than Corollary 4.10, although the latter is +more practical for concrete examples. We remind the reader that µ denotes the minimal norm +(Subsection 2.1). +Corollary 4.9. Assume ∆ = 2 and let d := det E(K)0. If QX represents an integer not divisible +by d in the interval [d·(2+2k −cmax), d·(2+2k −cmin)), then P1 ·P2 = k for some P1, P2 ∈ E(K). +Proof. We repeat the arguments in Corollary 4.5, in this case with the half-open interval. ■ +Corollary 4.10. Assume ∆ = 2. If there is a perfect square n2 ∈ +� +2+2k−cmax +µ +, 2+2k−cmin +µ +� +such that +n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K). +Proof. We repeat the arguments in Corollary 4.6, in this case with the half-open interval. ■ +4.3 +Necessary and sufficient conditions for ∆ ≤ 2 +For completeness, we present a unified statement of necessary and sufficient conditions assuming +∆ ≤ 2, which follows naturally from results in Subsections 4.1 and 4.2. +Lemma 4.11. Assume ∆ ≤ 2 and let k ∈ Z≥0. Then P1 · P2 = k for some P1, P2 ∈ E(K) if and +only if one of the following holds: +i) h(P) = 2 + 2k for some P ∈ E(K)0. +ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some P /∈ E(K)0. +iii) h(P) = 2 + 2k − cmin and � +v∈R contrv(P) = cmin for some P ∈ E(K). +Proof. If i) or iii) holds, then P · O = k directly by the height formula (2). But if ii) holds, it +suffices to to apply Lemma 4.3 when ∆ < 2 and by Lemma 4.8 when ∆ = 2. +Conversely, let P1·P2 = k. Without loss of generality, we may assume P2 = O, so that P1·O = k. +By the height formula (2), h(P1) = 2 + 2k − c, where c := � +v contrv(P1). +If c = 0, then P1 ∈ E(K)0 and h(P1) = 2+2k, so i) holds. Hence we let c ̸= 0, i.e. P1 /∈ E(K)0, +so that cmin ≤ c ≤ cmax by Lemma 2.7. In case c = cmin, then h(P1) = 2 + 2k − cmin and iii) holds. +Otherwise cmin < c ≤ cmax, which implies 2 + 2k − cmax ≤ h(P1) < 2 + 2k − cmin, so ii) holds. ■ +14 + +4.4 +Summary of sufficient conditions +For the sake of clarity, we summarize in a single proposition all sufficient conditions for having +P1 · P2 = k for some P1, P2 ∈ E(K) proven in this section. +Proposition 4.12. Let k ∈ Z≥0. If one of the following holds, then P1 · P2 = k for some P1, P2 ∈ +E(K). +1) h(P) = 2 + 2k for some P ∈ E(K)0. +2) h(P) = 2k for some P ∈ E(K)0 and E(K)tor is not trivial. +3) ∆ < 2 and there is a perfect square n2 ∈ +� +2+2k−cmax +µ +, 2+2k−cmin +µ +� +with n2µ /∈ Z, where µ is the +minimal norm (Subsection 2.1). In case ∆ = 2, consider the right half-open interval. +4) ∆ < 2 and the quadratic form QX represents an integer not divisible by d := det E(K)0 in the +interval [d · (2 + 2k − cmax), d · (2 + 2k − cmin)]. In case ∆ = 2, consider the right half-open +interval. +Proof. In 1) a height calculation gives 2 + 2k = h(P) = 2 + 2(P · O) − 0, so P · O = k. For +2), we apply Lemma 3.3 to conclude that P · Q = k for any Q ∈ E(K)tor \ {O}. In 3) we use +Corollary 4.6 when ∆ < 2 and Corollary 4.10 when ∆ = 2. In 4), we apply Corollary 4.5 if ∆ < 2 +and Corollary 4.9 if ∆ = 2. ■ +5 +Main Results +We prove the four main theorems of this paper, which are independent applications of the results +from Section 4. The first two are general attempts to describe when and how gap numbers occur: +Theorem 5.2 tells us that large Mordell-Weil groups prevent the existence of gaps numbers, more +precisely for Mordell-Weil rank r ≥ 5; in Theorem 5.4 we show that for small Mordell-Weil rank, +more precisely when r ≤ 2, then gap numbers occur with probability 1. The last two theorems, +on the other hand, deal with explicit values of gap numbers: Theorem 5.7 provides a complete +description of gap numbers in certain cases, while Theorem 5.8 is a classification of cases with a +1-gap. +5.1 +No gap numbers in rank r ≥ 5 +We show that if E(K) has rank r ≥ 5, then X is gap-free. Our strategy is to prove that for +every k ∈ Z≥0 there is some P ∈ E(K)0 such that h(P) = 2+2k, and by Proposition 4.12 1) we are +done. We accomplish this in two steps. First we show that this holds when there is an embedding +of A⊕ +1 or of A4 in E(K)0 (Lemma 5.1). Second, we show that if r ≥ 5, then such embedding exists, +hence X is gap-free (Theorem 5.2). +15 + +Lemma 5.1. Assume E(K)0 has a sublattice isomorphic to A⊕4 +1 +or A4. Then for every ℓ ∈ Z≥0 +there is P ∈ E(K)0 such that h(P) = 2ℓ. +Proof. +First assume A⊕4 +1 +⊂ E(K)0 and let P1, P2, P3, P4 be generators for each factor A1 in +A⊕4 +1 . Then h(Pi) = 2 and ⟨Pi, Pj⟩ = 0 for distinct i, j = 1, 2, 3, 4. +By Lagrange’s four-square +theorem [HW79, §20.5] there are integers a1, a2, a3, a4 such that a2 +1 + a2 +2 + a2 +3 + a2 +4 = ℓ. Defining +P := a1P1 + a2P2 + a3P3 + a4P4 ∈ A⊕4 +1 +⊂ E(K)0, we have +h(P) = 2a2 +1 + 2a2 +2 + 2a2 +3 + 2a2 +4 = 2ℓ. +Now let A4 ⊂ E(K)0 with generators P1, P2, P3, P4. +Then h(Pi) = 2 for i = 1, 2, 3, 4 and +⟨Pi, Pi+1⟩ = −1 for i = 1, 2, 3. We need to find integers x1, ..., x4 such that h(P) = 2ℓ, where +P := x1P1 + ... + x4P4 ∈ A4 ⊂ E(K)0. Equivalently, we need that +ℓ = 1 +2⟨P, P⟩ = x2 +1 + x2 +2 + x2 +3 + x2 +4 − x1x2 − x2x3 − x3x4. +Therefore ℓ must be represented by q(x1, ..., x4) := x2 +1 + x2 +2 + x2 +3 + x2 +4 − x1x2 − x2x3 − x3x4. We +prove that q represents all positive integers. Notice that q is positive-definite, since it is induced +by ⟨·, ·⟩. By Bhargava-Hanke’s 290-theorem [BH][Thm. 1], q represents all positive integers if and +only if it represents the following integers: +2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, +29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290. +The representation for each of the above is found in Table 5. ■ +We now prove the main theorem of this section. +Theorem 5.2. If r ≥ 5, then X is gap-free. +Proof. We show that for every k ≥ 0 there is P ∈ E(K)0 such that h(P) = 2 + 2k, so that by +Proposition 4.12 1) we are done. Using Lemma 5.1 it suffices to prove that E(K)0 has a sublattice +isomorphic to A⊕4 +1 +or A4. +The cases with r ≥ 5 are No. +1-7 (Table 8). +In No. +1-6, E(K)0 = E8, E7, E6, D6, D5, A5 +respectively. Each of these admit an A4 sublattice [Nis96, Lemmas 4.2,4.3]. In No. 7 we claim that +E(K)0 = D4 ⊕ A1 has an A⊕4 +1 +sublattice. This is the case because D4 admits an A⊕4 +1 +sublattice +[Nis96, Lemma 4.5 (iii)]. ■ +16 + +n +x1, x2, x3, x4 with x2 +1 + x2 +2 + x2 +3 + x2 +4 − x1x2 − x2x3 − x3x4 = n +1 +1, 0, 0, 0 +2 +1, 0, 1, 0 +3 +1, 1, 2, 0 +5 +1, 0, 2, 0 +6 +1, 1, −2, −1 +7 +1, 1, −2, 0 +10 +1, 0, 3, 0 +13 +2, 0, 3, 0 +14 +1, 2, 5, 1 +15 +1, 5, 5, 2 +17 +1, 0, 4, 0 +19 +1, 5, 3, −1 +21 +1, 5, 0, 0 +22 +1, 5, 0, −1 +23 +1, 6, 6, 2 +26 +1, 0, 5, 0 +29 +2, 0, 5, 0 +30 +1, 5, 0, −3 +31 +1, 3, −4, −2 +34 +3, 0, 5, 0 +35 +1, 2, −2, 4 +37 +1, 0, 6, 0 +42 +1, 1, −4, 3 +58 +3, 0, 7, 0 +93 +1, 1, −10, 0 +110 +1, −2, 3, −8 +145 +1, 0, 12, 0 +203 +1, −5, −9, 8 +290 +1, 0, 17, 0 +Table 5: Representation of the critical integers in Bhargava-Hanke’s 290-theorem. +5.2 +Gaps with probability 1 in rank r ≤ 2 +Fix a rational elliptic surface π : X → P1 with Mordell-Weil rank r ≤ 2. We prove that if k is +a uniformly random natural number, then k is a gap number with probability 1. More precisely, if +G := {k ∈ N | k is a gap number of X} is the set of gap numbers, then G ⊂ N has density 1, i.e. +d(G) := lim +n→∞ +#G ∩ {1, ..., n} +n += 1. +17 + +We adopt the following strategy. If k ∈ N \ G, then P1 · P2 = k for some P1, P2 ∈ E(K) and +by Corollary 4.2 the quadratic form QX represents some integer t depending on k. This defines a +function N\G → T, where T is the set of integers represented by QX. Since QX is a quadratic form +on r ≤ 2 variables, T has density 0 in N by Lemma 5.3. By analyzing the pre-images of N\G → T, +in Theorem 5.4 we conclude that d(N \ G) = d(T) = 0, hence d(G) = 1 as desired. +Lemma 5.3. Let Q be a positive-definite quadratic form on r = 1, 2 variables with integer coeffi- +cients. Then the set of integers represented by Q has density 0 in N. +Proof. Let S be the set of integers represented by Q. If d is the greatest common divisor of the +coefficients of Q, let S′ be the set of integers representable by the primitive form Q′ := 1 +d · Q. By +construction S′ is a rescaling of S, so d(S) = 0 if and only if d(S′) = 0. +If r = 1, then Q′(x1) = x2 +1 and S′ is the set of perfect squares, so clearly d(S′) = 0. If r = 2, +then Q′ is a binary quadratic form and the number of elements in S′ bounded from above by x > 0 +is given by C · +x +√log x + o(x) with C > 0 a constant and limx→∞ +o(x) +x += 0 [Ber12, p. 91]. Thus +d(S′) = lim +x→∞ +C +√log x + o(x) +x += 0. ■ +We now prove the main result of this section. +Theorem 5.4. Let π : X → P1 be a rational elliptic surface with Mordell-Weil rank r ≤ 2. Then +the set G := {k ∈ N | k is a gap number of X} of gap numbers of X has density 1 in N. +Proof. If r = 0, then the claim is trivial by Remark 2.3, hence we may assume r = 1, 2. We +prove that S := N \ G has density 0. +If S is finite, there is nothing to prove. +Otherwise, let +k1 < k2 < ... be the increasing sequence of all elements of S. By Corollary 4.2, for each n there is +some tn ∈ Jkn := [d · (2 + 2kn − cmax), d · (2 + 2kn)] represented by the quadratic form QX. Let T +be the set of integers represented by QX and define the function f : N \ G → T by kn �→ tn. Since +QX has r = 1, 2 variables, T has density 0 by Lemma 5.3. +For N > 0, let SN := S ∩ {1, ..., N} and TN := T ∩ {1, ..., N}. +Since T has density zero, +#TN = o(N), i.e. +#TN +N +→ 0 when N → ∞ and we need to prove that #SN = o(N). We analyze +the function f restricted to SN. Notice that as tn ∈ Jkn, then kn ≤ N implies tn ≤ d · (2 + 2kn) ≤ +d · (2 + 2N). Hence the restriction g := f|SN can be regarded as a function g : SN → Td·(2+2k). +We claim that #g−1(t) ≤ 2 for all t ∈ Td·(2+2N), in which case #SN ≤ 2 · #Td·(2+2N) = o(N) +and we are done. Assume by contradiction that g−1(t) contains three distinct elements, say kℓ1 < +kℓ2 < kℓ3 with t = tℓ1 = tℓ2 = tℓ3. Since tℓi ∈ Jkℓi for each i = 1, 2, 3, then t ∈ Jkℓ1 ∩ Jkℓ2 ∩ Jkℓ3. We +prove that Jkℓ1 and Jkℓ3 are disjoint, which yields a contradiction. Indeed, since kℓ1 < kℓ2 < kℓ3, +in particular kℓ3 − kℓ1 ≥ 2, therefore d · (2 + 2kℓ1) ≤ d · (2 + 2kℓ3 − 4). But cmax < 4 by Lemma 2.7, +so d · (2 + 2kℓ1) < d · (2 + 2kℓ3 − cmax), i.e. max Jkℓ1 < min Jkℓ3. Thus Jkℓ1 ∩ Jkℓ3 = ∅, as desired. ■ +5.3 +Identification of gaps when E(K) is torsion-free with rank r = 1 +The results in Subsections 5.1 and 5.2 concern the existence and the distribution of gap num- +bers. In the following subsections we turn our attention to finding gap numbers explicitly. In this +subsection we give a complete description of gap numbers assuming E(K) is torsion-free with rank +r = 1. Such descriptions are difficult in the general case, but our assumption guarantees that each +18 + +E(K), E(K)0 is generated by a single element and that ∆ < 2 by Lemma 2.8, which makes the +problem more accessible. +We organize this subsection as follows. First we point out some trivial facts about generators +of E(K), E(K)0 when r = 1 in Lemma 5.5. Next we state necessary and sufficient conditions for +having P1 · P2 = k when E(K) is torsion-free with r = 1 in Lemma 5.6. As an application of the +latter, we prove Theorem 5.7, which is the main result of the subsection. +Lemma 5.5. Let X be a rational elliptic surface with Mordell-Weil rank r = 1. If P generates the +free part of E(K), then +a) h(P) = µ. +b) 1/µ is an even integer. +c) E(K)0 is generated by P0 := (1/µ)P and h(P0) = 1/µ. +Proof. Item a) is clear. Items b), c) follow from the fact that E(K)0 is an even lattice and that +E(K) ≃ L∗ ⊕ E(K)tor, where L := E(K)0 [OS91, Main Thm.]. ■ +In what follows we use Lemma 5.5 and results from Section 4 to state necessary and sufficient +conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) in case E(K) is torsion-free with r = 1. +Lemma 5.6. Assume E(K) is torsion-free with rank r = 1. Then P1 · P2 = k for some P1, P2 ∈ +E(K) if and only if one of the following holds: +i) µ · (2 + 2k) is a perfect square. +ii) There is a perfect square n2 ∈ +� +2+2k−cmax +µ +, 2+2k−cmin +µ +� +such that µ · n /∈ Z. +Proof. By Lemma 5.5, E(K) is generated by some P with h(P) = µ and E(K)0 is generated by +P0 := n0P, where n0 := 1 +µ ∈ 2Z. +First assume that P1·P2 = k for some P1, P2. Without loss of generality we may assume P2 = O. +Let P1 = nP for some n ∈ Z. We show that P1 ∈ E(K)0 implies i) while P1 /∈ E(K)0 implies ii). +If P1 ∈ E(K)0, then n0 | n, hence P1 = nP = mP0, where m := +n +n0. By the height formula (2), +2 + 2k = h(P1) = h(mP0) = m2 · 1 +µ. Hence µ · (2 + 2k) = m2, i.e. i) holds. +If P1 /∈ E(K)0, then n0 ∤ n, hence µ · n = +n +n0 /∈ Z. Moreover, h(P1) = n2h(P) = n2µ and by +the height formula (2), n2µ = h(P) = 2 + 2k − c, where c := � +v contrv(P1) ̸= 0. The inequalities +cmin ≤ c ≤ cmax then give 2+2k−cmax +µ +≤ n2 ≤ 2+2k−cmin +µ +. Hence ii) holds. +Conversely, assume i) or ii) holds. Since E(K) is torsion-free, ∆ < 2 by Lemma 2.8, so we may +apply Lemma 4.3. If i) holds, then µ · (2 + 2k) = m2 for some m ∈ Z. Since mP0 ∈ E(K)0 and +h(mP0) = +m2 +µ += 2 + 2k, we are done by Lemma 4.3 i). +If ii) holds, the condition µ · n /∈ Z +is equivalent to n0 ∤ n, hence nP +/∈ E(K)0. +Moreover n2 ∈ +� +2+2k−cmax +µ +, 2+2k−cmin +µ +� +, implies +h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin]. By Lemma 4.3 ii), we are done. ■ +By applying Lemma 5.6 to all possible cases where E(K) is torsion-free with rank r = 1, +we obtain the main result of this subsection. +19 + +Theorem 5.7. If E(K) is torsion-free with rank r = 1, then all the gap numbers of X are described +in Table 6. +No. +T +k is a gap number ⇔ none of +the following are perfect squares +first gap numbers +43 +E7 +k + 1, 4k + 1 +1, 4 +45 +A7 +k+1 +4 , 16k, ..., 16k + 9 +8, 11 +46 +D7 +k+1 +2 , 8k + 1, ..., 8k + 4 +2, 5 +47 +A6 ⊕ A1 +k+1 +7 , 28k − 3, ..., 28k + 21 +12, 16 +49 +E6 ⊕ A1 +k+1 +3 , 12k + 1, ..., 12k + 9 +3, 7 +50 +D5 ⊕ A2 +k+1 +6 , 24k + 1, ..., 24k + 16 +6, 11 +55 +A4 ⊕ A3 +k+1 +10 , 40k − 4, ..., 40k + 25 +16, 20 +56 +A4 ⊕ A2 ⊕ A1 +k+1 +15 , 60k − 11, ..., 60k + 45 +22, 27 +Table 6: Description of gap numbers when E(K) is torsion-free with r = 1. +Proof. For the sake of brevity we restrict ourselves to No. 55. The other cases are treated similarly. +Here cmax = 2·3 +5 + 2·2 +4 = 11 +5 , cmin = min +� +4 +5, 3 +4 +� += 3 +4 and µ = 1/20. +By Lemma 5.6, k is a gap number if and only if neither i) nor ii) occurs. Condition i) is that +2+2k +20 += k+1 +10 +is a perfect square. Condition ii) is that +� +2+2k−cmax +µ +, 2+2k−cmin +µ +� += [40k − 4, 40k + 25] +contains some n2 with 20 ∤ n. We check that 20 ∤ n for every n such that n2 = 40k + ℓ, with +ℓ = −4, ..., 25. Indeed, if 20 | n, then 400 | n2 and in particular 40 | n2. Then 40 | (n2 − 40k) = ℓ, +which is absurd. ■ +5.4 +Surfaces with a 1-gap +In Subsection 5.3 we take each case in Table 6 and describe all its gap numbers. +In this +subsection we do the opposite, which is to fix a number and describe all cases having it as a gap +number. We remind the reader that our motivating problem (Section 1) was to determine when +there are sections P1, P2 such that P1 · P2 = 1, which induce a conic bundle having P1 + P2 as a +reducible fiber. The answer for this question is the main theorem of this subsection: +Theorem 5.8. Let X be a rational elliptic surface. Then X has a 1-gap if and only if r = 0 or +r = 1 and π has a III∗ fiber. +20 + +Our strategy for the proof is the following. We already know that a 1-gap exists whenever r = 0 +(Theorem 3.1) or when r = 1 and π has a III∗ fiber (Theorem 5.7, No. 43). Conversely, we need to +find P1, P2 with P1 · P2 = 1 in all cases with r ≥ 1 and T ̸= E7. +First we introduce two lemmas, which solve most cases with little computation, and leave the +remaining ones for the proof of Theorem 5.8. In both Lemma 5.9 and Lemma 5.11 our goal is to +analyze the narrow lattice E(K)0 and apply Proposition 4.12 to detect cases without a 1-gap. +Lemma 5.9. If one of the following holds, then h(P) = 4 for some P ∈ E(K)0. +a) The Gram matrix of E(K)0 has a 4 in its main diagonal. +b) There is an embedding of An ⊕ Am in E(K)0 for some n, m ≥ 1. +c) There is an embedding of An, Dn or En in E(K)0 for some n ≥ 3. +Proof. Case a) is trivial. Assuming b), we take generators P1, P2 from An, Am respectively with +h(P1) = h(P2) = 2. Since An, Am are in direct sum, ⟨P1, P2⟩ = 0, hence h(P1 + P2) = 4, as desired. +If c) holds, then the fact that n ≥ 3 allows us to choose two elements P1, P2 among the generators +of L1 = An, Dn or En such that h(P1) = h(P2) = 2 and ⟨P1, P2⟩ = 0. Thus h(P1 + P2) = 4 as +claimed. ■ +Corollary 5.10. In the following cases, X does not have a 1-gap. +• r ≥ 3 : all cases except possibly No. 20. +• r = 1, 2 : cases No. 25, 26, 30, 32-36, 38, 41, 42, 46, 52, 54, 60. +Proof. We look at column E(K)0 in Table 8 to find which cases satisfy one of the conditions a), +b), c) from Lemma 5.9. +a) Applies to No. 12, 17, 19, 22, 23, 25, 30, 32, 33, 36, 38, 41, 46, 52, 54, 60. +b) Applies to No. 10, 11, 14, 15, 18, 24, 26, 34, 35, 42. +c) Applies to No. 1-10, 13, 16, 21. +In particular, this covers all cases with r ≥ 3 (No. 1-24) except No. 20. By Lemma 5.9 in each +of these cases there is P ∈ E(K)0 with h(P) = 4 and we are done by Proposition 4.12 1). ■ +In the next lemma we also analyze E(K)0 to detect surfaces without a 1-gap. +Lemma 5.11. Assume E(K)0 ≃ An for some n ≥ 1 and that E(K) has nontrivial torsion part. +Then X does not have a 1-gap. This applies to cases No. 28, 39, 44, 48, 51, 57, 58 in Table 8. +Proof. Take a generator P of E(K)0 with h(P) = 2 and apply Proposition 4.12 2). ■ +21 + +We are ready to prove the main result of this subsection. +Proof of Theorem 5.8. We need to show that in all cases where r ≥ 1 and T ̸= E7 there are +P1, P2 ∈ E(K) such that P1 · P2 = 1. This corresponds to cases No. 1-61 except 43 in Table 8. +The cases where r = 1 and E(K) is torsion-free can be solved by Theorem 5.10, namely No. +45-47, 49, 50, 55, 56. Adding these cases to the ones treated in Corollary 5.10 and Lemma 5.11, +we have therefore solved the following: +No. 1-19, 21-26, 28, 30, 32-36, 38, 39, 41-52, 54-58, 60. +For the remaining cases, we apply Proposition 4.12 3), which involves finding perfect squares +in the interval +� +4−cmax +µ +, 4−cmin +µ +� +(see Table 7), considering the half-open interval in the cases with +∆ = 2 (No. 53, 61). +No. +T +E(K) +µ +I +n2 ∈ I +20 +A⊕2 +2 +⊕ A1 +A∗ +2 ⊕ ⟨1/6⟩ +1 +6 +[13, 23] +42 +27 +E6 +A∗ +2 +2 +3 +[4, 4] +22 +29 +A5 ⊕ A1 +A∗ +1 ⊕ ⟨1/6⟩ +1 +6 +[12, 21] +42 +31 +A4 ⊕ A2 +1 +15 +� +2 +1 +1 +8 +� +2 +15 +[16, 21] +42 +37 +A3 ⊕ A2 ⊕ A1 +A∗ +1 ⊕ ⟨1/12⟩ +1 +12 +[22, 28] +52 +40 +A⊕2 +2 +⊕ A⊕2 +1 +⟨1/6⟩⊕2 +1 +6 +[10, 21] +42 +53 +A5 ⊕ A⊕2 +1 +⟨1/6⟩ ⊕ Z/2Z +1 +6 +[9, 12] +32 +59 +A3 ⊕ A2 ⊕ A⊕2 +1 +⟨1/12⟩ ⊕ Z/2Z +1 +12 +[16, 42] +42, 52, 62 +61 +A⊕3 +2 +⊕ A1 +⟨1/6⟩ ⊕ Z/3Z +1 +6 +[9, 12] +32 +Table 7: Perfect squares in the interval I := +� +4−cmax +µ +, 4−cmin +µ +� +. +In No. 59 we have ∆ > 2, so a particular treatment is needed. Let T = Tv1 ⊕ Tv2 ⊕ Tv3 ⊕ Tv4 = +A3 ⊕ A2 ⊕ A1 ⊕ A1. If P generates the free part of E(K) and Q generates its torsion part, then +h(P) = +1 +12 and 4P + Q meets the reducible fibers at Θv1,2, Θv2,1, Θv3,1, Θv4,1 [Kur14][Example 1.7]. +By Table 1 and the height formula (2), +42 +12 = h(4P + Q) = 2 + 2(4P + Q) · O − 2 · 2 +4 +− 1 · 2 +3 +− 1 +2 − 1 +2, +hence (4P + Q) · O = 1, as desired. ■ +22 + +6 +Appendix +We reproduce part of the table in [OS91, Main Th.] with data on Mordell-Weil lattices of +rational elliptic surfaces with Mordell-Weil rank r ≥ 1. We only add columns cmax, cmin, ∆. +No. +r +T +E(K)0 +E(K) +cmax +cmin +∆ +1 +8 +0 +E8 +E8 +0 +0 +0 +2 +7 +A1 +E7 +E∗ +8 +1 +2 +1 +2 +0 +3 +6 +A2 +E6 +E∗ +6 +2 +3 +2 +3 +0 +4 +A⊕2 +1 +D6 +D∗ +6 +3 +2 +1 +1 +2 +5 +5 +A3 +D5 +D∗ +5 +1 +3 +4 +1 +4 +6 +A2 ⊕ A1 +A5 +A∗ +5 +7 +6 +1 +2 +2 +3 +7 +A⊕3 +1 +D4 ⊕ A1 +D∗ +4 ⊕ A∗ +1 +3 +2 +1 +2 +1 +8 +4 +A4 +A4 +A∗ +4 +6 +5 +4 +5 +2 +5 +9 +D4 +D4 +D∗ +4 +1 +1 +0 +10 +A3 ⊕ A1 +A3 ⊕ A1 +A∗ +3 ⊕ A∗ +1 +3 +2 +1 +2 +1 +11 +A⊕2 +2 +A⊕2 +2 +A∗ +2 +⊕2 +4 +3 +2 +3 +2 +3 +12 +A2 ⊕ A⊕2 +1 + + + + + +4 +−1 +0 +1 +−1 +2 +−1 +0 +0 +−1 +2 +−1 +1 +0 +−1 +2 + + + + + +1 +6 + + + + + +2 +1 +0 +−1 +1 +5 +3 +1 +0 +3 +6 +3 +−1 +1 +3 +5 + + + + + +5 +3 +1 +2 +7 +6 +13 +A⊕4 +1 +D4 +D∗ +4 ⊕ Z/2Z +2 +1 +2 +3 +2 +14 +A⊕4 +1 +A⊕4 +1 +A∗ +1 +⊕4 +2 +1 +2 +3 +2 +15 +3 +A5 +A2 ⊕ A1 +A∗ +2 ⊕ A∗ +1 +3 +2 +5 +6 +2 +3 +16 +D5 +A3 +A∗ +3 +5 +4 +1 +1 +4 +17 +A4 ⊕ A1 + + + +4 +−1 +1 +−1 +2 +−1 +1 +−1 +2 + + + +1 +10 + + + +3 +1 +−1 +1 +7 +3 +−1 +3 +7 + + + +17 +10 +1 +2 +6 +5 +18 +D4 ⊕ A1 +A⊕3 +1 +A∗ +1 +⊕3 +3 +2 +1 +2 +1 +19 +A3 ⊕ A2 + + + +2 +0 +−1 +0 +2 +−1 +−1 +−1 +4 + + + +1 +12 + + + +7 +1 +2 +1 +7 +2 +2 +2 +4 + + + +5 +3 +2 +3 +1 +23 + +20 +A⊕2 +2 +⊕ A1 +A2 ⊕ ⟨6⟩ +A∗ +2 ⊕ ⟨1/6⟩ +11 +6 +1 +2 +4 +3 +21 +A3 ⊕ A⊕2 +1 +A3 +A∗ +3 ⊕ Z/2Z +2 +1 +2 +3 +2 +22 +A3 ⊕ A⊕2 +1 +A1 ⊕ ⟨4⟩ +A∗ +1 ⊕ ⟨1/4⟩ +2 +1 +2 +3 +2 +23 +A2 ⊕ A⊕3 +1 +A1 ⊕ +� +4 +−2 +−2 +4 +� +A∗ +1 ⊕ 1 +6 +� +2 +1 +1 +2 +� +13 +6 +1 +2 +5 +3 +24 +A⊕5 +1 +A⊕3 +1 +A∗ +1 +⊕3 ⊕ Z/2Z +5 +2 +1 +2 +2 +25 +2 +A6 +� +4 +−1 +−1 +2 +� +1 +7 +� +2 +1 +1 +4 +� +12 +7 +6 +7 +6 +7 +26 +D6 +A⊕2 +1 +A∗ +1 +⊕2 +3 +2 +1 +1 +2 +27 +E6 +A2 +A∗ +2 +4 +3 +4 +3 +0 +28 +A5 ⊕ A1 +A2 +A∗ +2 ⊕ Z/2Z +2 +1 +2 +3 +2 +29 +A5 ⊕ A1 +A1 ⊕ ⟨6⟩ +A∗ +1 ⊕ ⟨1/6⟩ +2 +1 +2 +3 +2 +30 +D5 ⊕ A1 +A1 ⊕ ⟨4⟩ +A∗ +1 ⊕ ⟨1/4⟩ +7 +4 +1 +2 +5 +4 +31 +A4 ⊕ A2 +� +8 +−1 +−1 +2 +� +1 +15 +� +2 +1 +1 +8 +� +28 +15 +2 +3 +6 +5 +32 +D4 ⊕ A2 +� +4 +−2 +−2 +4 +� +1 +6 +� +2 +1 +1 +2 +� +5 +3 +2 +3 +1 +33 +A4 ⊕ A⊕2 +1 +� +6 +−2 +−2 +4 +� +1 +10 +� +2 +1 +1 +3 +� +11 +5 +1 +2 +17 +10 +34 +D4 ⊕ A⊕2 +1 +A⊕2 +1 +A∗ +1 +⊕2 +2 +1 +2 +3 +2 +35 +A⊕2 +3 +A⊕2 +1 +A∗ +1 +⊕2 ⊕ Z/2Z +2 +3 +4 +5 +4 +36 +A⊕2 +3 +⟨4⟩⊕2 +⟨1/4⟩⊕2 +2 +3 +4 +5 +4 +37 +A3 ⊕ A2 ⊕ A1 +A1 ⊕ ⟨12⟩ +A∗ +1 ⊕ ⟨1/12⟩ +13 +6 +1 +2 +5 +3 +38 +A3 ⊕ A⊕3 +1 +A1 ⊕ ⟨4⟩ +A∗ +1 ⊕ ⟨1/4⟩ ⊕ Z/2Z +5 +2 +1 +2 +2 +39 +A⊕3 +2 +A2 +A∗ +2 ⊕ Z/3Z +2 +2 +3 +4 +3 +40 +A⊕2 +2 +⊕ A⊕2 +1 +⟨6⟩⊕2 +⟨1/6⟩⊕2 +7 +3 +1 +2 +11 +6 +24 + +41 +A2 ⊕ A⊕4 +1 +� +4 +−2 +−2 +4 +� +1 +6 +� +2 +1 +1 +2 +� +8 +3 +1 +2 +13 +6 +42 +A⊕6 +1 +A⊕2 +1 +A∗ +1 +⊕2 ⊕ (Z/2Z)2 +3 +1 +2 +5 +2 +43 +1 +E7 +A1 +A∗ +1 +3 +2 +3 +2 +0 +44 +A7 +A1 +A∗ +1 ⊕ Z/2Z +2 +7 +8 +11 +8 +45 +A7 +⟨8⟩ +⟨1/8⟩ +2 +7 +8 +11 +8 +46 +D7 +⟨4⟩ +⟨1/4⟩ +7 +4 +1 +3 +4 +47 +A6 ⊕ A1 +⟨14⟩ +⟨1/14⟩ +31 +14 +1 +2 +12 +7 +48 +D6 ⊕ A1 +A1 +A∗ +1 +2 +3 +2 +1 +2 +49 +E6 ⊕ A1 +⟨6⟩ +⟨1/6⟩ +11 +6 +1 +2 +4 +3 +50 +D5 ⊕ A2 +⟨12⟩ +⟨1/12⟩ +23 +12 +2 +3 +5 +4 +51 +A5 ⊕ A2 +A1 +A∗ +1 ⊕ Z/3Z +13 +6 +2 +3 +3 +2 +52 +D5 ⊕ A⊕2 +1 +⟨4⟩ +⟨1/4⟩ ⊕ Z/2Z +9 +4 +1 +2 +7 +4 +53 +A5 ⊕ A⊕2 +1 +⟨6⟩ +⟨1/6⟩ ⊕ Z/2Z +5 +2 +1 +2 +2 +54 +D4 ⊕ A3 +⟨4⟩ +⟨1/4⟩ ⊕ Z/2Z +2 +3 +4 +5 +4 +55 +A4 ⊕ A3 +⟨20⟩ +⟨1/20⟩ +11 +5 +3 +4 +29 +20 +56 +A4 ⊕ A2 ⊕ A1 +⟨30⟩ +⟨1/30⟩ +71 +30 +1 +2 +28 +15 +57 +D4 ⊕ A⊕3 +1 +A1 +A∗ +1 +5 +2 +1 +2 +2 +58 +A⊕2 +3 +⊕ A1 +A1 +A∗ +1 ⊕ Z/4Z +5 +2 +1 +2 +2 +59 +A3 ⊕ A2 ⊕ A⊕2 +1 +⟨12⟩ +⟨1/12⟩ ⊕ Z/2Z +8 +3 +1 +2 +13 +6 +60 +A3 ⊕ A⊕4 +1 +⟨4⟩ +⟨1/4⟩ ⊕ Z/2Z +3 +1 +2 +5 +2 +61 +A⊕3 +2 +⊕ A1 +⟨6⟩ +⟨1/6⟩ ⊕ Z/3Z +5 +2 +1 +2 +2 +Table 8: +Mordell-Weil lattices of rational elliptic surfaces +with Mordell-Weil rank r ≥ 1. +25 + +References +[Ber12] P. Bernays. Über die Darstellung von positiven, ganzen Zahlen durch die primitive, binären +quadratischen Formen einer nicht-quadratischen Diskriminante. PhD thesis, Göttingen, +1912. +[BH] +M. Bhargava and J. Hanke. Universal quadratic forms and the 290-Theorem. Preprint at +http://math.stanford.edu/~vakil/files/290-Theorem-preprint.pdf. +[Cos] +R. D. Costa. +Classification of fibers of conic bundles on rational elliptic surfaces. +arXiv:2206.03549. +[Elk90] +N. D. Elkies. The Mordell-Weil lattice of a rational elliptic surface. Arbeitstagung Bonn, +1990. +[HW79] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Clarendon +Press, 1979. +[Kur14] Y. Kurumadani. Pencil of cubic curves and rational elliptic surfaces. +Master’s thesis, +Kyoto University, 2014. +[MP89] R. Miranda and U. Persson. Torsion groups of elliptic surfaces. Compositio Mathematica, +72(3):249–267, 1989. +[Nis96] +K. Nishiyama. The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups. +Japanese Journal of Mathematics, 22(2), 1996. +[OS91] +K. Oguiso and T. Shioda. The Mordell-Weil lattice of a rational elliptic surface. Com- +mentarii Mathematici Universitatis Sancti Pauli, 40, 1991. +[Shi89] +T. Shioda. The Mordell-Weil lattice and Galois representation, I, II, III. Proceedings of +the Japan Academy, 65(7), 1989. +[Shi90] +T. Shioda. On the Mordell-Weil lattices. Commentarii Mathematici Universitatis Sancti +Pauli, 39(7), 1990. +[SS10] +M. Schuett and T. Shioda. +Elliptic surfaces. +Advanced Studies in Pure Mathematics, +60:51–160, 2010. +[SS19] +M. Schuett and T. Shioda. Mordell-Weil Lattices, volume 70 of Ergebnisse der Mathematik +und ihrer Grenzgebiete. Springer, 2019. +26 + diff --git a/JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt b/JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0cd22abe3907f7a2289dfacac66cfd8786d4092 --- /dev/null +++ b/JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt @@ -0,0 +1,2414 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf,len=2413 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='03137v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='NT] 9 Jan 2023 Gaps on the intersection numbers of sections on a rational elliptic surface Renato Dias Costa Abstract Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If not, we say that k a gap number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We try to answer when gap numbers exist, how they are distributed and how to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 The Mordell-Weil Lattice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Gaps with probability 1 in rank r ≤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Identification of gaps when E(K) is torsion-free with rank r = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 Surfaces with a 1-gap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20 6 Appendix 23 1 1 Introduction Description of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface over an algebraically closed field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' a smooth, rational projective surface with a fibration π : X → P1 whose general fiber is a smooth curve of genus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume also that π is relatively minimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' no fiber contains an exceptional curve in its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use E/K to denote the generic fiber of π, which is an elliptic curve over the function field K := k(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the Mordell-Weil theorem, the set E(K) of K-points is a finitely generated Abelian group, whose rank we denote by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The points on E(K) are in bijective correspondence with the sections of π, as well as with the exceptional curves on X, so we use these terms interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This paper addresses the following question: given sections P1, P2 ∈ E(K), what values can the intersection number P1 · P2 possibly attain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Original motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The problem originates from a previous investigation of conic bundles on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' morphisms ϕ : X → P1 whose general fiber is a smooth curve of genus zero [Cos].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More specifically, one of the ways to produce a conic bundle is by finding a pair of sections P1, P2 ∈ E(K) with P1 · P2 = 1, so that the linear system |P1 + P2| induces a conic bundle ϕ|P1+P2| : X → P1 having P1 + P2 as a reducible fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We may ask under which conditions such a pair exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' An immediate necessary condition is that r ≥ 1, for if r = 0 any two distinct sections must be disjoint [SS19, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, given that r ≥ 1, does X admit such a pair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The first observation is that r ≥ 1 implies an infinite number of sections, so we should expect infinitely many values for P1·P2 as P1, P2 run through E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then the question is ultimately: what values may P1·P2 assume?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Mordell-Weil lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The computation of intersection numbers on a surface is a difficult prob- lem in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' However, as we are concerned with sections on an elliptic surface, the information we need is considerably more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The reason for this lies in the Mordell-Weil lattice, a concept first established in [Elk90], [Shi89], [Shi90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' It involves the definition of a Q-valued pair- ing ⟨·, ·⟩ on E(K), called the height pairing [SS19, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5], inducing a positive-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩), named the Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' A key aspect of its construction is the connection with the Néron-Severi lattice, so that the height pairing and the intersection pairing of sections are strongly intertwined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the case of rational elliptic surfaces, the possibilities for the Mordell-Weil lattice have already been classified in [OS91], which gives us a good starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Representation of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The use of Mordell-Weil lattices in our investigation leads to a classical problem in number theory, which is the representation of integers by positive-definite quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, the free part of E(K) is generated by r terms, so the height h(P) := ⟨P, P⟩ induces a positive-definite quadratic form on r variables with coefficients in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If O ∈ E(K) is the neutral section and R is the set of reducible fibers of π, then by the height formula (2) h(P) = 2 + 2(P · O) − � v∈R contrv(P), where the sum over v is a rational number which can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By clearing denominators, we see that the possible values of P · O depend on a certain range of integers represented by a positive-definite quadratic form with coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This point of view is explored in some parts of the paper, where we apply results such as the classical Lagrange’s four-square theorem [HW79, §20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5], the counting of integers represented by a binary quadratic form [Ber12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 91] and the more recent Bhargava-Hanke’s 290-theorem on universal quadratic forms [BH, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2 Statement of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Given k ∈ Z≥0 we investigate whether there is a pair of sections P1, P2 ∈ E(K) such that P1 · P2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If such a pair does not exist, we say that X has a k-gap, or that k is a gap number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Our first result is a complete identification of gap numbers in some cases: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If E(K) is torsion-free with rank r = 1, we have the following characterization of gap numbers on X according to the lattice T associated to the reducible fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T k is a gap number ⇔ none of the following are perfect squares E7 k + 1, 4k + 1 A7 k+1 4 , 16k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 16k + 9 D7 k+1 2 , 8k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 8k + 4 A6 ⊕ A1 k+1 7 , 28k − 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 28k + 21 E6 ⊕ A1 k+1 3 , 12k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 12k + 9 D5 ⊕ A2 k+1 6 , 24k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 24k + 16 A4 ⊕ A3 k+1 10 , 40k − 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 40k + 25 A4 ⊕ A2 ⊕ A1 k+1 15 , 60k − 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 60k + 45 We also explore the possibility of X having no gap numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that, in fact, this is always the case if the Mordell-Weil rank is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r ≥ 5, then X has no gap numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' On the other hand, for r ≤ 2 we show that gap numbers occur with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r ≤ 2, then the set of gap numbers of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' G := {k ∈ N | k is a gap number of X} has density 1 in N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' lim n→∞ #G ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', n} n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' At last we answer the question from the original motivation, which consists in classifying the rational elliptic surfaces with a 1-gap: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' X has a 1-gap if and only if r = 0 or r = 1 and π has a III∗ fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 3 Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The text is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Section 2 introduces the main objects, namely the Mordell-Weil lattice, the bounds cmax, cmin for the contribution term, the difference ∆ = cmax −cmin and the quadratic form QX induced by the height pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In Section 3 we explain the role of torsion sections in the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The key technical results are gathered in Section 4, where we state necessary and sufficient conditions for having P1 · P2 = k for a given k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Section 5 contains the main results of the paper, namely: the description of gap numbers when E(K) is torsion-free with r = 1 (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3), the absence of gap numbers for r ≥ 5 (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1), density of gap numbers when r ≤ 2 (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2) and the classification of surfaces with a 1-gap (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Section 6 is an appendix containing Table 8, which stores the relevant information about the Mordell-Weil lattices of rational elliptic surfaces with r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2 Preliminaries Throughout the paper X denotes a rational elliptic surface over an algebraically closed field k of any characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More precisely, X is a smooth rational projective surface with a fibration π : X → P1, with a section, whose general fiber is a smooth curve of genus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We assume moreover that π is relatively minimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' each fiber has no exceptional curve in its support) [SS19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The generic fiber of π is an elliptic curve E/K over K := k(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The set E(K) of K-points is called the Mordell-Weil group of X, whose rank is called the Mordell-Weil rank of X, denoted by r := rank E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In what follows we introduce the main objects of our investigation and stablish some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 The Mordell-Weil Lattice We give a brief description of the Mordell-Weil lattice, which is the central tool used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Although it can be defined on elliptic surfaces in general, we restrict ourselves to rational elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For more information on Mordell-Weil lattices, we refer the reader to the com- prehensive introduction by Schuett and Shioda [SS19] in addition to the original sources, namely [Elk90], [Shi89], [Shi90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We begin by noting that points in E(K) can be regarded as curves on X and by defining the lattice T and the trivial lattice Triv(X), which are needed to define the Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Sections, points on E(K) and exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The sections of π are in bijective cor- respondence with points on E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, since X is rational and relatively minimal, points on E(K) also correspond to exceptional curves on X [SS10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For this reason we identify sections of π, points on E(K) and exceptional curves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The lattice T and the trivial lattice Triv(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let O ∈ E(K) be the neutral section and R := {v ∈ P1 | π−1(v) is reducible} the set of reducible fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The components of a fiber π−1(v) are denoted by Θv,i, where Θv,0 is the only component intersected by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The Néron-Severi group NS(X) together with the intersection pairing is called the Néron-Severi lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4 We define the following sublattices of NS(X), which encode the reducible fibers of π: Tv := Z⟨Θv,i | i ̸= 0⟩ for v ∈ R, T := � v∈R Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Kodaira’s classification [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12], each Tv with v ∈ R is represented by a Dynkin diagram Am, Dm or Em for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We also define the trivial lattice of X, namely Triv(X) := Z⟨O, Θv,i | i ≥ 0, v ∈ R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Next we define the Mordell-Weil lattice and present the height formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In order to give E(K) a lattice structure, we cannot use the inter- section pairing directly, which only defines a lattice on NS(X) but not on E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is achieved by defining a Q-valued pairing, called the height pairing, given by ⟨·, ·⟩ : E(K) × E(K) → Q P, Q �→ −ϕ(P) · ϕ(Q), where ϕ : E(K) → NS(X) ⊗Z Q is defined from the orthogonal projection with respect to Triv(X) (for a detailed exposition, see [SS19, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, dividing by torsion elements we get a positive-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩) [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='20], called the Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The height formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The height pairing can be explicitly computed by the height formula [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For rational elliptic surfaces, it is given by ⟨P, Q⟩ = 1 + (P · O) + (Q · O) − (P · Q) − � v∈R contrv(P, Q), (1) h(P) := ⟨P, P⟩ = 2 + 2(P · O) − � v∈R contrv(P), (2) where contrv(P) := contrv(P, P) and contrv(P, Q) are given by Table 1 [SS19, Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1] assuming P, Q meet π−1(v) at Θv,i, Θv,j resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' with 0 < i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P or Q meets Θv,0, then contrv(P, Q) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The minimal norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since E(K) is finitely generated, there is a minimal positive value for h(P) as P runs through E(K) with h(P) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' It is called the minimal norm, denoted by µ := min{h(P) > 0 | P ∈ E(K)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The narrow Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' An important sublattice of E(K) is the narrow Mordell-Weil lattice E(K)0, defined as E(K)0 := {P ∈ E(K) | P intersects Θv,0 for all v ∈ R} = {P ∈ E(K) | contrv(P) = 0 for all v ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As a subgroup, E(K)0 is torsion-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' as a sublattice, it is a positive-definite even integral lattice with finite index in E(K) [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The importance of the narrow lattice can be explained by its considerable size as a sublattice and by the easiness to compute the height pairing on it, since all contribution terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' A complete classification of the lattices E(K) and E(K)0 on rational elliptic surfaces is found in [OS91, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5 Tv A1 E7 A2 E6 An−1 Dn+4 Type of π−1(v) III III∗ IV IV∗ In I∗ n contrv(P) 1 2 3 2 2 3 4 3 i(n−i) n � 1 (i = 1) 1 + n 4 (i > 1) contrv(P, Q) 1 3 2 3 i(n−j) n � 1 2 (i = 1) 1 2 + n 4 (i > 1) Table 1: Local contributions from reducible fibers to the height pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Gap numbers We introduce some convenient terminology to express the possibility of finding a pair of sections with a given intersection number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there are no sections P1, P2 ∈ E(K) such that P1 · P2 = k, we say that X has a k-gap or that k is a gap number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We say that X is gap-free if for every k ∈ Z≥0 there are sections P1, P2 ∈ E(K) such that P1 · P2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case the Mordell-Weil rank is r = 0, we have E(K) = E(K)tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, any two distinct sections are disjoint [SS19, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='30], hence every k ≥ 1 is a gap number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For positive rank, the description of gap numbers is less trivial, thus our focus on r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Bounds cmax, cmin for the contribution term We define the estimates cmax, cmin for the contribution term � v contrv(P) and state some simple facts about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We also provide an example to illustrate how they are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The need for these estimates comes from the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Suppose we are given a section P ∈ E(K) whose height h(P) is known and we want to determine P · O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case P ∈ E(K)0 we have a direct answer, namely P · O = h(P)/2 − 1 by the height formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' However if P /∈ E(K)0, the computation of P · O depends on the contribution term cP := � v∈R contrv(P), which by Table 1 depends on how P intersects the reducible fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Usually we do not have this intersection data at hand, which is why we need estimates for cP not depending on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If the set R of reducible fibers of π is not empty, we define cmax := � v∈R max{contrv(P) | P ∈ E(K)}, cmin := min {contrv(P) > 0 | P ∈ E(K), v ∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The case R = ∅ only occurs when X has Mordell-Weil rank r = 8 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1 in Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this case E(K)0 = E(K) and � v∈R contrv(P) = 0 ∀P ∈ E(K), hence we adopt the convention cmax = cmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use cmax, cmin as bounds for cP := � v contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For our purposes it is not necessary to know whether cP actually attains one of these bounds for some P, so that cmax, cmin should be understood as hypothetical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We state some facts about cmax, cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If π admits a reducible fiber, then: i) cmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) cmax < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' iii) cmin ≤ � v∈R contrv(P) ≤ cmax ∀P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For P ∈ E(K)0, only the second inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' iv) If � v∈R contrv(P) = cmin, then contrv′(P) = cmin for some v′ and contrv(P) = 0 for v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Item i) is immediate from the definition of cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For ii) it is enough to check the values of cmax directly in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For iii), the second inequality follows from the definition of cmax and clearly holds for any P ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P /∈ E(K)0, then cP := � v contrv(P) > 0, so contrv0(P) > 0 for some v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore cP ≥ contrv0(P) ≥ cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For iv), let � v contrv(P) = cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume by contradiction that there are distinct v1, v2 such that contrvi(P) > 0 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By definition of cmin we have cmin ≤ contrvi(P) for i = 1, 2 so cmin = � v contrv(P) ≥ contrv1(P) + contrv2(P) ≥ 2cmin, which is absurd because cmin > 0 by i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore there is only one v′ with contrv′(P) > 0, while contrv(P) = 0 for all v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, contrv′(P) = cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Explicit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Once we know the lattice T associated with the reducible fibers of π (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1), the computation of cmax, cmin is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For a fixed v ∈ R, the extreme values of the local contribution contrv(P) are given in Table 2, which is derived from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We provide an example to illustrate this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Tv max{contrv(P) | P ∈ E(K)} min{contrv(P) > 0 | P ∈ E(K)} An−1 ℓ(n−ℓ) n , where ℓ := �n 2 � n−1 n Dn+4 1 + n 4 1 E6 4 3 4 3 E7 3 2 3 2 Table 2: Extreme values of contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 7 Example: Let π with fiber configuration (I4, IV, III, I1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The reducible fibers are I4, IV, III, so T = A3 ⊕ A2 ⊕ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Table 2, the maximal contributions for A3, A2, A1 are 2·2 4 = 1, 2 3, 1 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The minimal positive contributions are 1·3 4 = 3 4, 2 3, 1 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then cmax = 1 + 2 3 + 1 2 = 13 6 , cmin = min �3 4, 2 3, 1 2 � = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 The difference ∆ = cmax − cmin In this section we explain why the value of ∆ := cmax − cmin is relevant to our discussion, specially in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We also verify that ∆ < 2 in most cases and identify the exceptional ones in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As noted in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, in case P /∈ E(K)0 and h(P) is known, the difficulty of determining P ·O lies in the contribution term cP := � v∈R contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, the range of possible values for cP determines the possibilities for P · O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This range is measured by the difference ∆ := cmax − cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence a smaller ∆ means a better control over the intersection number P · O, which is why ∆ plays an important role in determining possible intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 we assume ∆ ≤ 2 and state necessary and sufficient conditions for having a pair P1, P2 such that P1 · P2 = k for a given k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If however ∆ > 2, the existence of such a pair is not guaranteed a priori, so a case-by-case treatment is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Fortunately by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 the case ∆ > 2 is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The only cases with ∆ = 2 and ∆ > 2 are in Table 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular we have ∆ < 2 whenever E(K) is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T E(K) cmax cmin 24 A⊕5 1 A∗ 1 ⊕3 ⊕ Z/2Z 5 2 1 2 38 A3 ⊕ A⊕3 1 A∗ 1 ⊕ ⟨1/4⟩ ⊕ Z/2Z 5 2 1 2 53 A5 ⊕ A⊕2 1 ⟨1/6⟩ ⊕ Z/2Z 5 2 1 2 57 D4 ⊕ A⊕3 1 A∗ 1 ⊕ (Z/2Z)⊕2 5 2 1 2 58 A⊕2 3 ⊕ A1 A∗ 1 ⊕ Z/4Z 5 2 1 2 61 A⊕3 2 ⊕ A1 ⟨1/6⟩ ⊕ Z/3Z 5 2 1 2 Table 3: Cases with ∆ = 2 8 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T E(K) cmax cmin ∆ 41 A2 ⊕ A⊕4 1 1 6 � 2 1 1 2 � ⊕ Z/2Z 8 3 1 2 13 6 42 A⊕6 1 A∗ 1 ⊕2 ⊕ (Z/2Z)⊕2 3 1 2 5 2 59 A3 ⊕ A2 ⊕ A⊕2 1 ⟨1/12⟩ ⊕ Z/2Z 8 3 1 2 13 6 60 A3 ⊕ A⊕4 1 ⟨1/4⟩ ⊕ (Z/2Z)⊕2 3 1 2 5 2 Table 4: Cases with ∆ > 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By searching Table 8 for all cases with ∆ = 2 and ∆ > 2, we obtain Table 3 and Table 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Notice in particular that in both tables the torsion part of E(K) is always nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Consequently, if E(K) is torsion-free, then ∆ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 The quadratic form QX We define the positive-definite quadratic form with integer coefficients QX derived from the height pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The relevance of QX is due to the fact that some conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) can be stated in terms of what integers can be represented by QX (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The definition of QX consists in clearing denominators of the rational quadratic form induced by the height pairing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' the only question is how to find a scale factor that works in every case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More precisely, if E(K) has rank r ≥ 1 and P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr are generators of its free part, then q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) := h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr) is a quadratic form with coefficients in Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' we define QX by multiplying q by some integer d > 0 so as to produce coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We show that d may always be chosen as the determinant of the narrow lattice E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X with r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr be generators of the free part of E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Define QX(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) := (det E(K)0) · h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We check that the matrix representing QX has entries in Z, therefore QX has coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let A be the matrix representing the quadratic form QX, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) = xtAx, where x := (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then A has integer entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, QX has integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr be generators of the free part of E(K) and let L := E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The free part of E(K) is isomorphic to the dual lattice L∗ [OS91, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ], so we may find generators P 0 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', P 0 r of L such that the Gram matrix B0 := (⟨P 0 i , P 0 j ⟩)i,j of L is the inverse of the Gram matrix B := (⟨Pi, Pj⟩)i,j of L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 9 We claim that QX is represented by the adjugate matrix of B0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' the matrix adj(B0) such that B0 · adj(B0) = (det B0) · Ir, where Ir is the r × r identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, by construction B represents the quadratic form h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr), therefore QX(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) = (det E(K)0) · h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr) = (det B0) · xtBx = (det B0) · xt(B0)−1x = xtadj(B0)x, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' To prove that A := adj(B0) has integer coefficients, notice that the Gram matrix B0 of L = E(K)0 has integer coefficients (as E(K)0 is an even lattice), then so does A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ We close this subsection with a simple consequence of the definition of QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If h(P) = m for some P ∈ E(K), then QX represents d · m, where d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr be generators for the free part of E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P = a1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + arPr + Q, where ai ∈ Z and Q is a torsion element (possibly zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since torsion sections do not contribute to the height pairing, then h(P − Q) = h(P) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) = d · h(a1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + arPr) = d · h(P − Q) = d · m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 3 Intersection with a torsion section Before dealing with more technical details in Section 4, we explain how torsion sections can be of help in our investigation, specially in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We first note some general properties of torsion sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As the height pairing is positive- definite on E(K)/E(K)tor, torsion sections are inert in the sense that for each Q ∈ E(K)tor we have ⟨Q, P⟩ = 0 for all P ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, in the case of rational elliptic surfaces, torsion sections also happen to be mutually disjoint: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [MP89, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1] On a rational elliptic surface, Q1 · Q2 = 0 for any distinct Q1, Q2 ∈ E(K)tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, if O is the neutral section, then Q·O = 0 for all Q ∈ E(K)tor\\{O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As stated in [MP89, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 holds for elliptic surfaces over C even without assuming X is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' However, for an arbitrary algebraically closed field the rationality hypothesis is needed, and a proof can be found in [SS19, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By taking advantage of the properties above, we use torsion sections to help us find P1, P2 ∈ E(K) such that P1 · P2 = k for a given k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is particularly useful when ∆ ≥ 2, in which case E(K)tor is not trivial by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Given k ∈ Z≥0, suppose we can find P ∈ E(K)0 with height h(P) = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), P · O = k − 1 < k, which is not yet what we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the next lemma we show that replacing O with a torsion section Q ̸= O gives P · Q = k, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 10 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P ∈ E(K)0 such that h(P) = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then P · Q = k for all Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume there is some Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1, Q · O = 0 and by the height formula (2), 2k = 2 + 2(P · O) − 0, hence P · O = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use the height formula (1) for ⟨P, Q⟩ in order to conclude that P · Q = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since P ∈ E(K)0, it intersects the neutral component Θv,0 of every reducible fiber π−1(v), so contrv(P, Q) = 0 for all v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence 0 = ⟨P, Q⟩ = 1 + P · O + Q · O − P · Q − � v∈R contrv(P, Q) = 1 + (k − 1) + 0 − P · Q − 0 = k − P · Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 4 Existence of a pair of sections with a given intersection number Given k ∈ Z≥0, we state necessary and (in most cases) sufficient conditions for having P1 ·P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Necessary conditions are stated in generality in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1, while sufficient ones depend on the value of ∆ and are treated separately in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4, we collect all sufficient conditions proven in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 Necessary Conditions If k ∈ Z≥0, we state necessary conditions for having P1·P2 = k for some sections P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We note that the value of ∆ is not relevant in this subsection, although it plays a decisive role for sufficient conditions in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 · P2 = k for some P1, P2 ∈ E(K), then one of the following holds: i) h(P) = 2 + 2k for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Without loss of generality we may assume P2 is the neutral section, so that P1 · O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P1) = 2 + 2k − c, where c := � v contrv(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 ∈ E(K)0, then c = 0 and h(P1) = 2 + 2k, hence i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 /∈ E(K)0, then cmin ≤ c ≤ cmax by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But h(P1) = 2 + 2k − c, therefore 2 + 2k − cmax ≤ h(P1) ≤ 2 + 2k − cmin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 · P2 = k for some P1, P2 ∈ E(K), then QX represents some integer in [d · (2 + 2k − cmax), d · (2 + 2k)], where d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 and rephrase it in terms of QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If i) holds, then QX represents d · (2 + 2k) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But if ii) holds, then h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11, QX represents d · h(P) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In both i) and ii), QX represents some integer in [d · (2 + 2k − cmax), d · (2 + 2k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Sufficient conditions when ∆ ≤ 2 In this subsection we state sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) under the assumption that ∆ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, this covers almost all cases (more precisely, all but No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 41, 42, 59, 60 in Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We treat ∆ < 2 and ∆ = 2 separately, as the latter needs more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 The case ∆ < 2 We first prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, which gives sufficient conditions assuming ∆ < 2, then Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, which states sufficient conditions in terms of integers represented by QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is followed by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6, which is a simplified version of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ < 2 and let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let O ∈ E(K) be the neutral section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P) = 2 + 2(P · O) − c, where c := � v contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin], then 2 + 2k − cmax ≤ 2 + 2(P · O) − c ≤ 2 + 2k − cmin ⇒ c − cmax 2 ≤ P · O − k ≤ c − cmin 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore P · O − k is an integer in I := � c−cmax 2 , c−cmin 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that 0 is the only integer in I, so that P · O − k = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P · O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First notice that c ̸= 0, as P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 iii), cmin ≤ c ≤ cmax, consequently c−cmax 2 ≤ 0 ≤ c−cmin 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover ∆ < 2 implies that I has length cmax−cmin 2 = ∆ 2 < 1, so I contains no integer except 0 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 also applies when cmax = cmin, in which case the closed interval degen- erates into a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The following corollary of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 states a sufficient condition in terms of integers represented by the quadratic form QX (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ < 2 and let d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If QX represents an integer not divisible by d in the interval [d · (2+ 2k − cmax), d · (2+ 2k − cmin)], then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar ∈ Z such that QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)] with d ∤ QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P := a1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + arPr, where P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr are generators of the free part of E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then d ∤ QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) = d · h(P), which implies that h(P) /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular P /∈ E(K)0 since E(K)0 is an integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover h(P) = 1 dQX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) ∈ [2 + 2k − cmax, 2 + 2k − cmin] and we are done by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 12 The next corollary, although weaker than Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, is more practical for concrete examples and is frequently used in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' It does not involve finding integers represented by QX, but only finding perfect squares in an interval depending on the minimal norm µ (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � such that n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Take P ∈ E(K) such that h(P) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since h(nP) = n2µ /∈ Z, we must have nP /∈ E(K)0 as E(K)0 is an integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin] and we are done by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 The case ∆ = 2 The statement of sufficient conditions for ∆ = 2 is almost identical to the one for ∆ < 2: the only difference is that the closed interval Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 is substituted by a right half-open interval in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This change, however, is associated with a technical difficulty in the case when a section has minimal contribution term, thus the separate treatment for ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The results are presented in the following order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, which is a statement about sections whose contribution term is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Next we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, which states sufficient conditions for ∆ = 2, then Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there is P ∈ E(K) such that � v∈R contrv(P) = cmin, then P · Q = P · O + 1 for every Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If Q ∈ E(K)tor \\ {O}, then Q · O = 0 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, by the height formula (1), 0 = ⟨P, Q⟩ = 1 + P · O + 0 − P · Q − � v∈R contrv(P, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' (∗) Hence it suffices to show that contrv(P, Q) = 0 ∀v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 iv), contrv′(P) = cmin for some v′ and contrv(P) = 0 for all v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular P meets Θv,0, hence contrv(P, Q) = 0 for all v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus from (∗) we see that contrv′(P, Q) is an integer, which we prove is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We claim that Tv′ = A1, so that contrv′(P, Q) = 0 or 1 2 by Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this case, as contrv′(P, Q) is an integer, it must be 0, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' To see that Tv′ = A1 we analyse contrv′(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since ∆ = 2, then cmin = 1 2 by Table 3 and contrv′(P) = cmin = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Table 1, this only happens if Tv′ = An−1 and 1 2 = i(n−i) n for some 0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The only possibility is i = 1, n = 2 and Tv′ = A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ With the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 we are able to state sufficient conditions for ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2 and let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let O ∈ E(K) be the neutral section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P) = 2 + 2(P · O) − c, where c := � v contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We repeat the arguments from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, in this case with the right half-open interval, so that the hypothesis that h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin), implies that P · O − k is an integer in I′ := � c−cmax 2 , c−cmin 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since I′ is half-open with length cmax−cmin 2 = ∆ 2 = 1, then I′ contains exactly one integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If 0 ∈ I′, then P · O − k = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P · O = k and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence we assume 0 /∈ I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We claim that P ·O = k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First, notice that if c > cmin, then the inequalities cmin < c ≤ cmax give c−cmax 2 ≤ 0 < c−cmin 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 ∈ I′, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence c = cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since ∆ = 2, then I′ = [−1, 0), whose only integer is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus P · O − k = −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P · O = k − 1, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Finally, let Q ∈ E(K)tor \\ {O}, so that P · Q = P · O + 1 = k by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We remark that E(K)tor is not trivial by Table 3, therefore such Q exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ The following corollaries are analogues to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 adapted to ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Similarly to the case ∆ < 2, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 is stronger than Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10, although the latter is more practical for concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We remind the reader that µ denotes the minimal norm (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2 and let d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If QX represents an integer not divisible by d in the interval [d·(2+2k −cmax), d·(2+2k −cmin)), then P1 ·P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We repeat the arguments in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, in this case with the half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � such that n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We repeat the arguments in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6, in this case with the half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Necessary and sufficient conditions for ∆ ≤ 2 For completeness, we present a unified statement of necessary and sufficient conditions assuming ∆ ≤ 2, which follows naturally from results in Subsections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ ≤ 2 and let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then P1 · P2 = k for some P1, P2 ∈ E(K) if and only if one of the following holds: i) h(P) = 2 + 2k for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' iii) h(P) = 2 + 2k − cmin and � v∈R contrv(P) = cmin for some P ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If i) or iii) holds, then P · O = k directly by the height formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But if ii) holds, it suffices to to apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 when ∆ < 2 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 when ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, let P1·P2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Without loss of generality, we may assume P2 = O, so that P1·O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P1) = 2 + 2k − c, where c := � v contrv(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If c = 0, then P1 ∈ E(K)0 and h(P1) = 2+2k, so i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence we let c ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P1 /∈ E(K)0, so that cmin ≤ c ≤ cmax by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case c = cmin, then h(P1) = 2 + 2k − cmin and iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Otherwise cmin < c ≤ cmax, which implies 2 + 2k − cmax ≤ h(P1) < 2 + 2k − cmin, so ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 Summary of sufficient conditions For the sake of clarity, we summarize in a single proposition all sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) proven in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If one of the following holds, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1) h(P) = 2 + 2k for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2) h(P) = 2k for some P ∈ E(K)0 and E(K)tor is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 3) ∆ < 2 and there is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � with n2µ /∈ Z, where µ is the minimal norm (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case ∆ = 2, consider the right half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4) ∆ < 2 and the quadratic form QX represents an integer not divisible by d := det E(K)0 in the interval [d · (2 + 2k − cmax), d · (2 + 2k − cmin)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case ∆ = 2, consider the right half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In 1) a height calculation gives 2 + 2k = h(P) = 2 + 2(P · O) − 0, so P · O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For 2), we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 to conclude that P · Q = k for any Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In 3) we use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 when ∆ < 2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10 when ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In 4), we apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 if ∆ < 2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 if ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 5 Main Results We prove the four main theorems of this paper, which are independent applications of the results from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The first two are general attempts to describe when and how gap numbers occur: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 tells us that large Mordell-Weil groups prevent the existence of gaps numbers, more precisely for Mordell-Weil rank r ≥ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 we show that for small Mordell-Weil rank, more precisely when r ≤ 2, then gap numbers occur with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The last two theorems, on the other hand, deal with explicit values of gap numbers: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 provides a complete description of gap numbers in certain cases, while Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 is a classification of cases with a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 No gap numbers in rank r ≥ 5 We show that if E(K) has rank r ≥ 5, then X is gap-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Our strategy is to prove that for every k ∈ Z≥0 there is some P ∈ E(K)0 such that h(P) = 2+2k, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 1) we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We accomplish this in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we show that this holds when there is an embedding of A⊕ 1 or of A4 in E(K)0 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Second, we show that if r ≥ 5, then such embedding exists, hence X is gap-free (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 15 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume E(K)0 has a sublattice isomorphic to A⊕4 1 or A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then for every ℓ ∈ Z≥0 there is P ∈ E(K)0 such that h(P) = 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First assume A⊕4 1 ⊂ E(K)0 and let P1, P2, P3, P4 be generators for each factor A1 in A⊕4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then h(Pi) = 2 and ⟨Pi, Pj⟩ = 0 for distinct i, j = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lagrange’s four-square theorem [HW79, §20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5] there are integers a1, a2, a3, a4 such that a2 1 + a2 2 + a2 3 + a2 4 = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Defining P := a1P1 + a2P2 + a3P3 + a4P4 ∈ A⊕4 1 ⊂ E(K)0, we have h(P) = 2a2 1 + 2a2 2 + 2a2 3 + 2a2 4 = 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Now let A4 ⊂ E(K)0 with generators P1, P2, P3, P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then h(Pi) = 2 for i = 1, 2, 3, 4 and ⟨Pi, Pi+1⟩ = −1 for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We need to find integers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', x4 such that h(P) = 2ℓ, where P := x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + x4P4 ∈ A4 ⊂ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Equivalently, we need that ℓ = 1 2⟨P, P⟩ = x2 1 + x2 2 + x2 3 + x2 4 − x1x2 − x2x3 − x3x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore ℓ must be represented by q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', x4) := x2 1 + x2 2 + x2 3 + x2 4 − x1x2 − x2x3 − x3x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that q represents all positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Notice that q is positive-definite, since it is induced by ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Bhargava-Hanke’s 290-theorem [BH][Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1], q represents all positive integers if and only if it represents the following integers: 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The representation for each of the above is found in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ We now prove the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r ≥ 5, then X is gap-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We show that for every k ≥ 0 there is P ∈ E(K)0 such that h(P) = 2 + 2k, so that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 1) we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 it suffices to prove that E(K)0 has a sublattice isomorphic to A⊕4 1 or A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The cases with r ≥ 5 are No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-7 (Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-6, E(K)0 = E8, E7, E6, D6, D5, A5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Each of these admit an A4 sublattice [Nis96, Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 7 we claim that E(K)0 = D4 ⊕ A1 has an A⊕4 1 sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is the case because D4 admits an A⊕4 1 sublattice [Nis96, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 16 n x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' x4 with x2 1 + x2 2 + x2 3 + x2 4 − x1x2 − x2x3 − x3x4 = n 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' −9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8 290 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 Table 5: Representation of the critical integers in Bhargava-Hanke’s 290-theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Gaps with probability 1 in rank r ≤ 2 Fix a rational elliptic surface π : X → P1 with Mordell-Weil rank r ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that if k is a uniformly random natural number, then k is a gap number with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More precisely, if G := {k ∈ N | k is a gap number of X} is the set of gap numbers, then G ⊂ N has density 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' d(G) := lim n→∞ #G ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', n} n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 17 We adopt the following strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If k ∈ N \\ G, then P1 · P2 = k for some P1, P2 ∈ E(K) and by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 the quadratic form QX represents some integer t depending on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This defines a function N\\G → T, where T is the set of integers represented by QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since QX is a quadratic form on r ≤ 2 variables, T has density 0 in N by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By analyzing the pre-images of N\\G → T, in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 we conclude that d(N \\ G) = d(T) = 0, hence d(G) = 1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let Q be a positive-definite quadratic form on r = 1, 2 variables with integer coeffi- cients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then the set of integers represented by Q has density 0 in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let S be the set of integers represented by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If d is the greatest common divisor of the coefficients of Q, let S′ be the set of integers representable by the primitive form Q′ := 1 d · Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By construction S′ is a rescaling of S, so d(S) = 0 if and only if d(S′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r = 1, then Q′(x1) = x2 1 and S′ is the set of perfect squares, so clearly d(S′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r = 2, then Q′ is a binary quadratic form and the number of elements in S′ bounded from above by x > 0 is given by C · x √log x + o(x) with C > 0 a constant and limx→∞ o(x) x = 0 [Ber12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus d(S′) = lim x→∞ C √log x + o(x) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ We now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let π : X → P1 be a rational elliptic surface with Mordell-Weil rank r ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then the set G := {k ∈ N | k is a gap number of X} of gap numbers of X has density 1 in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r = 0, then the claim is trivial by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, hence we may assume r = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that S := N \\ G has density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If S is finite, there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Otherwise, let k1 < k2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' be the increasing sequence of all elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2, for each n there is some tn ∈ Jkn := [d · (2 + 2kn − cmax), d · (2 + 2kn)] represented by the quadratic form QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let T be the set of integers represented by QX and define the function f : N \\ G → T by kn �→ tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since QX has r = 1, 2 variables, T has density 0 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For N > 0, let SN := S ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', N} and TN := T ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since T has density zero, #TN = o(N), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' #TN N → 0 when N → ∞ and we need to prove that #SN = o(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We analyze the function f restricted to SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Notice that as tn ∈ Jkn, then kn ≤ N implies tn ≤ d · (2 + 2kn) ≤ d · (2 + 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence the restriction g := f|SN can be regarded as a function g : SN → Td·(2+2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We claim that #g−1(t) ≤ 2 for all t ∈ Td·(2+2N), in which case #SN ≤ 2 · #Td·(2+2N) = o(N) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume by contradiction that g−1(t) contains three distinct elements, say kℓ1 < kℓ2 < kℓ3 with t = tℓ1 = tℓ2 = tℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since tℓi ∈ Jkℓi for each i = 1, 2, 3, then t ∈ Jkℓ1 ∩ Jkℓ2 ∩ Jkℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that Jkℓ1 and Jkℓ3 are disjoint, which yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, since kℓ1 < kℓ2 < kℓ3, in particular kℓ3 − kℓ1 ≥ 2, therefore d · (2 + 2kℓ1) ≤ d · (2 + 2kℓ3 − 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But cmax < 4 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, so d · (2 + 2kℓ1) < d · (2 + 2kℓ3 − cmax), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' max Jkℓ1 < min Jkℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus Jkℓ1 ∩ Jkℓ3 = ∅, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Identification of gaps when E(K) is torsion-free with rank r = 1 The results in Subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 concern the existence and the distribution of gap num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the following subsections we turn our attention to finding gap numbers explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this subsection we give a complete description of gap numbers assuming E(K) is torsion-free with rank r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Such descriptions are difficult in the general case, but our assumption guarantees that each 18 E(K), E(K)0 is generated by a single element and that ∆ < 2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, which makes the problem more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We organize this subsection as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we point out some trivial facts about generators of E(K), E(K)0 when r = 1 in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Next we state necessary and sufficient conditions for having P1 · P2 = k when E(K) is torsion-free with r = 1 in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As an application of the latter, we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, which is the main result of the subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface with Mordell-Weil rank r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P generates the free part of E(K), then a) h(P) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' b) 1/µ is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' c) E(K)0 is generated by P0 := (1/µ)P and h(P0) = 1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Item a) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Items b), c) follow from the fact that E(K)0 is an even lattice and that E(K) ≃ L∗ ⊕ E(K)tor, where L := E(K)0 [OS91, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ In what follows we use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 and results from Section 4 to state necessary and sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) in case E(K) is torsion-free with r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume E(K) is torsion-free with rank r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then P1 · P2 = k for some P1, P2 ∈ E(K) if and only if one of the following holds: i) µ · (2 + 2k) is a perfect square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) There is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � such that µ · n /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, E(K) is generated by some P with h(P) = µ and E(K)0 is generated by P0 := n0P, where n0 := 1 µ ∈ 2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First assume that P1·P2 = k for some P1, P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Without loss of generality we may assume P2 = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1 = nP for some n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We show that P1 ∈ E(K)0 implies i) while P1 /∈ E(K)0 implies ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 ∈ E(K)0, then n0 | n, hence P1 = nP = mP0, where m := n n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), 2 + 2k = h(P1) = h(mP0) = m2 · 1 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence µ · (2 + 2k) = m2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 /∈ E(K)0, then n0 ∤ n, hence µ · n = n n0 /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, h(P1) = n2h(P) = n2µ and by the height formula (2), n2µ = h(P) = 2 + 2k − c, where c := � v contrv(P1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The inequalities cmin ≤ c ≤ cmax then give 2+2k−cmax µ ≤ n2 ≤ 2+2k−cmin µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, assume i) or ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since E(K) is torsion-free, ∆ < 2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, so we may apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If i) holds, then µ · (2 + 2k) = m2 for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since mP0 ∈ E(K)0 and h(mP0) = m2 µ = 2 + 2k, we are done by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If ii) holds, the condition µ · n /∈ Z is equivalent to n0 ∤ n, hence nP /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � , implies h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 ii), we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ By applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 to all possible cases where E(K) is torsion-free with rank r = 1, we obtain the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 19 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If E(K) is torsion-free with rank r = 1, then all the gap numbers of X are described in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T k is a gap number ⇔ none of the following are perfect squares first gap numbers 43 E7 k + 1, 4k + 1 1, 4 45 A7 k+1 4 , 16k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 16k + 9 8, 11 46 D7 k+1 2 , 8k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 8k + 4 2, 5 47 A6 ⊕ A1 k+1 7 , 28k − 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 28k + 21 12, 16 49 E6 ⊕ A1 k+1 3 , 12k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 12k + 9 3, 7 50 D5 ⊕ A2 k+1 6 , 24k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 24k + 16 6, 11 55 A4 ⊕ A3 k+1 10 , 40k − 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 40k + 25 16, 20 56 A4 ⊕ A2 ⊕ A1 k+1 15 , 60k − 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 60k + 45 22, 27 Table 6: Description of gap numbers when E(K) is torsion-free with r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For the sake of brevity we restrict ourselves to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The other cases are treated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Here cmax = 2·3 5 + 2·2 4 = 11 5 , cmin = min � 4 5, 3 4 � = 3 4 and µ = 1/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6, k is a gap number if and only if neither i) nor ii) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Condition i) is that 2+2k 20 = k+1 10 is a perfect square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Condition ii) is that � 2+2k−cmax µ , 2+2k−cmin µ � = [40k − 4, 40k + 25] contains some n2 with 20 ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We check that 20 ∤ n for every n such that n2 = 40k + ℓ, with ℓ = −4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, if 20 | n, then 400 | n2 and in particular 40 | n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then 40 | (n2 − 40k) = ℓ, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 Surfaces with a 1-gap In Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 we take each case in Table 6 and describe all its gap numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this subsection we do the opposite, which is to fix a number and describe all cases having it as a gap number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We remind the reader that our motivating problem (Section 1) was to determine when there are sections P1, P2 such that P1 · P2 = 1, which induce a conic bundle having P1 + P2 as a reducible fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The answer for this question is the main theorem of this subsection: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then X has a 1-gap if and only if r = 0 or r = 1 and π has a III∗ fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20 Our strategy for the proof is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We already know that a 1-gap exists whenever r = 0 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1) or when r = 1 and π has a III∗ fiber (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, we need to find P1, P2 with P1 · P2 = 1 in all cases with r ≥ 1 and T ̸= E7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we introduce two lemmas, which solve most cases with little computation, and leave the remaining ones for the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In both Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11 our goal is to analyze the narrow lattice E(K)0 and apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 to detect cases without a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If one of the following holds, then h(P) = 4 for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' a) The Gram matrix of E(K)0 has a 4 in its main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' b) There is an embedding of An ⊕ Am in E(K)0 for some n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' c) There is an embedding of An, Dn or En in E(K)0 for some n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Case a) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assuming b), we take generators P1, P2 from An, Am respectively with h(P1) = h(P2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since An, Am are in direct sum, ⟨P1, P2⟩ = 0, hence h(P1 + P2) = 4, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If c) holds, then the fact that n ≥ 3 allows us to choose two elements P1, P2 among the generators of L1 = An, Dn or En such that h(P1) = h(P2) = 2 and ⟨P1, P2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus h(P1 + P2) = 4 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the following cases, X does not have a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' r ≥ 3 : all cases except possibly No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' r = 1, 2 : cases No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 25, 26, 30, 32-36, 38, 41, 42, 46, 52, 54, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We look at column E(K)0 in Table 8 to find which cases satisfy one of the conditions a), b), c) from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' a) Applies to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12, 17, 19, 22, 23, 25, 30, 32, 33, 36, 38, 41, 46, 52, 54, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' b) Applies to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 10, 11, 14, 15, 18, 24, 26, 34, 35, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' c) Applies to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-10, 13, 16, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, this covers all cases with r ≥ 3 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-24) except No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 in each of these cases there is P ∈ E(K)0 with h(P) = 4 and we are done by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ In the next lemma we also analyze E(K)0 to detect surfaces without a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume E(K)0 ≃ An for some n ≥ 1 and that E(K) has nontrivial torsion part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then X does not have a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This applies to cases No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 28, 39, 44, 48, 51, 57, 58 in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Take a generator P of E(K)0 with h(P) = 2 and apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 21 We are ready to prove the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We need to show that in all cases where r ≥ 1 and T ̸= E7 there are P1, P2 ∈ E(K) such that P1 · P2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This corresponds to cases No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-61 except 43 in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The cases where r = 1 and E(K) is torsion-free can be solved by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10, namely No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 45-47, 49, 50, 55, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Adding these cases to the ones treated in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11, we have therefore solved the following: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-19, 21-26, 28, 30, 32-36, 38, 39, 41-52, 54-58, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For the remaining cases, we apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 3), which involves finding perfect squares in the interval � 4−cmax µ , 4−cmin µ � (see Table 7), considering the half-open interval in the cases with ∆ = 2 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 53, 61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T E(K) µ I n2 ∈ I 20 A⊕2 2 ⊕ A1 A∗ 2 ⊕ ⟨1/6⟩ 1 6 [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 23] 42 27 E6 A∗ 2 2 3 [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4] 22 29 A5 ⊕ A1 A∗ 1 ⊕ ⟨1/6⟩ 1 6 [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 21] 42 31 A4 ⊕ A2 1 15 � 2 1 1 8 � 2 15 [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 21] 42 37 A3 ⊕ A2 ⊕ A1 A∗ 1 ⊕ ⟨1/12⟩ 1 12 [22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 28] 52 40 A⊕2 2 ⊕ A⊕2 1 ⟨1/6⟩⊕2 1 6 [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 21] 42 53 A5 ⊕ A⊕2 1 ⟨1/6⟩ ⊕ Z/2Z 1 6 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12] 32 59 A3 ⊕ A2 ⊕ A⊕2 1 ⟨1/12⟩ ⊕ Z/2Z 1 12 [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 42] 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 62 61 A⊕3 2 ⊕ A1 ⟨1/6⟩ ⊕ Z/3Z 1 6 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12] 32 Table 7: Perfect squares in the interval I := � 4−cmax µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4−cmin µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 59 we have ∆ > 2, so a particular treatment is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let T = Tv1 ⊕ Tv2 ⊕ Tv3 ⊕ Tv4 = A3 ⊕ A2 ⊕ A1 ⊕ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P generates the free part of E(K) and Q generates its torsion part, then h(P) = 1 12 and 4P + Q meets the reducible fibers at Θv1,2, Θv2,1, Θv3,1, Θv4,1 [Kur14][Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Table 1 and the height formula (2), 42 12 = h(4P + Q) = 2 + 2(4P + Q) · O − 2 · 2 4 − 1 · 2 3 − 1 2 − 1 2, hence (4P + Q) · O = 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 22 6 Appendix We reproduce part of the table in [OS91, Main Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='] with data on Mordell-Weil lattices of rational elliptic surfaces with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We only add columns cmax, cmin, ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E(K)0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E(K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='cmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='cmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='A⊕3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⊕ A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨6⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨1/6⟩ ⊕ Z/3Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='Table 8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='Mordell-Weil lattices of rational elliptic surfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 25 References [Ber12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Bernays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Über die Darstellung von positiven, ganzen Zahlen durch die primitive, binären quadratischen Formen einer nicht-quadratischen Diskriminante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' PhD thesis, Göttingen, 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [BH] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Bhargava and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hanke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Universal quadratic forms and the 290-Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Preprint at http://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='stanford.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Advanced Studies in Pure Mathematics, 60:51–160, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [SS19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Schuett and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Shioda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Mordell-Weil Lattices, volume 70 of Ergebnisse der Mathematik und ihrer Grenzgebiete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} diff --git a/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/2301.02222v1.pdf.txt b/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/2301.02222v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d53fe3b7bd75a86f066c4026736a00c8a863c69 --- /dev/null +++ b/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/2301.02222v1.pdf.txt @@ -0,0 +1,2290 @@ +COMPUTING NONSURJECTIVE PRIMES ASSOCIATED TO GALOIS +REPRESENTATIONS OF GENUS 2 CURVES +BARINDER S. BANWAIT, ARMAND BRUMER, HYUN JONG KIM, ZEV KLAGSBRUN, JACOB MAYLE, +PADMAVATHI SRINIVASAN, AND ISABEL VOGT +Abstract. For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric en- +domorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely +many primes ℓ for which the Galois action on ℓ-torsion points of A is not maximal. Building on +work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are +Mitchell’s classification of maximal subgroups of PSp4(Fℓ), sampling of the characteristic polyno- +mials of Frobenius, and the Khare–Wintenberger modularity theorem. The algorithm has been +submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomor- +phism ring in the LMFDB, and the results incorporated into the homepage of each such curve on +a publicly-accessible branch of the LMFDB. +1. Introduction +Let C/Q be a smooth, projective, geometrically integral curve (referred to hereafter as a nice +curve) of genus 2, and let A be its Jacobian. We assume throughout that A admits no nontrivial +geometric endomorphisms; that is, we assume that End(AQ) = Z, and we refer to any abelian +variety satisfying this property as typical1. We also say that a nice curve is typical if its Jacobian is +typical. Let GQ ∶= Gal(Q/Q), let ℓ be a prime, and let A[ℓ] ∶= A(Q)[ℓ] denote the ℓ-torsion points +of A(Q). Let +ρA,ℓ ∶ GQ → Aut(A[ℓ]) +denote the Galois representation on A[ℓ]. +By fixing a basis for A[ℓ], and observing that A[ℓ] +admits a nondegenerate Galois-equivariant alternating bilinear form, namely the Weil pairing, we +may identify the codomain of ρA,ℓ with the general symplectic group GSp4(Fℓ). +In a letter to Vign´eras [Ser00, Corollaire au Th´eor`eme 3], Serre proved an open image theorem +for typical abelian varieties of dimensions 2 or 6, or of odd dimension, generalizing his celebrated +open image theorem for elliptic curves [Ser72]. More precisely, the set of nonsurjective primes ℓ for +which the representation ρA,ℓ is not surjective — i.e., the set of primes ℓ for which ρA,ℓ(GQ) is +contained in a proper subgroup of GSp4(Fℓ) — is finite. +In the elliptic curve case, Serre subsequently provided a conditional upper bound in terms of the +conductor of E on this finite set [Ser81, Th´eor`eme 22]; this bound has since been made unconditional +[Coj05, Kra95]. There are also algorithms to compute the finite set of nonsurjective primes [Zyw15], +and practical implementations in Sage [CL12]. +Serre’s open image theorem for typical abelian surfaces was made explicit by Dieulefait [Die02] +who described an algorithm that returns a finite set of primes containing the set of nonsurjective +primes. In a different direction Lombardo [Lom16, Theorem 1.3] provided an upper bound on the +nonsurjective primes involving the stable Faltings height of A. +Date: January 6, 2023. +2010 Mathematics Subject Classification. 11F80 (primary), 11G10, 11Y16 (secondary). +1Abelian varieties with extra endomorphisms define a thin set (in the sense of Serre) in Ag and as such are not +the typically arising case. +1 +arXiv:2301.02222v1 [math.NT] 5 Jan 2023 + +2 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +In this paper we develop Algorithms 3.1 and 4.1, which together allow for the exact determination +of the nonsurjective primes for C, yielding our main result as follows. +Theorem 1.1. Let C/Q be a typical genus 2 curve whose Jacobian A has conductor N. +(1) Algorithm 3.1 produces a finite list PossiblyNonsurjectivePrimes(C) that provably contains all +nonsurjective primes. +(2) For a given bound B > 0, Algorithm 4.1 produces a sublist LikelyNonsurjectivePrimes(C;B) +of PossiblyNonsurjectivePrimes(C) that contains all the nonsurjective primes. +If B is suffi- +ciently large, then the elements of LikelyNonsurjectivePrimes(C;B) are precisely the nonsurjec- +tive primes of A. +The two common ingredients in Algorithms 3.1 and 4.1 are Mitchell’s 1914 classification of +maximal subgroups of PSp4(Fℓ) [Mit14] and sampling of characteristic polynomials of Frobenius +elements. Indeed, ρA,ℓ is nonsurjective precisely when its image is contained in one of the proper +maximal subgroups of GSp4(Fℓ). The (integral) characteristic polynomial of Frobenius at a good +prime p is computationally accessible since it is determined by counting points on C over Fpr for +small r. The reduction of this polynomial modulo ℓ gives the characteristic polynomial of the action +of the Frobenius element on A[ℓ]. By the Chebotarev density theorem, the images of the Frobenius +elements for varying primes p equidistribute over the conjugacy classes of ρA,ℓ(GQ) and hence let +us explore the image. +Algorithm 3.1 makes use of the fact that if the image of ρA,ℓ is nonsurjective, then the character- +istic polynomials of Frobenius at auxiliary primes p will be constrained modulo ℓ. Using this idea, +Dieulefait worked out the constraints imposed by each type of maximal subgroup for ρA,ℓ(GQ) to +be contained in that subgroup. Our Algorithm 3.1 combines Dieulefait’s conditions, with some +modest improvements, to produce a finite list PossiblyNonsurjectivePrimes(C). +Algorithm 4.1 then weeds out the extraneous surjective primes from PossiblyNonsurjectivePrimes(C). +Equipped with the prime ℓ, the task here is try to generate enough different elements in the image +to rule out containment in any proper maximal subgroup. The key input is a purely group-theoretic +condition (Proposition 4.2) that guarantees that a subgroup is all of GSp4(Fℓ) if it contains par- +ticular types of elements. This algorithm is probabilistic and depends on the choice of a parameter +B which, if sufficiently large, provably establishes nonsurjectivity. The parameter B is a cut-off for +the number of Frobenius elements that we use to sample the conjugacy classes of ρA,ℓ(GQ). +As an illustration of the interplay between theory and practice, analyzing the “worst case” run +time of each step in Algorithm 3.1 yields a new theoretical bound, conditional on the Generalized +Riemann Hypothesis (GRH), on the product of all nonsurjective primes in terms of the conductor. +Theorem 1.2. Let C/Q be a typical genus 2 curve with conductor N. Assuming the Generalized +Riemann Hypothesis (GRH), we have, for any ϵ > 0, +∏ +ℓ nonsurjective +ℓ ≪ exp(N1/2+ϵ), +where the implied constant is absolute and effectively computable. +While we believe this bound to be far from asymptotically optimal, it is the first bound in the +literature expressed in terms of the (effectively computable) conductor. +Naturally one wants to find the sufficiently large value of B in Theorem 1.1(2), which the next +result gives, conditional on GRH. +Theorem 1.3. Let C/Q be a typical genus 2 curve, B be a positive integer, and q be the largest +prime in LikelyNonsurjectivePrimes(C;B). Assuming GRH, the set LikelyNonsurjectivePrimes(C;B) +is precisely the set of nonsurjective primes of C, provided that +B ≥ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5) +2 . + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +3 +The proof of Theorem 1.3 involves an explicit Chebotarev bound due to Bach and Sorenson +[BS96] that is dependent on GRH. An unconditional version of Theorem 1.3 can be given using an +unconditional Chebotarev result (for instance [KW22]), though the bound for B will be exponential +in q. In addition, if we assume both GRH and the Artin Holomorphy Conjecture (AHC), then a +version of Theorem 1.3 holds with the improved asymptotic bound B ≫ q11 log2(qNA), but without +an explicit constant. +Unfortunately, the bound from Theorem 1.3 is prohibitively large to use in practice. By way of +illustration, consider the smallest (with respect to conductor) typical genus 2 curve, which has a +model +y2 + (x3 + 1)y = x2 + x, +and label 249.a.249.1 in the L-functions and modular forms database (LMFDB) [LMF22]. The +output of Algorithm 3.1 is the set {2,3,5,7,83}. Applying Algorithm 4.1 with B = 100 rules out +the prime 83, suggesting that 7 is the largest nonsurjective prime. Subsequently applying Theorem +1.3 with q = 7 yields the value B = 3.578 × 1023 for which LikelyNonsurjectivePrimes(C;B) coincides +with the set of nonsurjective primes associated with C. With this value of B, our implementation of +the algorithm was still running after 24 hours, after which we terminated it. Even if the version of +Theorem 1.3 that relies on AHC could be made explicit, the value of q11 log2(qNA) in this example +is on the order of 1011, which would still be a daunting prospect. +To execute the combined algorithm on all typical genus 2 curves in the LMFDB - which at the +time of writing constitutes 63,107 curves - we have decided to take a fixed value of B = 1000 in +Algorithm 4.1. The combined algorithm then takes about 4 hours on MIT’s Lovelace computer, +a machine with 2 AMD EPYC 7713 2GHz processors, each with 64 cores, and a total of 2TB of +memory. The result of this computation of nonsurjective primes for these curves is available to +view on the homepage of each curve in the LMFDB beta: +https://beta.lmfdb.org +In addition, the combined algorithm has been run on a much larger set of 1,823,592 curves +provided to us by Andrew Sutherland. See Section 6 for the results of this computation. +Algorithm 4.1 samples the characteristic polynomial of Frobenius Pp(t) for each prime p of +good reduction for the curve up to a particular bound and applies Tests 4.4 and 4.5 to Pp(t). +Assuming that ρA,ℓ is surjective, we expect that the outcome of these tests should be independent +for sufficiently large primes. More precisely, +Theorem 1.4. Let C/Q be a typical genus 2 curve with Jacobian A and suppose ℓ is an odd prime +such that ρA,ℓ is surjective. There is an effective bound B0 such that for any B > B0, if we sample +the characteristic polynomials of Frobenius Pp(t) for n primes p ∈ [B,2B] chosen uniformly and +independently at random, the probability that none of these pass Tests 4.4 or 4.5 is less than 3⋅( 9 +10) +n. +Remark 1. In fact, for each prime ℓ satisfying the conditions of Theorem 1.4, there is an explicit +constant cℓ ≤ +9 +10 tending to 3 +4 as ℓ → ∞ which may be computed using Corollary 5.3 such that +bound of 3 ⋅ ( 9 +10) +n in Theorem 1.4 can be replaced by 3 ⋅ cn +ℓ . +The combined algorithm to probabilistically determine the nonsurjective primes of a nice genus +2 curve over Q has been implemented in Sage [The20], and it will appear in a future release of this +software2. Until then, the implementation is available at the following repository: +https://github.com/ivogt/abeliansurfaces +The README.md file contains detailed instructions on its use. This repository also contains other +scripts in both Sage and Magma [BCP97] useful for verifying some of the results of this work; any +filenames used in the sequel will refer to the above repository. +2see https://trac.sagemath.org/ticket/30837 for the ticket tracking this integration. + +4 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +Outline of this paper. In Section 2, we begin by reviewing the properties of the characteristic +polynomial of Frobenius with a view towards computational aspects. We also recall the classification +of maximal subgroups of GSp4(Fℓ). In Section 3, we explain Algorithm 3.1 and establish Theorem +1.1(1); that is, for each of the maximal subgroups of GSp4(Fℓ) listed in Section 2.4, we generate a +list of primes that provably contains all primes ℓ for which the mod ℓ image of Galois is contained +in this maximal subgroup. Theorem 1.2 is also proved in this section (Subsection 3.3). In Section 4, +we first prove a group-theoretic criterion (Proposition 4.2) for a subgroup of GSp4(Fℓ) to equal +GSp4(Fℓ). Then, for each ℓ in the finite list from Section 3, we ascertain whether the characteristic +polynomials of the Frobenius elements sampled satisfy the group-theoretic criterion; Theorem 1.1(2) +and Theorem 1.3 also follow from this study. In Section 5 we prove Theorem 1.4 concerning the +probability of output error, assuming that Frobenius elements distribute in ρA,ℓ(GQ) as they would +in a randomly chosen element of GSp4(Fℓ). Finally, in Section 6, we close with remarks concerning +the execution of the algorithm on the large dataset of genus 2 curves mentioned above, and highlight +some interesting examples that arose therein. +Acknowledgements. This work was started at a workshop held remotely ‘at’ the Institute for +Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, in May +2020, and was supported by a grant from the Simons Foundation (546235) for the collaboration +‘Arithmetic Geometry, Number Theory, and Computation’. +It has also been supported by the +National Science Foundation under Grant No. DMS-1929284 while the authors were in residence +at ICERM during a Collaborate@ICERM project held in May 2022. We are grateful to Noam Elkies +for providing interesting examples of genus 2 curves in the literature, Davide Lombardo for helpful +discussions related to computing geometric endomorphism rings, and to Andrew Sutherland for +providing a dataset of Hecke characteristic polynomials that were used for executing our algorithm +on all typical genus 2 curves in the LMFDB, as well as making available the larger dataset of +approximately 2 million curves that we ran our algorithm on. +2. Preliminaries +2.1. Notation. Let A be an abelian variety of dimension g defined over Q. By conductor we mean +the Artin conductor N = NA of A. We write Nsq for the largest integer such that N2 +sq ∣ N. +Let ℓ be a prime. We write TℓA for the ℓ-adic Tate module of A: +TℓA ≃ lim +←� +n +A[ℓn]. +This is a free Zℓ-module of rank 2g. +For each prime p, we write Frobp ∈ Gal(Q/Q) for an absolute Frobenius element associated to p. +By a good prime p for an abelian variety A, we mean a prime p for which A has good reduction, or +equivalently p ∤ NA. If p is a good prime for A, then the trace ap of the action of Frobp on TℓA is +an integer. See Section 2.2 for a discussion of the characteristic polynomial of Frobenius. +By a typical abelian variety A, we mean an abelian variety with geometric endomorphism ring +Z. A typical genus 2 curve is a nice curve whose Jacobian is a typical abelian surface. +Let V be a 4-dimensional vector space over Fℓ endowed with a nondegenerate skew-symmetric +bilinear form ⟨⋅,⋅⟩. A subspace W ⊆ V is called isotropic (for ⟨⋅,⋅⟩) if ⟨w1,w2⟩ = 0 for all w1,w2 ∈ W. +A subspace W ⊆ V is called nondegenerate (for ⟨⋅,⋅⟩) if ⟨⋅,⋅⟩ restricts to a nondegenerate form on +W. The general symplectic group of (V,⟨⋅,⋅⟩) is defined as +GSp(V,⟨⋅,⋅⟩) ∶= {M ∈ GL(V ) ∶ ∃ mult(M) ∈ F× +ℓ ∶ ⟨Mv,Mw⟩ = mult(M)⟨v,w⟩ ∀ v,w ∈ V }. +The map M ↦ mult(M) is a surjective homomorphism from GSp(V,⟨⋅,⋅⟩) to F× +ℓ called the similitude +character; its kernel is the symplectic group, denoted Sp(V,⟨⋅,⋅⟩). +Usually the bilinear form is +understood from the context, in which case one drops ⟨⋅,⋅⟩ from the notation; moreover, for our + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +5 +purposes, we will have fixed a basis for V , one in which the bilinear form is represented by the +nonsingular skew-symmetric matrix +J ∶= ( 0 +I2 +−I2 +0 ), +where I2 is the 2 × 2 identity matrix. +By a subquotient W of a Galois module U, we mean a Galois module W that admits a surjection +U ′ ↠ W from a subrepresentation U ′ of U. +Since we are chiefly concerned with computing the sets LikelyNonsurjectivePrimes(C;B) and +PossiblyNonsurjectivePrimes(C) for a fixed curve C, we will henceforth, for ease of notation, drop +the C from the notation for these sets. +2.2. Integral characteristic polynomial of Frobenius. The theoretical result underlying the +whole approach is the following. +Theorem 2.1 (Weil, see [ST68, Theorem 3]). Let A be an abelian variety of dimension g defined +over Q and let p be a prime of good reduction for A. Then there exists a monic integral polynomial +Pp(t) ∈ Z[t] of degree 2g with constant coefficient pg such that for any ℓ ≠ p, the polynomial Pp(t) +modulo ℓ is the characteristic polynomial of the action of Frobp on TℓA. Furthermore, every root +of Pp(t) has complex absolute value p1/2. +The polynomials Pp(t) are computationally accessible by counting points on C over Fpr r = 1,2. +See [Poo17, Chapter 7] for more details. +In fact, Pp(t) can be accessed via the frobenius_ +polynomial command in Sage. In particular, we denote the trace of Frobenius by ap. By the +Grothendieck-Lefschetz trace formula, if A = JacX, p is a prime of good reduction for X, and +λ1,...,λ2g are the roots of Pp(t), then +#X(Fpr) = pr + 1 − +2g +∑ +i=1 +λr +i . +2.3. The Weil pairing and consequences on the characteristic polynomial of Frobenius. +The nondegenerate Weil pairing gives an isomorphism (of Galois modules): +(1) +TℓA ≃ (TℓA)∨ ⊗Zℓ Zℓ(1). +The Galois character acting on Zℓ(1) is the ℓ-adic cyclotomic character, which we denote by cycℓ. +The integral characteristic polynomial for the action of Frobp on Zℓ(1) is simply t−p. The integral +characteristic polynomial for the action of Frobp on (TℓA)∨ is the reversed polynomial +P ∨ +p (t) = Pp(1/t) ⋅ t2g/pg +whose roots are the inverses of the roots of Pp(t). +We now record a few easily verifiable consequences of the nondegeneracy of the Weil pairing +when dim(A) = 2. +Lemma 2.2. +(i) The roots of Pp(t) come in pairs that multiply out to p. In particular, Pp(t) has no root with +multiplicity 3. +(ii) Pp(t) = t4 − apt3 + bpt2 − papt + p2 for some ap,bp ∈ Z. +(iii) If the trace of an element of GSp4(Fℓ) is 0 mod ℓ, then its characteristic polynomial is re- +ducible modulo ℓ. In particular, this applies to Pp(t) when ap ≡ 0 (mod ℓ). +(iv) If A[ℓ] is a reducible GQ-module, then Pp(t) is reducible modulo ℓ. +Proof. Parts (i) and (ii) are immediate from the fact that the non-degenerate Weil pairing allows +us to pair up the four roots of Pp(t) into two pairs that each multiply out to p. + +6 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +For part (iii), suppose that M ∈ GSp4(Fℓ) has tr(M) = 0. Then the characteristic polynomial +PM(t) of M is of the form t4 +bt2 +c2. When the discriminant of PM is 0 modulo ℓ, the polynomial +PM has repeated roots and is hence reducible. So assume that the discriminant of PM is nonzero +modulo ℓ. When ℓ ≠ 2, the result follows from [Car56, Theorem 1]. When ℓ = 2, a direct computation +shows that the characteristic polynomial of a trace 0 element of GSp4(F2) is either (t + 1)4 or +(t2 + t + 1)2, which are both reducible. +Part (iv) is immediate from Theorem 2.1 since Pp(t) mod ℓ by definition is the characteristic +polynomial for the action of Frobp on A[ℓ]. +□ +2.4. Maximal subgroups of GSp4(Fℓ). Mitchell [Mit14] classified the maximal subgroups of +PSp4(Fℓ) in 1914. This can be used to deduce the following classification of maximal subgroups of +GSp4(Fℓ) with surjective similitude character. +Lemma 2.3 (Mitchell). Let V be a 4-dimensional Fℓ-vector space endowed with a nondegener- +ate skew-symmetric bilinear form ω. Then any proper subgroup G of GSp(V,ω) with surjective +similitude character is contained in one of the following types of maximal subgroups. +(1) Reducible maximal subgroups +(a) Stabilizer of a 1-dimensional isotropic subspace for ω. +(b) Stabilizer of a 2-dimensional isotropic subspace for ω. +(2) Irreducible subgroups governed by a quadratic character +Normalizer Gℓ of the group Mℓ that preserves each summand in a direct sum decomposition +V1 ⊕ V2 of V , where V1 and V2 are jointly defined over Fℓ and either: +(a) both nondegenerate for ω; or +(b) both isotropic for ω. +Moreover, Mℓ is an index 2 subgroup of Gℓ. +(3) Stabilizer of a twisted cubic +GL(W) acting on Sym3 W ≃ V , where W is a 2-dimensional Fℓ-vector space. +(4) Exceptional subgroups See Table A for explicit generators for the groups described below. +(a) When ℓ ≡ ±3 (mod 8): a group whose image G1920 in PGSp(V,ω) has order 1920. +(b) When ℓ ≡ ±5 (mod 12) and ℓ ≠ 7: a group whose image G720 in PGSp(V,ω) has order 720. +(c) When ℓ = 7: a group whose image G5040 in PGSp(V,ω) has order 5040. +Remark 2. We have chosen to label the maximal subgroups in the classification using invariant +subspaces for the symplectic pairing ω on V , following the more modern account due to Aschbacher +(see [Lom16, Section 3.1]; for a more comprehensive treatment see [KL90]). For the convenience of +the reader, we record the correspondence between Mitchell’s original labels and ours below. +Mitchell’s label +Label in Lemma 2.3 +Group having an invariant point and plane +1a +Group having an invariant parabolic congruence +1b +Group having an invariant hyperbolic or elliptic congruence +2a +Group having an invariant quadric +2b +Table 1. Dictionary between maximal subgroup labels in [Die02]/[Mit14] and Lemma 2.3 +Remark 3. The maximal subgroups in (1) are the analogues of the Borel subgroup of GL2(Fℓ). +The maximal subgroups in (2) when the two subspaces V,V ′ in the direct sum decomposition +are individually defined over Fℓ are the analogues of normalizers of the split Cartan subgroup of +GL2(Fℓ). When the two subspaces V,V ′ are not individually defined over Fℓ instead, the maximal +subgroups in (2) are analogues of the normalizers of the non-split Cartan subgroups of GL2(Fℓ). + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +7 +Remark 4. We briefly explain why the action of GL2(Fℓ) on Sym3(F2 +ℓ) preserves a nondegenerate +symplectic form. It suffices to show that the restriction to SL2(Fℓ) fixes a vector in ⋀2 Sym3(F2 +ℓ). +This follows by character theory. If W is the standard 2-dimensional representation of SL2, then +we have ⋀2(Sym3 W) ≃ Sym4 W ⊕ 1 as representations of SL2. +Remark 5. One can extract explicit generators of the exceptional maximal subgroups from Mitchell’s +original work3. Indeed [Mit14, the proof of Theorem 8, page 390] gives four explicit matrices that +generate a G1920 (which is unique up to conjugacy in PGSp4(Fℓ)). Mitchell’s description of the +other exceptional groups is in terms of certain projective linear transformations called skew perspec- +tivities attached to a direct sum decomposition V = V1 ⊕ V2 into 2-dimensional subspaces. A skew +perspectivity of order n with axes V1 and V2 is the projective linear transformation that scales V1 by +a primitive nth root of unity and fixes V2. This proof also gives the axes of the skew perspectivities +of order 2 and 3 that generate the remaining exceptional groups [Mit14, pages 390-391]. Table 5 +lists generators of (one representative of the conjugacy class of) each of the exceptional maximal +subgroup extracted from Mitchell’s descriptions. In the file exceptional.m publicly available +with our code, we verify that Magma’s list of conjugacy classes of maximal subgroups of GSp4(Fℓ) +agree with those described in Lemma 2.3 for 3 ≤ ℓ ≤ 47. +Remark 6. The classification of exceptional maximal subgroups of PSp4(Fℓ) is more subtle than +that of PGSp4(Fℓ), because of the constraint on the similitude character of matrices in PSp4(Fℓ). +While the similitude character is not well-defined on PGSp4(Fℓ) (multiplication by a scalar c ∈ F× +ℓ +scales the similitude character by c2) it is well-defined modulo squares. The group PSp4(Fℓ) is the +kernel of this natural map: +1 → PSp4(Fℓ) → PGSp4(Fℓ) +mult +��→ F× +ℓ /(F× +ℓ )2 ≃ {±1} → 1. +An exceptional subgroup of PGSp4(Fℓ) gives rise to an exceptional subgroup of PSp4(Fℓ) of either +the same size or half the size depending on the image of mult restricted to that subgroup, which +in turn depends on the congruence class of ℓ. For this reason, the maximal exceptional subgroups +of PSp4(Fℓ) in Mitchell’s original classification (also recalled in Dieulefait [Die02, Section 2.1]) can +have order 1920 or 960 and 720 or 360 depending on the congruence class of ℓ, and 2520 (for +ℓ = 7). Such an exceptional subgroup gives rise to a maximal exceptional subgroup of PGSp4(Fℓ) +only when mult is surjective (i.e., its intersection with PSp4(Fℓ) is index 2), which explains the +restricted congruence classes of ℓ for which they arise. +We now record a lemma that directly follows from the structure of maximal subgroups described +above. This lemma will be used in Section 4 to devise a criterion for a subgroup of GSp4(Fℓ) to be +the entire group. For an element T in GSp4(Fℓ), let tr(T), mid(T), mult(T) denote the trace of +T, the middle coefficient of the characteristic polynomial of T, and the similitude character applied +to T respectively4. For a scalar λ, we have +tr(λT) = λtr(T), +mid(λT) = λ2 mid(T), +mult(λT) = λ2 mult(T). +Hence the quantities tr(T)2/mult(T) and mid(T)/mult(T) are well-defined on PGSp4(Fℓ). For +ℓ > 2 and ∗ ∈ {720,1920,5040}, define +(2) +Cℓ,∗ ∶= {( tr(T)2 +mult(T), mid(T) +mult(T)) ∣ T ∈ an exceptional subgroup of GSp4(Fℓ) of projective order ∗} +Lemma 2.4. +(1) In cases 2a and 2b of Lemma 2.3: +3Mitchell’s notation for PGSp4(Fℓ) is Aν(ℓ) and for PSp4(Fℓ) is A1(ℓ). +4Explicitly, the characteristic polynomial of T is therefore t4 − tr(T)t3 + mid(T)t2 − mult(T) tr(T)t + mult(T)2. + +8 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +(a) every element in Gℓ ∖ Mℓ has trace 0, and, +(b) the group Mℓ stabilizes a non-trivial linear subspace of F +4 +ℓ. +(2) Every element that is contained in a maximal subgroup corresponding to the stabilizer of a +twisted cubic has a reducible characteristic polynomial. +(3) For ∗ ∈ {1920,720}, the set Cℓ,∗ defined in (2) equals the reduction modulo ℓ of the elements of +the set C∗ below. +C1920 = {(0,−2),(0,−1),(0,0),(0,1),(0,2),(1,1),(2,1),(2,2),(4,2),(4,3),(8,4),(16,6)} +C720 = {(0,1),(0,0),(4,3),(1,1),(16,6),(0,2),(1,0),(3,2),(0,−2)} +We also have +C7,5040 = {(0,0),(0,1),(0,2),(0,5),(0,6),(1,0),(1,1),(2,6),(3,2),(4,3),(5,3),(6,3)}. +Proof. +(1) In cases 2a and 2b of Lemma 2.3, since any element of the normalizer Gℓ that is not in Mℓ +switches elements in the two subspaces V1 and V2 (i.e. maps elements in the subspace V1 +in the decomposition V1 ⊕ V2 to elements in V2 and vice-versa), it follows that any element +in Gℓ ∖ Mℓ has trace zero. +(2) The conjugacy class of maximal subgroups corresponding to the stabilizer of a twisted cubic +comes from the embedding GL2(Fℓ) +ι�→ GSp4(Fℓ) induced by the natural action of GL2(Fℓ) +on the space of monomials of degree 3 in 2 variables. If M is a matrix in GL2(Fℓ) with +eigenvalues λ,µ (possibly repeated), then the eigenvalues of ι(M) are λ3,µ3,λ2µ,λµ2 and +hence the characteristic polynomial of ι(M) factors as (T 2 −(λ3 +µ3)T +λ3µ3)(T 2 −(λ2µ+ +λµ2)T + λ3µ3) over Fℓ which is reducible over Fℓ. +(3) This follows from the description of the maximal subgroups given in Table 5. Each case +(except G5040 that only occurs for ℓ = 7) depends on a choice of a root of a quadratic +polynomial. In the file exceptional statistics.sage, we generate the corresponding +finite subgroups over the appropriate quadratic number field to compute C∗. It follows that +the corresponding values for the subgroup G∗ in GSp4(Fℓ) can be obtained by reducing +these values modulo ℓ. Since the group G5040 only appears for ℓ = 7, we directly compute +the set C7,5040. +□ +Remark 7. The condition in Lemma 2.4(3) is the analogue of the condition [Ser72, Proposition 19 +(iii)] used to rule out exceptional maximal subgroups of GL2(Fℓ). +We end this subsection by including the following lemma, to further highlight the similarities +between the above classification of maximal subgroups of GSp4(Fℓ) and the more familiar classi- +fication of maximal subgroups of GL2(Fℓ). This lemma is not used elsewhere in the article and is +thus for expositional purposes only. +Lemma 2.5. +(1) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are indi- +vidually defined over Fℓ is isomorphic to +{(m1,m2) ∈ GL2(Fℓ)2 ∣ det(m1) = det(m2)}. +In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ2 − 1)2. +(2) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are individually +defined over Fℓ is isomorphic to +{(m1,m2) ∈ GL2(Fℓ)2 ∣ mT +1 m2 = λI, for some λ ∈ F∗ +ℓ }. +In particular, the order of Mℓ is ℓ(ℓ − 1)2(ℓ2 − 1). + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +9 +(3) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are not +individually defined over Fℓ is isomorphic to +{m ∈ GL2(Fℓ2) ∣ det(m) ∈ F∗ +ℓ }. +In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ4 − 1). +(4) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are not indi- +vidually defined over Fℓ is isomorphic to GU2(Fℓ2), i.e., +{m ∈ GL2(Fℓ2) ∣ mT ι(m) = λI, for some λ ∈ F∗ +ℓ }, +where ι denotes the natural extension of the Galois automorphism of Fℓ2/Fℓ to GL2(Fℓ2). In +particular, the order of Mℓ is ℓ(ℓ2 − 1)2. +Proof. Given a direct sum decomposition V1 ⊕ V2 of a vector space V over Fq, we get a natural +embedding of Aut(V1) × Aut(V2) (≅ GL2(Fq)2) into Aut(V ) (≅ GL4(Fq)), whose image consists of +automorphisms that preserve this direct sum decomposition. We will henceforth refer to elements +of Aut(V1) × Aut(V2) as elements of Aut(V ) using this embedding. To understand the subgroup +Mℓ of GSp4(Fq) in cases (1) and (2) where the two subspaces in the direct sum decomposition are +individually defined over Fq, we need to further impose the condition that the automorphisms in +the image of the map Aut(V1) × Aut(V2) → Aut(V ) preserve the symplectic form ω on V up to a +scalar. +In (1), without any loss of generality, the two nondegenerate subspaces V1 and V2 can be chosen +to be orthogonal complements under the nondegenerate pairing ω, and so by Witt’s theorem, in a +suitable basis for V1⊕V2 obtained by concatenating a basis of V1 and a basis of V2, the nondegenerate +symplectic pairing ω has the following block-diagonal shape: +B ∶= +⎡⎢⎢⎢⎢⎢⎢⎢⎣ +0 +1 +−1 +0 +0 +1 +−1 +0 +⎤⎥⎥⎥⎥⎥⎥⎥⎦ +. +The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing +up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which boils down to +det(m1) = λ = det(m2). +Similarly, in (2), without any loss of generality, by Witt’s theorem, in a suitable basis for V1 ⊕V2 +obtained by concatenating a basis of the isotropic subspace V1 and a basis of the isotropic subspace +V2, the nondegenerate symplectic pairing ω has the following block-diagonal shape. +B ∶= +⎡⎢⎢⎢⎢⎢⎢⎢⎣ +0 +1 +1 +0 +0 +−1 +−1 +0 +⎤⎥⎥⎥⎥⎥⎥⎥⎦ +. +The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing +up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which again boils down +to mT +1 m2 = λI. +If we have a subspace W defined over Fq2 but not defined over Fq, and we let W denote the +conjugate subspace and further assume that W ⊕W gives a direct sum decomposition of V , then we +get a natural embedding of Aut(W) (≅ GL2(Fq2)) into Aut(V ) (≅ GL4(Fq)) whose image consists +of automorphisms that commute with the natural involution of V ⊗ Fq2 induced by the Galois +automorphism of Fq2 over Fq. The proofs of cases (3) and (4) are analogous to the cases (1) and (2) +respectively, by using the direct sum decomposition W ⊕W and letting m2 = ι(m1). The condition +that det(m1) = det(m2) in (1) becomes the condition det(m1) = det(m2) = detm1 = det(m1), or + +10 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +equivalently, that det(m1) ∈ Fq in (3). Similarly, the condition that mT +1 m2 = λI in (2) becomes the +condition that mT +1 ι(m1) = λI in (4). +□ +2.5. Image of inertia and (tame) fundamental characters. Dieulefait [Die02] used Mitchell’s +work described in the previous subsection to classify the maximal subgroups of GSp4(Fℓ) that could +occur as the image of ρA,ℓ . This was achieved via an application of a fundamental result of Serre +and Raynaud that strongly constrains the action of inertia at ℓ, and which we now recall. +Fix a prime ℓ > 3 that does not divide the conductor N of A. Let Iℓ be an inertia subgroup +at ℓ. Let ψn∶Iℓ → F× +ℓn denote a (tame) fundamental character of level n. The n Galois-conjugate +fundamental characters ψn,1,...,ψn,n of level n are given by ψn,i ∶= ψℓi +n . Recall that the fundamental +character of level 1 is simply the mod ℓ cyclotomic character cycℓ, and that the product of all +fundamental characters of a given level is the cyclotomic character. +Theorem 2.6 (Serre [Ser72], Raynaud [Ray74], cf. [Die02][Theorem 2.1). Let ℓ be a semistable +prime for A. Let V /Fℓ be an n-dimensional Jordan–H¨older factor of the Iℓ-module A[ℓ]. Then V +admits a 1-dimensional Fℓn-vector space structure such that ρA,ℓ∣Iℓ acts on V via the character +ψd1 +n,1⋯ψdn +n,n +with each di equal to either 0 or 1. +On the other hand, the following fundamental result of Grothendieck constrains the action of +inertia at semistable primes p ≠ ℓ. +Theorem 2.7 (Grothendieck [GRR72, Expos´e IX, Prop 3.5]). Let A be an abelian variety over a +number field K. Then A has semistable reduction at p ≠ ℓ if and only if the action of Ip ⊂ GK on +TℓA is unipotent of length 2. +Combining these two results allows one fine control of the determinant of a subquotient of A[ℓ]; +this will be used in Section 3. +Corollary 2.8. Let A/Q be an abelian surface, and let Xℓ be a Jordan–H¨older factor of the Fℓ[GQ]- +module A[ℓ] ⊗ Fℓ. If ℓ is a semistable prime, then +detXℓ ≃ ϵ ⋅ cycx +ℓ +for some character ϵ∶GQ → Fℓ that is unramified at ℓ and some 0 ≤ x ≤ dimXℓ. Moreover, ϵ120 = 1. +Proof. The first part follows immediately from Theorem 2.6. +For the fact that ϵ120 = 1, every +abelian surface attains semistable reduction over an extension K/Q with [K ∶ Q] dividing 120 by +[LV14a, Theorem 7.2], and so this follows from Theorem 2.7 since there are no nontrivial unramified +characters of GQ. +□ +We can now state Dieulefait’s classification of maximal subgroups of GSp4(Fℓ) that can occur +as the image ρA,ℓ(GQ) for a semistable prime ℓ > 7. +Proposition 2.9 ([Die02]). Let A be the Jacobian of a genus 2 curve defined over Q with Weil +pairing ω on A[ℓ]. If ℓ > 7 is a semistable prime, then ρA,ℓ(GQ) is either all of GSp(A[ℓ],ω) or it +is contained in one of the maximal subgroups of Types (1) or (2) in Lemma 2.3. +See also [Lom16, Proposition 3.15] for an expanded exposition of why the image of GQ cannot +be contained in maximal subgroup of Type (3) for a semistable prime ℓ > 7. +Remark 8. However, if ℓ is a prime of additive reduction, or if ℓ ≤ 7, then the image of GQ may also +be contained in any of the four types of maximal subgroups described in Lemma 2.3. Nevertheless, +by [LV22, Theorem 6.6], for any prime ℓ > 24, we have that the exponent of the projective image is +bounded exp(PρA,ℓ) ≥ (ℓ−1)/12. Since exp(G1920) = 2exp(S6) = 120 and exp(G720) = exp(S5) = 60, +the exceptional maximal subgroups cannot occur as ρA,ℓ(GQ) for ℓ > 1441. + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +11 +2.6. A consequence of the Chebotarev density theorem. Let K/Q be a finite Galois exten- +sion with Galois group G = Gal(K/Q) and absolute discriminant dK. Let S ⊆ G be a nonempty +subset that is closed under conjugation. By the Chebotarev density theorem, we know that +(3) +lim +x→∞ +∣{p ≤ x ∶ p is unramified in K and Frobp ∈ S}∣ +∣{p ≤ x}∣ += ∣S∣ +∣G∣. +Let p be the least prime such that p is unramified in K and Frobp ∈ S. There are effective versions +of the Chebotarev density theorem that give bounds on p. The best known unconditional bounds +are polynomial in dK [LMO79, AK19, KW22]. Under GRH, the best known bounds are polynomial +in log dK. In particular Bach and Sorenson [BS96] showed that under GRH, +(4) +p ≤ (4log dK + 2.5[K ∶ Q] + 5)2. +The present goal is to give an effective version of the Chebotarev density theorem in the context +of abelian surfaces. We will use a corollary of (4) that is noted in [MW21] which allows for the +avoidance of a prescribed set of primes by taking a quadratic extension of K. We do this because +we will take K = Q(A[ℓ]), and p being unramified in K is not sufficient to imply that p is a prime +of good reduction for A. Lastly, we will use that by [Ser81, Proposition 6], if K/Q is finite Galois, +then +(5) +log dK ≤ ([K ∶ Q] − 1)log rad(dK) + [K ∶ Q]log([K ∶ Q]), +where radn = ∏p∣n p denotes the radical of an integer n. +Lemma 2.10. Let A/Q be a typical principally polarized abelian surface with conductor NA. Let q +be a prime. Let S ⊆ ρA,q(GQ) be a nonempty subset that is closed under conjugation. Let p be the +least prime of good reduction for A such that p ≠ q and ρA,q(Frobp) ∈ S. Assuming GRH, we have +p ≤ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5) +2 . +Proof. Let K = Q(A[q]). Then K/Q is Galois and +[K ∶ Q] ≤ ∣GSp4(Fq)∣ = q4(q4 − 1)(q2 − 1)(q − 1) ≤ q11. +As raddK is the product of primes that ramify in Q(A[q]), the criterion of N´eron-Ogg-Shafarevich +for abelian varieties [ST68, Theorem 1] implies that rad(dK) divides rad(qNA). Let ˜K ∶= K(√m) +where m ∶= rad(2NA). Note that the primes that ramify in ˜K are precisely 2, q, and the primes of +bad reduction for A. Thus rad(d ˜ +K) = rad(2qNA). Moreover [ ˜K ∶ Q] ≤ 2q11 and by (5), +log(d ˜ +K) ≤ (2q11 − 1)log rad(2qNA) + 22q11 log(2q). +Applying [MW21, Corollary 6] to the field ˜K, we get that (under GRH) there exists a prime p +satisfying the claimed bound, that does not divide m, and for which ρA,q(Frobp) ∈ S. +□ +3. Finding a finite set containing all nonsurjective primes +In this section we describe Algorithm 3.1 referenced in Theorem 1.1(1). This algorithm produces +a finite list PossiblyNonsurjectivePrimes that provably includes all nonsurjective primes ℓ. We also +prove Theorem 1.2. +Since our goal is to produce a finite list (from which we will later remove extraneous primes) it +is harmless to include the finitely many bad primes as well as 2,3,5,7. Using Proposition 2.9, it +suffices to find conditions on ℓ > 7 for which ρA,ℓ(GQ) could be contained in one of the maximal +subgroups of type (1) and (2) in Lemma 2.3. We first find primes ℓ for which ρA,ℓ has (geometrically) +reducible image (and hence is contained in a maximal subgroup in case (1) of Lemma 2.3 or in a +subgroup Mℓ in case (2)). To treat the geometrically irreducible cases, we then make use of the +observation from Lemma 2.4 1a that every element outside of an index 2 subgroup has trace 0. + +12 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +Algorithm 3.1. Given a typical genus 2 curve C/Q with conductor N and Jacobian A, compute +a finite list PossiblyNonsurjectivePrimes of primes as follows. +(1) Initialize PossiblyNonsurjectivePrimes = [2,3,5,7]. +(2) Add to PossiblyNonsurjectivePrimes all primes dividing N. +(3) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗ Fℓ could be reducible via +Algorithms 3.3, 3.6, and 3.10. +(4) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗Fℓ could be irreducible but +nonsurjective via Algorithm 3.13. +(5) Return PossiblyNonsurjectivePrimes. +At a very high-level, each of the subalgorithms of Algorithm 3.1 makes use of a set of auxiliary +good primes p. We compute the integral characteristic polynomial of Frobenius Pp(t) and use it to +constrain those ℓ ≠ p for which the image could have a particular shape. +Remark 9. Even though robust methods to compute the conductor N of a genus 2 curve are not +implemented at the time of writing, the odd-part Nodd of N can be computed via genus2red +function of PARI and the genus2reduction module of SageMath, both based on an algorithm +of Liu [Liu94]. Moreover, [BK94, Theorem 6.2] bounds the 2-exponent of N above by 20 and hence +N can be bounded above by 220Nodd. While these algorithms can be run only with the bound +220Nodd, it will substantially increase the run-time of the limiting Algorithm 3.10. +We now explain each of these steps in detail. +3.1. Good primes that are not geometrically irreducible. In this section we describe the +conditions that ℓ must satisfy for the base-extension A[ℓ] ∶= A[ℓ] ⊗Fℓ Fℓ to be reducible. In this +case, the representation A[ℓ] is an extension +(6) +0 → Xℓ → A[ℓ] → Yℓ → 0 +of a (quotient) representation Yℓ by a (sub) representation Xℓ. Recall that Nsq denotes the largest +square divisor of N. +Lemma 3.2. Let ℓ be a prime of good reduction for A and suppose that A[ℓ] sits in sequence (6). +Let p ≠ ℓ be a good prime for A and let f denote the order of p in (Z/NsqZ)×. Then there exists +0 ≤ x ≤ dimXℓ and 0 ≤ y ≤ dimYℓ such that Frobgcd(f,120) +p +acts on detXℓ by pgcd(f,120)x, respectively +on detYℓ by pgcd(f,120)y. +Proof. Since ℓ is a good prime and Xℓ is composed of Jordan–H¨older factors of A[ℓ], Corollary 2.8 +constrains its determinant. We have detXℓ = ϵcycx +ℓ for some character ϵ∶GQ → Fℓ unramified at ℓ, +and 0 ≤ x ≤ dimXℓ, and ϵ120 = 1. Hence Frob120 +p +acts on detXℓ by cycℓ(Frobp)120x = p120x. +In fact, we can do slightly better. Since detA[ℓ] ≃ cyc2 +ℓ, we have detYℓ ≃ ϵ−1 cyc2−x +ℓ +. Since the +conductor is multiplicative in extensions, we conclude that cond(ϵ)2 ∣ N. By class field theory, +the character ϵ factors through (Z/cond(ϵ)Z)×, and hence through (Z/NsqZ)×, sending Frobp +to p (mod Nsq). Since pf ≡ 1 (mod Nsq), we have that ϵ(Frobp)gcd(f,120) = 1, and we see that +Frobgcd(f,120) +p +acts on detXℓ by pgcd(f,120)x. Exchanging the roles of Xℓ and Yℓ, we deduce the +analogous statement for Yℓ. +□ +This is often enough information to find all ℓ for which A[ℓ] has a nontrivial subquotient. Namely, +by Theorem 2.1, every root of Pp(t) has complex absolute value p1/2. Thus the gcd(f,120)-th power +of each root has complex absolute value pgcd(f,120)/2, and hence is never integrally equal to 1 or +pgcd(f,120). Since Lemma 3.2 guarantees that this equality must hold modulo ℓ for any good prime +ℓ for which A[ℓ] is reducible with a 1-dimensional subquotient, we always get a nontrivial condition +on ℓ. Some care must be taken to rule out ℓ for which A[ℓ] only has 2-dimensional subquotient(s). + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +13 +3.1.1. Odd-dimensional subquotient. Let p be a good prime. +Given a polynomial P(t) and an +integer f, write P (f)(t) for the polynomial whose roots are the fth powers of roots of P(t). +Universal formulas for such polynomials in terms of the coefficients of P(t) are easy to compute, +and are implemented in our code in the case where P is a degree 4 polynomial whose roots multiply +in pairs to pα, and f ∣ 120. +Algorithm 3.3. Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of +p in (Z/NsqZ)× and write f′ = gcd(f,120). Compute an integer Modd as follows. +(1) Choose a nonempty finite set T of auxiliary good primes p ∤ N. +(2) For each p, compute +Rp ∶= P (f′) +p +(1). +(3) Let Modd = gcdp∈T (pRp) over all auxiliary primes. +Return the list of prime divisors ℓ of Modd. +Proposition 3.4. Any good prime ℓ for which A[ℓ] has an odd-dimensional subrepresentation is +returned by Algorithm 3.3. +Proof. Since A[ℓ] is 4-dimensional and has an odd-dimensional subrepresentation, it has a 1- +dimensional subquotient. For any p ∈ T , Lemma 3.2 shows that Frobf′ +p acts on detXℓ by either pf′ +or by 1. Thus, the action of Frobf′ +p on A[ℓ] has an eigenvalue that is congruent to pf′ or 1 modulo +ℓ, and so P (f′) +p +(t) has a root that is congruent to 1 or pf′ modulo ℓ. Since the roots of P (f′)(t) +multiply in pairs to pf′, we have P (f′) +p +(pf′) = p2f′P (f′) +p +(1). Hence ℓ divides p ⋅ P (f′) +p +(1) = pRp. +□ +Using Theorem 2.1, we can give a theoretical bound on the “worst case” of this step of the +algorithm using only one auxiliary prime p. Of course, taking the greatest common divisor over +multiple auxiliary primes will likely remove extraneous factors, and in practice this step of the +algorithm runs substantially faster than other steps. +Proposition 3.5. Algorithm 3.3 terminates. More precisely, if p is any good prime for A, then +0 ≠ ∣Modd∣ ≪ p240 +where the implied constant is absolute. +Proof. This follows from the fact that the coefficient of ti in P (f′) +p +(t) has magnitude on the order +of p(2−i)f′ and f′ ≤ 120. +□ +3.1.2. Two-dimensional subquotients. We now assume that A[ℓ] is reducible, but does not have +any odd-dimensional subquotients. +In particular, it has an irreducible subrepresentation Xℓ of +dimension 2, with irreducible quotient Yℓ of dimension 2. If A[ℓ] is reducible but indecomposable, +then Xℓ is the unique subrepresentation of A[ℓ] and Y ∨ +ℓ ⊗ cycℓ is the unique subrepresentation +of A[ℓ] +∨ ⊗ cycℓ. The isomorphism TℓA ≃ (TℓA)∨ ⊗ cycℓ from (1) yields an isomorphism A[ℓ] ≃ +(A[ℓ])∨ ⊗ cycℓ and hence Xℓ ≃ Y ∨ +ℓ ⊗ cycℓ. Otherwise, A[ℓ] ≃ Xℓ ⊕ Yℓ and so the nondegeneracy of +the Weil pairing gives +Xℓ ⊕ Yℓ ≃ (X∨ +ℓ ⊗ cycℓ) ⊕ (Y ∨ +ℓ ⊗ cycℓ). +Therefore either: +(a) Xℓ ≃ Y ∨ +ℓ ⊗ cycℓ and Yℓ ≃ X∨ +ℓ ⊗ cycℓ, or +(b) Xℓ ≃ X∨ +ℓ ⊗ cycℓ and Yℓ ≃ Y ∨ +ℓ ⊗ cycℓ and A[ℓ] ≃ Xℓ ⊕ Yℓ. +We call the first case related 2-dimensional subquotients and the second case self-dual 2-dimensional +subrepresentations. +We will see that the ideas of Lemma 3.2 easily extend to treat the related +subquotient case; we will use the validity of Serre’s conjecture to treat the self-dual case. In the + +14 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +case that A[ℓ] is decomposable, the above two cases correspond respectively to the index 2 subgroup +Mℓ in cases (2a) (the isotropic case) and (2b) (the nondegenerate case) of Lemma 2.3. +3.1.3. Related two-dimensional subquotients. Let p be a good prime. Let Pp(t) ∶= t4−at3+bt2−pat+p2 +be the characteristic polynomial of Frobp acting on A[ℓ]. Suppose that α and β are the eigenvalues +of Frobp acting on the subrepresentation Xℓ. Then, since Xℓ ≃ Y ∨ +ℓ ⊗ cycℓ, the eigenvalues of the +action of Frobp on Yℓ are p/α and p/β. The action of Frobp on detXℓ is therefore by a product of +two of the roots of Pp(t) that do not multiply to p. Note that there are four such pairs of roots of +Pp(t) that do not multiply to p. Let Qp(t) be the quartic polynomial whose roots are the products +of pairs of roots of Pp(t) that do not multiply to p. By design, the roots of Qp(t) have complex +absolute value p, but are not equal to p. (It is elementary to work out that +Qp(t) = t4 − (b − 2p)t3 + p(a2 − 2b + 2p)t2 − p2(b − 2p)t + p4 +and is a quartic whose roots multiply in pairs to p2.) +Algorithm 3.6. Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of +p in (Z/NsqZ)× and write f′ = gcd(f,120). Compute an integer Mrelated as follows. +(1) Choose a finite set T of auxiliary good primes p ∤ N; +(2) For each p, compute the product +Rp ∶= Q(f′) +p +(1)Q(f′) +p +(pf′) +(3) Let Mrelated = gcdp∈T (pRp). +Return the list of prime divisors ℓ of Mrelated. +Proposition 3.7. Any good prime ℓ for which A[ℓ] has related two-dimensional subquotients is +returned by Algorithm 3.6. +Proof. Proceed similarly as in the proof of Proposition 3.4 — in particular, ℓ divides Q(f′) +p +(1), +Q(f′) +p +(pf′), or Q(f′) +p +(p2f′) and hence ℓ divides pRp since Q(f′) +p +(p2f′) = p4f′Q(f′) +p +(1). +□ +A theoretical “worst case” analysis yields the following. +Proposition 3.8. Algorithm 3.6 terminates. More precisely, if q is the smallest surjective prime +for A, then a good prime p for which Rp is nonzero is bounded by a function of q. Assuming GRH, +p ≪ q22 log2(qN), +where the implied constants are absolute and effectively computable. Moreover, for such a prime p, +∣Mrelated∣ ≪ p961 ≪ q21142 log1922(qN), +where the implied constants are absolute. +Proof. By Serre’s open image theorem for genus 2 curves, such a prime q exists, and by Lemma +2.10, the prime p can be chosen such that Rp is nonzero modulo q. Finally, +Mrelated ≤ pRp = pQ(f′)(1)Q(f′)(pf′) ≪ p8f′+1 ≪ p961, +since the coefficient of ti in Q(f′)(t) has magnitude on the order of p(4−i)f′ and f′ ≤ 120. +□ + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +15 +3.1.4. Self-dual two-dimensional subrepresentations. In this case, both subrepresentations Xℓ and +Yℓ are absolutely irreducible 2-dimensional Galois representations with determinant the cyclotomic +character cycℓ. It follows that the representations are odd (i.e., the determinant of complex con- +jugation is −1.) Therefore, by the Khare–Wintenberger theorem (formerly Serre’s conjecture on +the modularity of mod-ℓ Galois representations) [Kha06, KW09a, KW09b], both Xℓ and Yℓ are +modular; that is, for i = 1,2, there exist newforms fi ∈ Snew +ki (Γ1(Ni),ϵi) such that +Xℓ ≅ ρf1,ℓ and Yℓ ≅ ρf2,ℓ. +Furthermore, by the multiplicativity of Artin conductors, we obtain the divisibility N1N2 ∣ N. +Lemma 3.9. Both f1 and f2 have weight two and trivial Nebentypus; that is, k1 = k2 = 2, and +ϵ1 = ϵ2 = 1. +Proof. From Theorem 2.6, we have that Xℓ∣Iℓ and Yℓ∣Iℓ must each be conjugate to either of the +following subgroups of GL2(Fℓ): +(1 +∗ +0 +cycℓ +) or (ψ2 +0 +0 +ψℓ +2 +). +The assertion of weight 2 now follows from [Ser87, Proposition 3]. (Alternatively, one may use +Proposition 4 of loc. cit., observing that Xℓ and Yℓ are finite and flat as group schemes over Zℓ +because ℓ is a prime of good reduction.) +From Section 1 of loc. cit., the Nebentypus ϵi of fi satisfies, for all p ∤ ℓN, +detXℓ(Frobp) = p ⋅ ϵi(p), +where this equality is viewed inside F +× +ℓ . The triviality follows. +□ +We therefore have newforms fi ∈ Snew +2 +(Γ0(Ni)) such that +(7) +A[ℓ] ≃ ρf1,ℓ ⊕ ρf2,ℓ. +We may assume without loss of generality that N1 ≤ +√ +N. Let p ∤ N be an auxiliary prime. We +obtain from equation (7) that the integral characteristic polynomial of Frobenius factors: +Pp(t) ≡ (t2 − ap(f1)t + p)(t2 − ap(f2)t + p) +mod ℓ; +here we use the standard property that, for f a normalised eigenform with trivial Nebentypus, +ρf,ℓ(Frobp) satisfies the polynomial equation t2 − ap(f)t + p for p ≠ ℓ. In particular, we have +Res(Pp(t),t2 − ap(f1)t + p) ≡ 0 +mod ℓ. +This serves as the basis of the algorithm to find all primes ℓ in this case. +Algorithm 3.10. Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer +Mself-dual as follows. +(1) Compute the set S of divisors d of N with d ≤ +√ +N. +(2) For each d ∈ S: +(a) compute the Hecke L-polynomial +Qd(t) ∶= ∏ +f +(t2 − ap(f)t + p), +where the product is taken over the finitely many newforms in Snew +2 +(Γ0(d)); +(b) choose a finite set T of auxiliary primes p ∤ N; +(c) for each auxiliary prime p, compute the resultant +Rp(d) ∶= Res(Pp(t),Qd(t)); + +16 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +(d) Take the greatest common divisor +M(d) ∶= gcd +p∈T +(pRp(d)). +(3) Let Mself-dual ∶= ∏d∈S M(d). +Return the list of prime divisors ℓ of Mself-dual. +Proposition 3.11. Any good prime ℓ for which A[ℓ] has self-dual two-dimensional subrepresenta- +tions is returned by Algorithm 3.10. +Proof. If ℓ is in T for any d ∈ S, then ℓ is in the output because Mself-dual is a multiple of M(d) +which in turn is a multiple of any element of T . Otherwise, as explained before Algorithm 3.10, +there is some N1 ∈ S and some newform f1 ∈ Snew +2 +(Γ0(N1)) such that Res(Pp(t),t2 − apf1t + p) ≡ 0 +(mod ℓ) for every p ∈ T . In particular, Rp(N1) ≡ 0 (mod ℓ), so ℓ divides M(N1) and Mself-dual. +□ +We can again do a “worst case” theoretical analysis of this algorithm to conclude the following. +As this indicates, this is by far the limiting step of the algorithm. +Proposition 3.12. Algorithm 3.10 terminates. More precisely, if q is the smallest surjective prime +for A, then a good prime p for which Rp(d) is nonzero is bounded by a function of q. Assuming GRH, +p ≪ q22 log2(qN), where the implied constant is absolute and effectively computable. Moreover, for +such a prime p, we have +∣Rp(d)∣ ≪ (2p1/2)8 dim Snew +2 +(Γ0(d)) ≪ (4p)(d+1)/3, +and so all together +∣Mself-dual∣ ≪ (4q)N1/2+ϵ, +where the implied constants are absolute. +Proof. As in Proposition 3.8, we use Serre’s open image theorem and the Effective Chebotarev +Theorem. If Rp(d) is zero integrally, then in particular Rp(d) ≡ 0 (mod q) and Pp(t) is reducible +modulo q. Since GSp4(Fq) contains elements that do not have reducible characteristic polynomial, +Lemma 2.10 implies that such elements are the image of Frobp for p bounded as claimed. +The resultant Rp(d) is the product of the pairwise differences of the roots of Pp(t) and Qd(t), +which all have complex absolute value p1/2. Hence the pairwise differences have absolute value +at most 2p1/2. +Moreover dimSnew +2 +(Γ0(d)) ≤ (d + 1)/12 by [Mar05, Theorem 2]. +Since there +are 8dimSnew +2 +(Γ0(d)) such terms multiplied to give Rp(d), the bound for Rp(d) follows. Since +Mself-dual = ∏ d∣N +d≤ +√ +N +pRp(d), it suffices to bound +∑ +d∣N +d≤ +√ +N +d + 4 +3 +≤ +∑ +d∣N +d≤ +√ +N +√ +N + 4 +3 +≤ σ0(N) +√ +N + 4 +3 +. +Since σ0(N) ≪ Nϵ by [Apo76, (31) on page 296], we obtain the claimed bound. +□ +Remark 10. The polynomial Qd(t) in step (2) of Algorithm 3.10 is closely related to the charac- +teristic polynomial Hd(t) of the Hecke operator Tp acting on the space S2(Γ0(d)), which may be +computed via modular symbols computations. One may recover Qd(t) from Hd(t) by first homoge- +nizing H with an auxiliary variable z (say) to obtain Hd(t,z), and setting t = 1+pz2 (an observation +we made in conjunction with Joseph Wetherell). In our computation of nonsurjective primes for +the database of genus 2 curves with conductor at most 220 (including those in the LMFDB), we +only needed to use polynomials Qd(t) for level up to 210 (since step (1) of the Algorithm has a +√ +N term). We are grateful to Andrew Sutherland for providing us with a precomputed dataset +for these levels resulting from the creation of an extensive database of modular forms going well +beyond what was previously available [BBB+21]. + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +17 +Remark 11. Our Sage implementation uses two auxiliary primes in Step 2(b) of the above algorithm. +Increasing the number of such primes yields smaller supersets at the expense of longer runtime. +3.2. Good primes that are geometrically irreducible. Let φ be any quadratic Dirichlet char- +acter φ∶(Z/NZ)× → {±1}. Our goal in this subsection is to find all good primes ℓ governed by φ, +by which we mean that +tr(ρA,ℓ(Frobp)) ≡ ap ≡ 0 +mod ℓ +whenever φ(p) = −1. +We will consider the set of all quadratic Dirichlet character φ∶(Z/NZ)× → {±1}. Using the struc- +ture theorem for finite abelian groups and the fact that φ factors through (Z/NZ)×/((Z/NZ)×)2, +this set has the structure of an F2-vector space of dimension +d(N) ∶= ω(N) + +⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ +0 +∶ v2(N) = 0 +−1 +∶ v2(N) = 1 +0 +∶ v2(N) = 2 +1 +∶ v2(N) ≥ 3, +where ω(m) denotes the number of prime factors of m and v2(m) is the 2-adic valuation of m. In +particular, d(N) ≤ ω(N) + 1. +Algorithm 3.13. Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mquad +as follows. +(1) Compute the set S of quadratic Dirichlet characters φ∶(Z/NZ)× → {±1}. +(2) For each φ ∈ S: +(a) Choose a nonempty finite set T of “auxiliary” primes p ∤ N for which ap ≠ 0 and φ(p) = −1. +(b) Take the greatest common divisor +Mφ ∶= gcd +p∈T +(pap), +over all auxiliary primes p. +(3) Let Mquad ∶= ∏φ∈S Mφ. +Return the list of prime divisors ℓ of Mquad. +Proposition 3.14. Any good prime ℓ for which A[ℓ] is governed by a quadratic character is +returned by Algorithm 3.13. +Proof. Suppose that A[ℓ] is governed by the quadratic character φ∶(Z/NZ)× → {±1}. Then for +every good prime p ≠ ℓ for which φ(p) = −1, the prime ℓ must divide the integral trace of Frobenius +ap. Hence ℓ divides Mφ and Mquad. +□ +Proposition 3.15. Algorithm 3.13 terminates. More precisely, if q is the smallest surjective prime +for A, then a good prime p for which φ(p) = −1 and ap is nonzero is bounded by a function of q. +Assuming GRH, p ≪ 22d(N)q22 log2(qN), where the implied constant is absolute and effectively +computable. Moreover, we have +∏ +φ∈S +∏ +ℓ governed +by φ +ℓ ≪ (23d(N)q33 log3(qN))2−21−d(N) ≪ 26ω(N)q66 log6(qN), +where the implied constant is absolute and effectively computable. +Proof. We imitate the proof of [LV14b, Lemma 21] in our setting. Let V be the d-dimensional +F2-vector space of quadratic Dirichlet characters of modulus N (equivalently, quadratic Galois +characters unramified outside of N). Let ρV ∶GK → V ∨ denote the representation sending Frobp to +the linear functional φ ↦ φ(p). Since the character for PGSp4(Fq)/PSp4(Fq) is the abelianization + +18 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +of PρA,q, we conclude in the same way as [LV14b, Proof of Lemma 21] that for any α ∈ V ∨, there +exists an Xα ∈ GSp4(Fq) with tr(Xα) ≠ 0 such that (α,Xα) is in the image of ρV × ρA,ℓ. +Apply the effective Chebotarev density theorem to the Galois extension corresponding to ρV × +ρA,q. This has degree at most 2d(N)∣GSp4(Fq)∣ and is unramified outside of qN. Therefore, assum- +ing GRH and combining (4) and (5), there exists a prime +pα ≪ 22d(N)q22 log2(qN) +for which (α,Xα) = (ρV (Frobpα),ρA,q(Frobpα)). Let φ be a character not in the kernel of α. Any +exceptional prime ℓ governed by φ must divide pαapα, which is nonzero because it is nonzero modulo +q. This proves that the algorithm terminates, since every φ is not in the kernel of precisely half +of all α ∈ V ∨. We now bound the size of the product of all ℓ governed by a character in S. If ℓ is +governed by φ, then ℓ divides the quantity +p∣ap∣ ≤ p3/2 ≪ 23d(N)q33 log3(qN). +Taking the product over all nonzero α in V (of which there are 2d(N) − 1), each ℓ will show up half +the time, so we obtain: +⎛ +⎜⎜⎜ +⎝ +∏ +ℓ governed +by φ ∈ S +ℓ +⎞ +⎟⎟⎟ +⎠ +2d(N)−1 +≪ (23d(N)q33 log3(qN)) +2d(N)−1 +, +which implies the result by taking the (2d(N)−1)th root of both sides. +□ +Putting all of these pieces together, we obtain the following. +Proof of Theorem 1.1(1). If ρA,ℓ is nonsurjective, ℓ > 7, and ℓ ∤ N, then Proposition 2.9 implies +that ρA,ℓ(GQ) must be in one of the maximal subgroups of Type (1) or (2) listed in Lemma +2.3. If it is contained in one of the reducible subgroups, i.e. the subgroups of Type (1), then +ρA,ℓ(GQ) (and, hence, ρA,ℓ(GQ) ⊗ Fℓ) is reducible, and so ℓ is added to PossiblyNonsurjectivePrimes +in Step (3) by Propositions 3.4, 3.7, and 3.11. +If ρA,ℓ(GQ) is contained in one of the index 2 +subgroups Mℓ of an irreducible subgroup of Type (2) listed in Lemma 2.3, then again ℓ is added to +PossiblyNonsurjectivePrimes in Step (3), since Mℓ ⊗ Fℓ is always reducible by Lemma 2.4(1b). +Hence we may assume that ρA,ℓ(GQ) is contained in one of the irreducible maximal subgroups +Gℓ of Type (2) listed in Lemma 2.3, but not in the index 2 subgroup Mℓ. The normalizer character +GQ +ρA,ℓ +��→ Gℓ → Gℓ/Mℓ = {±1} +is nontrivial and unramified outside of N, and so it corresponds to a quadratic Dirichlet character +φ∶(Z/NZ)× → {±1}. Lemma 2.4(1a) shows that tr(g) = 0 in Fℓ for any g ∈ Gℓ ∖ Mℓ. Consequently, +ℓ is governed by φ (in the language of Section 3.2), so ℓ is added to PossiblyNonsurjectivePrimes in +Step (4) by Proposition 3.14. +□ +3.3. Bounds on Serre’s open image theorem. In this section we combine the theoretical worst +case bounds in the Algorithms 3.3, 3.6, 3.10, and 3.13 to give a bound on the smallest surjective +good prime q, and the product of all nonsurjective primes, thereby establishing Theorem 1.2. +Corollary 3.16. Let A/Q be a typical genus 2 Jacobian of conductor N. Assuming GRH, we have +∏ +ℓ nonsurjective +ℓ ≪ exp(N1/2+ϵ), +where the implied constant is absolute and effectively computable. + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +19 +Proof. Let q be the smallest surjective good prime for A, which is finite by Serre’s open image +theorem. Multiplying the bounds in Propositions 3.5, 3.8, 3.12, and 3.15 by the conductor N, the +product of all nonsurjective primes is bounded by a function of q and N of the following shape +(8) +∏ +ℓ nonsurjective +ℓ ≪ qN1/2+ϵ. +On the other hand, since q is the smallest surjective prime by definition, the product of all primes +less than q divides the product of all nonsurjective primes. Using [Ser81, Lemme 11], we have +exp(q) ≪ ∏ +ℓ 0. Since LikelyNonsurjectivePrimes(B) is a sub- +list of PossiblyNonsurjectivePrimes, which contains all nonsurjective primes by Theorem 1.1(1), any +prime not in PossiblyNonsurjectivePrimes is surjective. Now consider ℓ ∈ PossiblyNonsurjectivePrimes +and not in LikelyNonsurjectivePrimes(B). If ℓ = 2, then by Proposition 4.3, ρA,2 is surjective. If +ℓ > 2, this means that we found primes p1,p2,p3,p4,p5 ≤ B each distinct from ℓ and of good reduc- +tion for A for which ρA,ℓ(Frobp1) ∈ Cα, ρA,ℓ(Frobp2) ∈ Cβ, ρA,ℓ(Frobp3) ∈ Cγ1, ρA,ℓ(Frobp4) ∈ Cγ2, +and ρA,ℓ(Frobp4) ∈ Cγ3. Note that by (1), the similitude factor mult(ρA,ℓ(Frobp)) is p. Therefore, +by Lemma 2.4(3), it follows that ρA,ℓ(GQ) is not contained in an exceptional maximal subgroup. +The surjectivity of ρA,ℓ now follows from Proposition 4.2. +Finally, we will show that if B is sufficiently large (as quantified by Theorem 1.3), then any +prime ℓ in PossiblyNonsurjectivePrimes is nonsurjective. Since the sets Cα, Cβ, Cγ1, Cγ2 and Cγ3 +are nonempty by Proposition 5.1 below and closed under conjugation, it follows by Lemma 2.10, +there exist primes p1,p2,p3,p4,p5 ≤ B as above. +□ + +22 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +Remark 15. If we assume both GRH and AHC, Ram Murty and Kumar Murty [MM97, p. 52] noted +(see also [FJ20, Theorem 2.3]) that the bound (4) can be replaced with p ≪ (log dK)2 +∣S∣ +. Proposition +5.1, which follows, shows that the sets Cα, Cβ, and Cγ have size at least ∣ GSp4(Fℓ)∣ +10 +. This can be +used to prove the ineffective version of Theorem 1.3 which relies on AHC noted in the introduction +in a manner similar to the proof of Theorem 1.3. +5. The probability of success +In this section we prove Theorem 1.4, by studying the probability that a matrix chosen uniformly +at random from GSp4(Fℓ) is contained in each of Cα, Cβ, and Cγ defined in Section 4.3. Let αℓ, βℓ, +and γℓ respectively be the probabilities that a matrix chosen uniformly at random from GSp4(Fℓ) +is contained in Cα, Cβ, or Cγ. +Proposition 5.1. Let M be a matrix chosen uniformly at random from GSp4(Fℓ) with ℓ odd. Then +(i) The probability that M ∈ Cα is given by +αℓ = 1 +4 − +1 +2(ℓ2 + 1). +(ii) The probability that M ∈ Cβ is given by +βℓ = 3 +8 − +3 +4(ℓ − 1) + +1 +2(ℓ − 1)2 . +(iii) The probability that M ∈ Cγ is +γℓ ≥ 1 − +3ℓ +ℓ2 + 1. +Remark 16. Magma code that directly verifies the sizes of Cα,Cβ,Cγ (i.e. computes αℓ,βℓ,γℓ) for +small ℓ may be found in helper_scripts/SanityCheckProbability.m in the repository. +[Shi82] characterizes all conjugacy classes of elements of GSp4(Fℓ) for ℓ odd, grouping them into +26 different types. For each type γ, Shinoda further computes the number N(γ) of conjugacy +classes of type γ and the size of the centralizer ∣CGSp4(Fℓ)(γ)∣, which is the size of the centralizer +∣CGSp4(Fℓ)(M)∣ of M in GSp4(Fℓ) for any M in a conjugacy class of type γ. The size ∣C(γ)∣ of any +conjugacy class of type γ can then easily be computed as +∣C(γ)∣ = +∣GSp4(Fℓ)∣ +∣CGSp4(Fℓ)(γ)∣ +and the probability that a uniformly chosen M ∈ GSp4(Fℓ) has conjugacy type γ is then given by +(9) +N(γ)∣C(γ)∣ +∣GSp4(Fℓ)∣ = +N(γ) +∣CGSp4(Fℓ)(γ)∣. +To prove Proposition 5.1, we will need to examine a handful of types of conjugacy classes of +GSp4(Fℓ). +There is only a single conjugacy type γ whose characteristic polynomials are irreducible. This +type is denoted K0 in [Shi82] where it is shown there that N(K0) = (ℓ−1)(ℓ2−1) +4 +and ∣CGSp4(Fℓ)(K0)∣ = +(ℓ − 1)(ℓ2 + 1). +While there is only one way for a polynomial to be irreducible, there are several ways for a +quartic polynomial to have a root of odd order. However, only some of these can occur if f(t) is +the characteristic polynomial of a matrix M ∈ GSp4(Fℓ) and we only need to concern ourselves +with the following three possibilities: +(a) f(t) splits completely over Fℓ; +(b) f(t) has two roots over Fℓ, both of which occur with multiplicity one; and +(c) f(t) has two simple roots and one double root over Fℓ. + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +23 +Cases (a) and (b) correspond to the conjugacy types H0 and J0 in [Shi82] respectively. +In +contrast, there are two types of conjugacy classes for which f(t) has two simple roots and one +double root, which are denoted E0 and E1 in [Shi82]. +The number of conjugacy classes and centralizer size for each of these conjugacy types is given by +Table 2, along with the associated probability that a uniform random M ∈ GSp4(Fℓ) has conjugacy +type γ computed using (9). +Type γ in [Shi82] +N(γ) +∣CGSp4(Fℓ)(γ)∣ +Associated Probability +K0 (Irreducible) +(ℓ−1)(ℓ2−1) +4 +(ℓ2 + 1)(ℓ − 1) +1 +4 − +1 +2(ℓ2+1) +H0 (Split) +(ℓ−1)(ℓ−3)2 +8 +(ℓ − 1)3 +1 +8 − +1 +2(ℓ−1) + +1 +2(ℓ−1)2 +J0 (Two Simple Roots) +(ℓ−1)3 +4 +(ℓ + 1)(ℓ − 1)2 +1 +4 − +1 +2(ℓ+1) +E0 (One Double Root) +(ℓ−1)(ℓ−3) +2 +ℓ(ℓ − 1)2(ℓ2 − 1) +1 +2ℓ(ℓ2−1) − +1 +ℓ(ℓ−1)(ℓ2−1) +E1 (One Double Root) +(ℓ−1)(ℓ−3) +2 +ℓ(ℓ − 1)2 +1 +2ℓ − +1 +ℓ(ℓ−1) +Table 2. Number of conjugacy classes and centralizer sizes for each conjugacy class +type in [Shi82]. +Proof of Proposition 5.1. Part (i) is simply the entry in Table 2 in the last column corresponding +to the “K0 (Irreducible)” type. +We now establish part (ii). As indicated in the discussion above Table 2, the only conjugacy +classes of matrices in GSp4(Fℓ) whose characteristic polynomials have some linear factors of odd +multiplicity are those of the types H0,J0,E0,E1. However, for part (ii) since we are only interested +in matrices M also having non-zero trace, it is insufficient to simply sum over the rightmost entries +in the bottom four rows of Table 2. From [Shi82, Table 2], we see that the elements of E0 and E1 +have trace c(a+1)2 +a +for some c,a ∈ F× +ℓ with a ≠ ±1. In particular, it follows that elements of types E0 +and E1 have nonzero traces. The elements of J0 have trace (c+a)(c+aℓ) +c +where c ∈ F× +ℓ and a ∈ Fℓ2 ∖Fℓ. +Therefore, the elements of J0 also have nonzero trace. +It remains to analyze which conjugacy classes of Type H0 have nonzero trace. Following [Shi82], +the +(ℓ−1)(ℓ−3)2 +8 +conjugacy classes of type H0 correspond to quadruples of distinct elements in +a1,a2,b1,b2 ∈ F× +ℓ satisfying a1b1 = a2b2 modulo the action of swapping any of a1 with b1, a2 with +b2, or a1,b1 with a2,b2. The eigenvalues of any matrix in the conjugacy class are a1, a2, b1, and b2. +Consequently the matrix has trace zero only if either a2 = −a1 and b2 = −b1 or b1 = −a2 and b2 = −a1. +This accounts for (ℓ−1)(ℓ−3) +4 +of the (ℓ−1)(ℓ−3)2 +8 +conjugacy classes of type H0, leaving (ℓ−1)(ℓ−3)(ℓ−5) +8 +conjugacy classes with non-zero trace. As a result, the probability that a matrix M ∈ GSp4(Fℓ) +chosen uniformly at random has non-zero trace and totally split characteristic polynomial is +(10) +(ℓ − 1)(ℓ − 3)(ℓ − 5) +8(ℓ − 1)3 += 1 +8 − +3 +4(ℓ − 1) + +1 +(ℓ − 1)2 . +To obtain part (ii), we add (10) to the entries in the rightmost column of the final three rows of +Table 2, getting +(1 +8 − +3 +4(ℓ − 1) + +1 +(ℓ − 1)2 ) + (1 +4 − +1 +2(ℓ + 1)) + ( +1 +2ℓ(ℓ2 − 1) − +1 +ℓ(ℓ − 1)(ℓ2 − 1)) + ( 1 +2ℓ − +1 +ℓ(ℓ − 1)) += 3 +8 − +3 +4(ℓ − 1) + +1 +2(ℓ − 1)2 . + +24 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +To prove (iii), we start by noting that for any pair (u,v), the cardinality of the set +{t4 − at3 + bt2 − amt + m2 ∶ a,b ∈ Fℓ,m ∈ F× +ℓ and (a2 +m , b +m) = (u,v)} +is at most ℓ − 1. +By [Cha97, Theorem 3.5], the number of matrices in GSp4(Fℓ) with a given +characteristic polynomial is at most (ℓ+3)8. Assuming ℓ ≠ 7, by combining these observations, and +noting that ∣Cℓ,720 ∪ Cℓ,1920∣ ≤ 14, we obtain the bound +γℓ ≥ 1 − 14(ℓ − 1)(ℓ + 3)8 +∣GSp4(Fℓ)∣ +. +For ℓ > 17, this implies the claimed bound. For 3 ≤ ℓ ≤ 17, we directly check the claim using +Magma. +□ +Lemma 5.2. Let C/Q be a typical genus 2 curve with Jacobian A and suppose ℓ is an odd prime +such that ρA,ℓ is surjective. For any ϵ > 0, there exists an effective constant B0 (with B0 > ℓNA) +such that for any B > B0 and each δ ∈ {α,β,γ}, we have +∣∣{p prime ∶ B ≤ p ≤ 2B and ρA,ℓ(Frobp) ∈ Cδ}∣ +∣{p prime ∶ B ≤ p ≤ 2B}∣ +− δℓ∣ < ϵ. +Proof. Let G = Gal(Q(A[ℓ])/Q) and S ⊆ G be any subset that is closed under conjugation. By +taking B to be sufficiently large, we have that B > ℓNA and can make +∣∣{p prime ∶ B ≤ p ≤ 2B and Frobp ∈ S}∣ +∣{p prime ∶ B ≤ p ≤ 2B}∣ +− ∣S∣ +∣G∣∣ +arbitrarily small by (3). +Moreover, the previous statement can be made effective by using an +effective version of the Chebotarev density theorem. The result then follows because each of the +sets Cα, Cβ, and Cγ is closed under conjugation. +□ +For positive integers n and B > ℓNA, let P(B,n) be the probability that n primes p1,...,pn +(possibly non-distinct) chosen uniformly at random in the interval [B,2B] have the property that +ρA,ℓ(Frobpi) /∈ Cα for each i +or +ρA,ℓ(Frobpi) /∈ Cβ for each i +or +ρA,ℓ(Frobpi) /∈ Cγ for each i. +Corollary 5.3. Suppose C and ℓ are as in Lemma 5.2 and let n be a positive integer. For any +ϵ > 0, there exists an effective constant B0 (with B0 > ℓNA) such that for all B > B0, we have +P(B,n) < (1 − αℓ)n + (1 − βℓ)n + (1 − γℓ)n + ϵ. +Proof. For δ ∈ {α,β,γ}, let Xδ be the event that none of the ρA,ℓ(Frobpi) are contained in Cδ. We +then have +P(Xα ∪ Xβ ∪ Xγ) ≤ P(Xα) + P(Xβ) + P(Xγ) +The result then follows by Lemma 5.2, which shows that there exists a B0 such that the probabilities +of Xα, Xβ, and Xγ can be made arbitrarily close to (1−αℓ)n, (1−βℓ)n, and (1−γℓ)n respectively. +□ +Proof of Theorem 1.4. The claim made by Theorem 1.4 is that P(B,n) < 3⋅( 9 +10) +n for B sufficiently +large. By Proposition 5.1, we have 1 − αℓ ≤ 4 +5, 1 − βℓ ≤ 7 +8, and 1 − γℓ ≤ 9 +10 for all ℓ odd. The result +then follows from Corollary 5.3 because (4 +5) +n + (7 +8) +n + ( 9 +10) +n < 3 ⋅ ( 9 +10) +n. +□ +6. Results of computation and interesting examples +We report on the results of running our algorithm on a dataset of 1,823,592 typical genus 2 +curves with conductor bounded by 220 that are part of a new dataset of approximately 5 million +curves currently being prepared for addition into the LMFDB. Running our algorithm on all of +these curves in parallel took about 45 hours on MIT’s Lovelace computer (see the Introduction for +the hardware specification of this machine). + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +25 +We first show in Table 3 how many of these curves were nonsurjective at particular primes, +indicating also if this can be explained by the existence of a rational torsion point of that prime +order. We found 31 as the largest nonsurjective prime, which occurred for the curve +(11) +y2 + (x + 1)y = x5 + 23x4 − 48x3 + 85x2 − 69x + 45 +of conductor 72 ⋅ 312 and discriminant 72 ⋅ 319 (the prime 2 was also nonsurjective here). +The +Jacobian of this curve does not admit a nontrivial rational 31-torsion point, so unlike many other +instances of nonsurjective primes we observed, this one cannot be explained by the presence of +rational torsion. One could ask if it might be explained by the existence of a Q-rational 31-isogeny +(as suggested by Algorithm 3.1, since 31 is returned by Algorithm 3.6). This seems to be the case +- see forthcoming work of van Bommel, Chidambaram, Costa, and Kieffer [vBCCK22] where the +isogeny class of this curve (among others) is computed. +nonsurj. prime +No. of curves w/ torsion +No. of curves w/o torsion +Example curve +2 +1,100,706 +462,616 +464.a.464.1 +3 +79,759 +98,750 +277.a.277.2 +5 +12,040 +10,809 +16108.b.64432.1 +7 +1,966 +2,213 +295.a.295.2 +11 +167 +210 +4288.b.548864.1 +13 +108 +310 +439587.d.439587.1 +17 +22 +61 +1996.b.510976.1 +19 +10 +20 +1468.6012928 +23 +2 +8 +1696.1736704 +29 +1 +5 +976.999424 +31 +0 +1 +47089.1295541485872879 +Table 3. Nonsurjective primes in the dataset, and whether they are explained by +torsion, with examples from the LMFDB dataset if available, else a string of the +form “conductor.discrimnant”. +We also observed (see Table 4) that the vast majority of curves had less than 3 nonsurjective +primes. +No. of nonsurj. primes +No. of curves +Example curve +Nonsurj. primes of example +0 +211,620 +743.a.743.1 +– +1 +1,455,473 +1923.a.1923.1 +5 (torsion) +2 +155,186 +976.a.999424.1 +2, 29(torsion) +3 +1,313 +15876.a.15876.1 +2, 3, 5 +Table 4. Frequency count of nonsurjective primes in the dataset, with examples +from the LMFDB dataset. +Instructions for obtaining the entire results file may be found in the README.md file of the +repository. +Remark 17. It would be interesting to know if there is a uniform upper bound on the largest prime +ℓ that could occur as a nonsurjective prime for the Jacobian of a genus 2 curve defined over Q, + +26 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +analogous to the conjectural bound of 37 for the largest nonsurjective prime for elliptic curves +defined over Q (see e.g. [BPR13, Introduction]). As the example of (11) shows, this bound - if it +exists - would have to be at least 31. +We conclude with a few examples that illustrate where Algorithm 3.1 fails when the abelian +surface has extra (geometric) endomorphisms. +Example 6.1. The Jacobian A of the genus 2 curve 3125.a.3125.1 on the LMFDB given by y2+y = +x5 has End(AQ) = Z but End(AQ) = Z[ζ5]. Let φ be the Dirichlet character of modulus 5 defined +by the Legendre symbol +φ∶(Z/5Z)× → {±1}, +2 ↦ −1. +In this case, Algorithm 3.13 fails to find an auxilliary prime p < 1000 for which ap ≠ 0 and φ(p) = −1. +This is consistent with the endomorphism calculation, since the trace of ρA,ℓ(Frobp) is 0 for all +primes p that do not split completely in Q(ζp) and any inert prime in Q( +√ +5) automatically does +not split completely in Q(ζ5). +Example 6.2. The modular curve X1(13) (169.a.169.1) has genus 2 and its Jacobian J1(13) has +CM by Z[ζ3] over Q. As in [MT74, Claim 2, page 45], for any prime ℓ that splits as ππ in Q(ζ3), the +representation J1(13)[ℓ] splits as a direct sum Vπ ⊕Vπ of two 2-dimensional subrepresentations that +are dual to each other. (A similar statement holds for J1(13)[ℓ]⊗Fℓ Fℓ, and so this representation is +never absolutely irreducible.) As expected, Algorithm 3.6 fails to find an auxiliary prime p < 1000 +for which Rp is nonzero. +Example 6.3. The first (ordered by conductor) curve whose Jacobian J admits real multiplication +over Q is the curve 529.a.529.1; indeed, this Jacobian is isogenous to the Jacobian of the modular +curve X0(23). Since there is a single Galois orbit of newforms - call it f - of level Γ0(23) and weight +2, we have that J is isogenous to the abelian variety Af associated to f, and thus we expect the +integer Mself-dual output by Algorithm 3.10 to be zero for any auxiliary prime, which is indeed the +case. +References +[AdRK13] +Sara Arias-de Reyna and Christian Kappen. Abelian varieties over number fields, tame ramification and +big Galois image. Math. Res. Lett., 20(1):1–17, 2013. +[AK19] +Jeoung-Hwan Ahn and Soun-Hi Kwon. An explicit upper bound for the least prime ideal in the Cheb- +otarev density theorem. Ann. Inst. Fourier (Grenoble), 69(3):1411–1458, 2019. +[Apo76] +Tom M. Apostol. Introduction to analytic number theory. Undergraduate Texts in Mathematics. 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On the surjectivity of mod ℓ representations associated to elliptic curves, 2015. +arXiv:1508.07661. + +COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 +29 +Appendix A. Exceptional maximal subgroups of GSp4(Fℓ) +ℓ +type +choices +generators +ℓ ≡ 5 (mod 8) +G1920 +b2 = −1 in Fℓ +⎛ +⎜⎜⎜ +⎝ +1 +0 +0 +−1 +0 +1 +−1 +0 +0 +1 +1 +0 +1 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +1 +0 +0 +b +0 +1 +b +0 +0 +b +1 +0 +b +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +1 +0 +0 +−1 +0 +1 +1 +0 +0 +−1 +1 +0 +1 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +1 +0 +1 +0 +0 +1 +0 +1 +−1 +0 +1 +0 +0 +−1 +0 +1 +⎞ +⎟⎟⎟ +⎠ +ℓ ≡ 3 (mod 8) +G1920 +b2 = −2 in Fℓ +⎛ +⎜⎜⎜ +⎝ +1 +0 +0 +−1 +0 +1 +−1 +0 +0 +1 +1 +0 +1 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +0 +0 +0 +b +0 +0 +b +0 +0 +b +2 +0 +b +0 +0 +2 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +1 +0 +0 +−1 +0 +1 +1 +0 +0 +−1 +1 +0 +1 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +1 +0 +1 +0 +0 +1 +0 +1 +−1 +0 +1 +0 +0 +−1 +0 +1 +⎞ +⎟⎟⎟ +⎠ +ℓ ≡ 7 (mod 12) +G720 +a2 + a + 1 = 0 in Fℓ +⎛ +⎜⎜⎜ +⎝ +a +0 +0 +0 +0 +a +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +a +0 +0 +0 +0 +1 +0 +0 +0 +0 +a +0 +0 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +a +0 +−a − 1 +a + 1 +0 +a +−a − 1 +−a − 1 +−a − 1 +−a − 1 +−1 +0 +a + 1 +−a − 1 +0 +−1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +0 +−1 +0 +0 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +1 +0 +⎞ +⎟⎟⎟ +⎠ +ℓ ≡ 5 (mod 12) +G720 +b2 = −1 in Fℓ +⎛ +⎜⎜⎜ +⎝ +−1 +0 +0 +−1 +0 +−1 +−1 +0 +0 +1 +0 +0 +1 +0 +0 +0 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +0 +0 +0 +1 +0 +−1 +−1 +0 +0 +1 +0 +0 +−1 +0 +0 +−1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +−b − 1 +b +2b +−2b + 1 +b +b − 1 +2b + 1 +2b +b +b − 1 +−b − 2 +−b +−b − 1 +b +−b +b − 2 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +0 +−b +−2b +0 +b +0 +0 +2b +−2b +0 +0 +−b +0 +2b +b +0 +⎞ +⎟⎟⎟ +⎠ +ℓ = 7 +G5040 +a = 2 satisfies +a2 + a + 1 = 0 +⎛ +⎜⎜⎜ +⎝ +2 +0 +0 +0 +0 +2 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +2 +0 +0 +0 +0 +1 +0 +0 +0 +0 +2 +0 +0 +0 +0 +1 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +6 +0 +5 +2 +0 +6 +5 +5 +5 +5 +4 +0 +2 +5 +0 +4 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +0 +6 +0 +0 +1 +0 +0 +0 +0 +0 +0 +6 +0 +0 +1 +0 +⎞ +⎟⎟⎟ +⎠ +, +⎛ +⎜⎜⎜ +⎝ +4 +6 +0 +0 +6 +6 +0 +0 +0 +0 +4 +1 +0 +0 +1 +6 +⎞ +⎟⎟⎟ +⎠ +Table 5. Explicit generators for each exceptional maximal subgroup in GSp4(Fℓ) +(up to conjugacy). The matrices described in Table 5 depend on an auxiliary choice +of a parameter denoted either a and b in each case. In each row, any one choice of +the corresponding a and b satisfying the equations described in the table suffices. + +30 +BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT +Barinder S. Banwait, Department of Mathematics & Statistics, Boston University, Boston, MA +Email address: barinder@bu.edu +URL: https://barinderbanwait.github.io/ +Armand Brumer, Department of Mathematics, Fordham University, New York, NY +Email address: brumer@fordham.edu +Hyun Jong Kim, Department of Mathematics, University of Wisconsin-Madison, Madison, WI +Email address: hyunjong.kim@math.wisc.edu +URL: https://sites.google.com/wisc.edu/hyunjongkim +Zev Klagsbrun, Center for Communications Research, San Diego, CA +Email address: zdklags@ccr-lajolla.org +Jacob Mayle, Department of Mathematics, Wake Forest University, Winston-Salem, NC +Email address: maylej@wfu.edu +Padmavathi Srinivasan, ICERM, Providence, RI +Email address: padmavathi srinivasan@brown.edu +URL: https://padmask.github.io/ +Isabel Vogt, Department of Mathematics, Brown University, Providence, RI +Email address: ivogt.math@gmail.com +URL: https://www.math.brown.edu/ivogt/ + diff --git a/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt b/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a26446916897574f7ad931eba70f4df1d453a103 --- /dev/null +++ b/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt @@ -0,0 +1,1361 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf,len=1360 +page_content='COMPUTING NONSURJECTIVE PRIMES ASSOCIATED TO GALOIS REPRESENTATIONS OF GENUS 2 CURVES BARINDER S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' BANWAIT, ARMAND BRUMER, HYUN JONG KIM, ZEV KLAGSBRUN, JACOB MAYLE, PADMAVATHI SRINIVASAN, AND ISABEL VOGT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric en- domorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely many primes ℓ for which the Galois action on ℓ-torsion points of A is not maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Building on work of Dieulefait, we give a practical algorithm to compute this finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The key inputs are Mitchell’s classification of maximal subgroups of PSp4(Fℓ), sampling of the characteristic polyno- mials of Frobenius, and the Khare–Wintenberger modularity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The algorithm has been submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomor- phism ring in the LMFDB, and the results incorporated into the homepage of each such curve on a publicly-accessible branch of the LMFDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Introduction Let C/Q be a smooth, projective, geometrically integral curve (referred to hereafter as a nice curve) of genus 2, and let A be its Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We assume throughout that A admits no nontrivial geometric endomorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, we assume that End(AQ) = Z, and we refer to any abelian variety satisfying this property as typical1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We also say that a nice curve is typical if its Jacobian is typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let GQ ∶= Gal(Q/Q), let ℓ be a prime, and let A[ℓ] ∶= A(Q)[ℓ] denote the ℓ-torsion points of A(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ρA,ℓ ∶ GQ → Aut(A[ℓ]) denote the Galois representation on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By fixing a basis for A[ℓ], and observing that A[ℓ] admits a nondegenerate Galois-equivariant alternating bilinear form, namely the Weil pairing, we may identify the codomain of ρA,ℓ with the general symplectic group GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In a letter to Vign´eras [Ser00, Corollaire au Th´eor`eme 3], Serre proved an open image theorem for typical abelian varieties of dimensions 2 or 6, or of odd dimension, generalizing his celebrated open image theorem for elliptic curves [Ser72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, the set of nonsurjective primes ℓ for which the representation ρA,ℓ is not surjective — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', the set of primes ℓ for which ρA,ℓ(GQ) is contained in a proper subgroup of GSp4(Fℓ) — is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the elliptic curve case, Serre subsequently provided a conditional upper bound in terms of the conductor of E on this finite set [Ser81, Th´eor`eme 22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' this bound has since been made unconditional [Coj05, Kra95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' There are also algorithms to compute the finite set of nonsurjective primes [Zyw15], and practical implementations in Sage [CL12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Serre’s open image theorem for typical abelian surfaces was made explicit by Dieulefait [Die02] who described an algorithm that returns a finite set of primes containing the set of nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In a different direction Lombardo [Lom16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3] provided an upper bound on the nonsurjective primes involving the stable Faltings height of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 11F80 (primary), 11G10, 11Y16 (secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 1Abelian varieties with extra endomorphisms define a thin set (in the sense of Serre) in Ag and as such are not the typically arising case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='02222v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='NT] 5 Jan 2023 2 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT In this paper we develop Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1, which together allow for the exact determination of the nonsurjective primes for C, yielding our main result as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve whose Jacobian A has conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 produces a finite list PossiblyNonsurjectivePrimes(C) that provably contains all nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For a given bound B > 0, Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 produces a sublist LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) of PossiblyNonsurjectivePrimes(C) that contains all the nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If B is suffi- ciently large, then the elements of LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) are precisely the nonsurjec- tive primes of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The two common ingredients in Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 are Mitchell’s 1914 classification of maximal subgroups of PSp4(Fℓ) [Mit14] and sampling of characteristic polynomials of Frobenius elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Indeed, ρA,ℓ is nonsurjective precisely when its image is contained in one of the proper maximal subgroups of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The (integral) characteristic polynomial of Frobenius at a good prime p is computationally accessible since it is determined by counting points on C over Fpr for small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The reduction of this polynomial modulo ℓ gives the characteristic polynomial of the action of the Frobenius element on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By the Chebotarev density theorem, the images of the Frobenius elements for varying primes p equidistribute over the conjugacy classes of ρA,ℓ(GQ) and hence let us explore the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 makes use of the fact that if the image of ρA,ℓ is nonsurjective, then the character- istic polynomials of Frobenius at auxiliary primes p will be constrained modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using this idea, Dieulefait worked out the constraints imposed by each type of maximal subgroup for ρA,ℓ(GQ) to be contained in that subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Our Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 combines Dieulefait’s conditions, with some modest improvements, to produce a finite list PossiblyNonsurjectivePrimes(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 then weeds out the extraneous surjective primes from PossiblyNonsurjectivePrimes(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Equipped with the prime ℓ, the task here is try to generate enough different elements in the image to rule out containment in any proper maximal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The key input is a purely group-theoretic condition (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2) that guarantees that a subgroup is all of GSp4(Fℓ) if it contains par- ticular types of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This algorithm is probabilistic and depends on the choice of a parameter B which, if sufficiently large, provably establishes nonsurjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The parameter B is a cut-off for the number of Frobenius elements that we use to sample the conjugacy classes of ρA,ℓ(GQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As an illustration of the interplay between theory and practice, analyzing the “worst case” run time of each step in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 yields a new theoretical bound, conditional on the Generalized Riemann Hypothesis (GRH), on the product of all nonsurjective primes in terms of the conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve with conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming the Generalized Riemann Hypothesis (GRH), we have, for any ϵ > 0, ∏ ℓ nonsurjective ℓ ≪ exp(N1/2+ϵ), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' While we believe this bound to be far from asymptotically optimal, it is the first bound in the literature expressed in terms of the (effectively computable) conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Naturally one wants to find the sufficiently large value of B in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(2), which the next result gives, conditional on GRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve, B be a positive integer, and q be the largest prime in LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, the set LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) is precisely the set of nonsurjective primes of C, provided that B ≥ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 3 The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 involves an explicit Chebotarev bound due to Bach and Sorenson [BS96] that is dependent on GRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' An unconditional version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 can be given using an unconditional Chebotarev result (for instance [KW22]), though the bound for B will be exponential in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In addition, if we assume both GRH and the Artin Holomorphy Conjecture (AHC), then a version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 holds with the improved asymptotic bound B ≫ q11 log2(qNA), but without an explicit constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Unfortunately, the bound from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 is prohibitively large to use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By way of illustration, consider the smallest (with respect to conductor) typical genus 2 curve, which has a model y2 + (x3 + 1)y = x2 + x, and label 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 in the L-functions and modular forms database (LMFDB) [LMF22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The output of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 is the set {2,3,5,7,83}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Applying Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 with B = 100 rules out the prime 83, suggesting that 7 is the largest nonsurjective prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Subsequently applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 with q = 7 yields the value B = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='578 × 1023 for which LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) coincides with the set of nonsurjective primes associated with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' With this value of B, our implementation of the algorithm was still running after 24 hours, after which we terminated it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Even if the version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 that relies on AHC could be made explicit, the value of q11 log2(qNA) in this example is on the order of 1011, which would still be a daunting prospect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' To execute the combined algorithm on all typical genus 2 curves in the LMFDB - which at the time of writing constitutes 63,107 curves - we have decided to take a fixed value of B = 1000 in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The combined algorithm then takes about 4 hours on MIT’s Lovelace computer, a machine with 2 AMD EPYC 7713 2GHz processors, each with 64 cores, and a total of 2TB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The result of this computation of nonsurjective primes for these curves is available to view on the homepage of each curve in the LMFDB beta: https://beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='lmfdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='org In addition, the combined algorithm has been run on a much larger set of 1,823,592 curves provided to us by Andrew Sutherland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See Section 6 for the results of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 samples the characteristic polynomial of Frobenius Pp(t) for each prime p of good reduction for the curve up to a particular bound and applies Tests 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5 to Pp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming that ρA,ℓ is surjective, we expect that the outcome of these tests should be independent for sufficiently large primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve with Jacobian A and suppose ℓ is an odd prime such that ρA,ℓ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' There is an effective bound B0 such that for any B > B0, if we sample the characteristic polynomials of Frobenius Pp(t) for n primes p ∈ [B,2B] chosen uniformly and independently at random, the probability that none of these pass Tests 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5 is less than 3⋅( 9 10) n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In fact, for each prime ℓ satisfying the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4, there is an explicit constant cℓ ≤ 9 10 tending to 3 4 as ℓ → ∞ which may be computed using Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 such that bound of 3 ⋅ ( 9 10) n in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 can be replaced by 3 ⋅ cn ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The combined algorithm to probabilistically determine the nonsurjective primes of a nice genus 2 curve over Q has been implemented in Sage [The20], and it will appear in a future release of this software2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Until then, the implementation is available at the following repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='com/ivogt/abeliansurfaces The README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='md file contains detailed instructions on its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This repository also contains other scripts in both Sage and Magma [BCP97] useful for verifying some of the results of this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' any filenames used in the sequel will refer to the above repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2see https://trac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='sagemath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='org/ticket/30837 for the ticket tracking this integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 4 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT Outline of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 2, we begin by reviewing the properties of the characteristic polynomial of Frobenius with a view towards computational aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We also recall the classification of maximal subgroups of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 3, we explain Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 and establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, for each of the maximal subgroups of GSp4(Fℓ) listed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4, we generate a list of primes that provably contains all primes ℓ for which the mod ℓ image of Galois is contained in this maximal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 is also proved in this section (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 4, we first prove a group-theoretic criterion (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2) for a subgroup of GSp4(Fℓ) to equal GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then, for each ℓ in the finite list from Section 3, we ascertain whether the characteristic polynomials of the Frobenius elements sampled satisfy the group-theoretic criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(2) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 also follow from this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 5 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 concerning the probability of output error, assuming that Frobenius elements distribute in ρA,ℓ(GQ) as they would in a randomly chosen element of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Finally, in Section 6, we close with remarks concerning the execution of the algorithm on the large dataset of genus 2 curves mentioned above, and highlight some interesting examples that arose therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This work was started at a workshop held remotely ‘at’ the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, in May 2020, and was supported by a grant from the Simons Foundation (546235) for the collaboration ‘Arithmetic Geometry, Number Theory, and Computation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It has also been supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' DMS-1929284 while the authors were in residence at ICERM during a Collaborate@ICERM project held in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We are grateful to Noam Elkies for providing interesting examples of genus 2 curves in the literature, Davide Lombardo for helpful discussions related to computing geometric endomorphism rings, and to Andrew Sutherland for providing a dataset of Hecke characteristic polynomials that were used for executing our algorithm on all typical genus 2 curves in the LMFDB, as well as making available the larger dataset of approximately 2 million curves that we ran our algorithm on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be an abelian variety of dimension g defined over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By conductor we mean the Artin conductor N = NA of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We write Nsq for the largest integer such that N2 sq ∣ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ℓ be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We write TℓA for the ℓ-adic Tate module of A: TℓA ≃ lim ←� n A[ℓn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This is a free Zℓ-module of rank 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For each prime p, we write Frobp ∈ Gal(Q/Q) for an absolute Frobenius element associated to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By a good prime p for an abelian variety A, we mean a prime p for which A has good reduction, or equivalently p ∤ NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If p is a good prime for A, then the trace ap of the action of Frobp on TℓA is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 for a discussion of the characteristic polynomial of Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By a typical abelian variety A, we mean an abelian variety with geometric endomorphism ring Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A typical genus 2 curve is a nice curve whose Jacobian is a typical abelian surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V be a 4-dimensional vector space over Fℓ endowed with a nondegenerate skew-symmetric bilinear form ⟨⋅,⋅⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A subspace W ⊆ V is called isotropic (for ⟨⋅,⋅⟩) if ⟨w1,w2⟩ = 0 for all w1,w2 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A subspace W ⊆ V is called nondegenerate (for ⟨⋅,⋅⟩) if ⟨⋅,⋅⟩ restricts to a nondegenerate form on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The general symplectic group of (V,⟨⋅,⋅⟩) is defined as GSp(V,⟨⋅,⋅⟩) ∶= {M ∈ GL(V ) ∶ ∃ mult(M) ∈ F× ℓ ∶ ⟨Mv,Mw⟩ = mult(M)⟨v,w⟩ ∀ v,w ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The map M ↦ mult(M) is a surjective homomorphism from GSp(V,⟨⋅,⋅⟩) to F× ℓ called the similitude character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' its kernel is the symplectic group, denoted Sp(V,⟨⋅,⋅⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Usually the bilinear form is understood from the context, in which case one drops ⟨⋅,⋅⟩ from the notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' moreover, for our COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 5 purposes, we will have fixed a basis for V , one in which the bilinear form is represented by the nonsingular skew-symmetric matrix J ∶= ( 0 I2 −I2 0 ), where I2 is the 2 × 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By a subquotient W of a Galois module U, we mean a Galois module W that admits a surjection U ′ ↠ W from a subrepresentation U ′ of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since we are chiefly concerned with computing the sets LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) and PossiblyNonsurjectivePrimes(C) for a fixed curve C, we will henceforth, for ease of notation, drop the C from the notation for these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Integral characteristic polynomial of Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The theoretical result underlying the whole approach is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 (Weil, see [ST68, Theorem 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be an abelian variety of dimension g defined over Q and let p be a prime of good reduction for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then there exists a monic integral polynomial Pp(t) ∈ Z[t] of degree 2g with constant coefficient pg such that for any ℓ ≠ p, the polynomial Pp(t) modulo ℓ is the characteristic polynomial of the action of Frobp on TℓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Furthermore, every root of Pp(t) has complex absolute value p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The polynomials Pp(t) are computationally accessible by counting points on C over Fpr r = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See [Poo17, Chapter 7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In fact, Pp(t) can be accessed via the frobenius_ polynomial command in Sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, we denote the trace of Frobenius by ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By the Grothendieck-Lefschetz trace formula, if A = JacX, p is a prime of good reduction for X, and λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=',λ2g are the roots of Pp(t), then #X(Fpr) = pr + 1 − 2g ∑ i=1 λr i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The Weil pairing and consequences on the characteristic polynomial of Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The nondegenerate Weil pairing gives an isomorphism (of Galois modules): (1) TℓA ≃ (TℓA)∨ ⊗Zℓ Zℓ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The Galois character acting on Zℓ(1) is the ℓ-adic cyclotomic character, which we denote by cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The integral characteristic polynomial for the action of Frobp on Zℓ(1) is simply t−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The integral characteristic polynomial for the action of Frobp on (TℓA)∨ is the reversed polynomial P ∨ p (t) = Pp(1/t) ⋅ t2g/pg whose roots are the inverses of the roots of Pp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now record a few easily verifiable consequences of the nondegeneracy of the Weil pairing when dim(A) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (i) The roots of Pp(t) come in pairs that multiply out to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, Pp(t) has no root with multiplicity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (ii) Pp(t) = t4 − apt3 + bpt2 − papt + p2 for some ap,bp ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (iii) If the trace of an element of GSp4(Fℓ) is 0 mod ℓ, then its characteristic polynomial is re- ducible modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, this applies to Pp(t) when ap ≡ 0 (mod ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (iv) If A[ℓ] is a reducible GQ-module, then Pp(t) is reducible modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Parts (i) and (ii) are immediate from the fact that the non-degenerate Weil pairing allows us to pair up the four roots of Pp(t) into two pairs that each multiply out to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 6 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT For part (iii), suppose that M ∈ GSp4(Fℓ) has tr(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then the characteristic polynomial PM(t) of M is of the form t4 +bt2 +c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When the discriminant of PM is 0 modulo ℓ, the polynomial PM has repeated roots and is hence reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' So assume that the discriminant of PM is nonzero modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When ℓ ≠ 2, the result follows from [Car56, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When ℓ = 2, a direct computation shows that the characteristic polynomial of a trace 0 element of GSp4(F2) is either (t + 1)4 or (t2 + t + 1)2, which are both reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Part (iv) is immediate from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 since Pp(t) mod ℓ by definition is the characteristic polynomial for the action of Frobp on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Maximal subgroups of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Mitchell [Mit14] classified the maximal subgroups of PSp4(Fℓ) in 1914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This can be used to deduce the following classification of maximal subgroups of GSp4(Fℓ) with surjective similitude character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 (Mitchell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V be a 4-dimensional Fℓ-vector space endowed with a nondegener- ate skew-symmetric bilinear form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then any proper subgroup G of GSp(V,ω) with surjective similitude character is contained in one of the following types of maximal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Reducible maximal subgroups (a) Stabilizer of a 1-dimensional isotropic subspace for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) Stabilizer of a 2-dimensional isotropic subspace for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) Irreducible subgroups governed by a quadratic character Normalizer Gℓ of the group Mℓ that preserves each summand in a direct sum decomposition V1 ⊕ V2 of V , where V1 and V2 are jointly defined over Fℓ and either: (a) both nondegenerate for ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' or (b) both isotropic for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, Mℓ is an index 2 subgroup of Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Stabilizer of a twisted cubic GL(W) acting on Sym3 W ≃ V , where W is a 2-dimensional Fℓ-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (4) Exceptional subgroups See Table A for explicit generators for the groups described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (a) When ℓ ≡ ±3 (mod 8): a group whose image G1920 in PGSp(V,ω) has order 1920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) When ℓ ≡ ±5 (mod 12) and ℓ ≠ 7: a group whose image G720 in PGSp(V,ω) has order 720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (c) When ℓ = 7: a group whose image G5040 in PGSp(V,ω) has order 5040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We have chosen to label the maximal subgroups in the classification using invariant subspaces for the symplectic pairing ω on V , following the more modern account due to Aschbacher (see [Lom16, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' for a more comprehensive treatment see [KL90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For the convenience of the reader, we record the correspondence between Mitchell’s original labels and ours below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Mitchell’s label Label in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 Group having an invariant point and plane 1a Group having an invariant parabolic congruence 1b Group having an invariant hyperbolic or elliptic congruence 2a Group having an invariant quadric 2b Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Dictionary between maximal subgroup labels in [Die02]/[Mit14] and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The maximal subgroups in (1) are the analogues of the Borel subgroup of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The maximal subgroups in (2) when the two subspaces V,V ′ in the direct sum decomposition are individually defined over Fℓ are the analogues of normalizers of the split Cartan subgroup of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When the two subspaces V,V ′ are not individually defined over Fℓ instead, the maximal subgroups in (2) are analogues of the normalizers of the non-split Cartan subgroups of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 7 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We briefly explain why the action of GL2(Fℓ) on Sym3(F2 ℓ) preserves a nondegenerate symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It suffices to show that the restriction to SL2(Fℓ) fixes a vector in ⋀2 Sym3(F2 ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This follows by character theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If W is the standard 2-dimensional representation of SL2, then we have ⋀2(Sym3 W) ≃ Sym4 W ⊕ 1 as representations of SL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' One can extract explicit generators of the exceptional maximal subgroups from Mitchell’s original work3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Indeed [Mit14, the proof of Theorem 8, page 390] gives four explicit matrices that generate a G1920 (which is unique up to conjugacy in PGSp4(Fℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Mitchell’s description of the other exceptional groups is in terms of certain projective linear transformations called skew perspec- tivities attached to a direct sum decomposition V = V1 ⊕ V2 into 2-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A skew perspectivity of order n with axes V1 and V2 is the projective linear transformation that scales V1 by a primitive nth root of unity and fixes V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This proof also gives the axes of the skew perspectivities of order 2 and 3 that generate the remaining exceptional groups [Mit14, pages 390-391].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Table 5 lists generators of (one representative of the conjugacy class of) each of the exceptional maximal subgroup extracted from Mitchell’s descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the file exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='m publicly available with our code, we verify that Magma’s list of conjugacy classes of maximal subgroups of GSp4(Fℓ) agree with those described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 for 3 ≤ ℓ ≤ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The classification of exceptional maximal subgroups of PSp4(Fℓ) is more subtle than that of PGSp4(Fℓ), because of the constraint on the similitude character of matrices in PSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' While the similitude character is not well-defined on PGSp4(Fℓ) (multiplication by a scalar c ∈ F× ℓ scales the similitude character by c2) it is well-defined modulo squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The group PSp4(Fℓ) is the kernel of this natural map: 1 → PSp4(Fℓ) → PGSp4(Fℓ) mult ��→ F× ℓ /(F× ℓ )2 ≃ {±1} → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' An exceptional subgroup of PGSp4(Fℓ) gives rise to an exceptional subgroup of PSp4(Fℓ) of either the same size or half the size depending on the image of mult restricted to that subgroup, which in turn depends on the congruence class of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For this reason, the maximal exceptional subgroups of PSp4(Fℓ) in Mitchell’s original classification (also recalled in Dieulefait [Die02, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1]) can have order 1920 or 960 and 720 or 360 depending on the congruence class of ℓ, and 2520 (for ℓ = 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Such an exceptional subgroup gives rise to a maximal exceptional subgroup of PGSp4(Fℓ) only when mult is surjective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', its intersection with PSp4(Fℓ) is index 2), which explains the restricted congruence classes of ℓ for which they arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now record a lemma that directly follows from the structure of maximal subgroups described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This lemma will be used in Section 4 to devise a criterion for a subgroup of GSp4(Fℓ) to be the entire group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For an element T in GSp4(Fℓ), let tr(T), mid(T), mult(T) denote the trace of T, the middle coefficient of the characteristic polynomial of T, and the similitude character applied to T respectively4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For a scalar λ, we have tr(λT) = λtr(T), mid(λT) = λ2 mid(T), mult(λT) = λ2 mult(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence the quantities tr(T)2/mult(T) and mid(T)/mult(T) are well-defined on PGSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For ℓ > 2 and ∗ ∈ {720,1920,5040}, define (2) Cℓ,∗ ∶= {( tr(T)2 mult(T), mid(T) mult(T)) ∣ T ∈ an exceptional subgroup of GSp4(Fℓ) of projective order ∗} Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) In cases 2a and 2b of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3: 3Mitchell’s notation for PGSp4(Fℓ) is Aν(ℓ) and for PSp4(Fℓ) is A1(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 4Explicitly, the characteristic polynomial of T is therefore t4 − tr(T)t3 + mid(T)t2 − mult(T) tr(T)t + mult(T)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 8 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT (a) every element in Gℓ ∖ Mℓ has trace 0, and, (b) the group Mℓ stabilizes a non-trivial linear subspace of F 4 ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) Every element that is contained in a maximal subgroup corresponding to the stabilizer of a twisted cubic has a reducible characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) For ∗ ∈ {1920,720}, the set Cℓ,∗ defined in (2) equals the reduction modulo ℓ of the elements of the set C∗ below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' C1920 = {(0,−2),(0,−1),(0,0),(0,1),(0,2),(1,1),(2,1),(2,2),(4,2),(4,3),(8,4),(16,6)} C720 = {(0,1),(0,0),(4,3),(1,1),(16,6),(0,2),(1,0),(3,2),(0,−2)} We also have C7,5040 = {(0,0),(0,1),(0,2),(0,5),(0,6),(1,0),(1,1),(2,6),(3,2),(4,3),(5,3),(6,3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) In cases 2a and 2b of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, since any element of the normalizer Gℓ that is not in Mℓ switches elements in the two subspaces V1 and V2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' maps elements in the subspace V1 in the decomposition V1 ⊕ V2 to elements in V2 and vice-versa), it follows that any element in Gℓ ∖ Mℓ has trace zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) The conjugacy class of maximal subgroups corresponding to the stabilizer of a twisted cubic comes from the embedding GL2(Fℓ) ι�→ GSp4(Fℓ) induced by the natural action of GL2(Fℓ) on the space of monomials of degree 3 in 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If M is a matrix in GL2(Fℓ) with eigenvalues λ,µ (possibly repeated), then the eigenvalues of ι(M) are λ3,µ3,λ2µ,λµ2 and hence the characteristic polynomial of ι(M) factors as (T 2 −(λ3 +µ3)T +λ3µ3)(T 2 −(λ2µ+ λµ2)T + λ3µ3) over Fℓ which is reducible over Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) This follows from the description of the maximal subgroups given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Each case (except G5040 that only occurs for ℓ = 7) depends on a choice of a root of a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the file exceptional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='sage, we generate the corresponding finite subgroups over the appropriate quadratic number field to compute C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It follows that the corresponding values for the subgroup G∗ in GSp4(Fℓ) can be obtained by reducing these values modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the group G5040 only appears for ℓ = 7, we directly compute the set C7,5040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4(3) is the analogue of the condition [Ser72, Proposition 19 (iii)] used to rule out exceptional maximal subgroups of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We end this subsection by including the following lemma, to further highlight the similarities between the above classification of maximal subgroups of GSp4(Fℓ) and the more familiar classi- fication of maximal subgroups of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This lemma is not used elsewhere in the article and is thus for expositional purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are indi- vidually defined over Fℓ is isomorphic to {(m1,m2) ∈ GL2(Fℓ)2 ∣ det(m1) = det(m2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ2 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are individually defined over Fℓ is isomorphic to {(m1,m2) ∈ GL2(Fℓ)2 ∣ mT 1 m2 = λI, for some λ ∈ F∗ ℓ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ(ℓ − 1)2(ℓ2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 9 (3) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are not individually defined over Fℓ is isomorphic to {m ∈ GL2(Fℓ2) ∣ det(m) ∈ F∗ ℓ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ4 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (4) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are not indi- vidually defined over Fℓ is isomorphic to GU2(Fℓ2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', {m ∈ GL2(Fℓ2) ∣ mT ι(m) = λI, for some λ ∈ F∗ ℓ }, where ι denotes the natural extension of the Galois automorphism of Fℓ2/Fℓ to GL2(Fℓ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ(ℓ2 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a direct sum decomposition V1 ⊕ V2 of a vector space V over Fq, we get a natural embedding of Aut(V1) × Aut(V2) (≅ GL2(Fq)2) into Aut(V ) (≅ GL4(Fq)), whose image consists of automorphisms that preserve this direct sum decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will henceforth refer to elements of Aut(V1) × Aut(V2) as elements of Aut(V ) using this embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' To understand the subgroup Mℓ of GSp4(Fq) in cases (1) and (2) where the two subspaces in the direct sum decomposition are individually defined over Fq, we need to further impose the condition that the automorphisms in the image of the map Aut(V1) × Aut(V2) → Aut(V ) preserve the symplectic form ω on V up to a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In (1), without any loss of generality, the two nondegenerate subspaces V1 and V2 can be chosen to be orthogonal complements under the nondegenerate pairing ω, and so by Witt’s theorem, in a suitable basis for V1⊕V2 obtained by concatenating a basis of V1 and a basis of V2, the nondegenerate symplectic pairing ω has the following block-diagonal shape: B ∶= ⎡⎢⎢⎢⎢⎢⎢⎢⎣ 0 1 −1 0 0 1 −1 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which boils down to det(m1) = λ = det(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Similarly, in (2), without any loss of generality, by Witt’s theorem, in a suitable basis for V1 ⊕V2 obtained by concatenating a basis of the isotropic subspace V1 and a basis of the isotropic subspace V2, the nondegenerate symplectic pairing ω has the following block-diagonal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' B ∶= ⎡⎢⎢⎢⎢⎢⎢⎢⎣ 0 1 1 0 0 −1 −1 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which again boils down to mT 1 m2 = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If we have a subspace W defined over Fq2 but not defined over Fq, and we let W denote the conjugate subspace and further assume that W ⊕W gives a direct sum decomposition of V , then we get a natural embedding of Aut(W) (≅ GL2(Fq2)) into Aut(V ) (≅ GL4(Fq)) whose image consists of automorphisms that commute with the natural involution of V ⊗ Fq2 induced by the Galois automorphism of Fq2 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The proofs of cases (3) and (4) are analogous to the cases (1) and (2) respectively, by using the direct sum decomposition W ⊕W and letting m2 = ι(m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition that det(m1) = det(m2) in (1) becomes the condition det(m1) = det(m2) = detm1 = det(m1), or 10 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT equivalently, that det(m1) ∈ Fq in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Similarly, the condition that mT 1 m2 = λI in (2) becomes the condition that mT 1 ι(m1) = λI in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Image of inertia and (tame) fundamental characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Dieulefait [Die02] used Mitchell’s work described in the previous subsection to classify the maximal subgroups of GSp4(Fℓ) that could occur as the image of ρA,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This was achieved via an application of a fundamental result of Serre and Raynaud that strongly constrains the action of inertia at ℓ, and which we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Fix a prime ℓ > 3 that does not divide the conductor N of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let Iℓ be an inertia subgroup at ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ψn∶Iℓ → F× ℓn denote a (tame) fundamental character of level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The n Galois-conjugate fundamental characters ψn,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=',ψn,n of level n are given by ψn,i ∶= ψℓi n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Recall that the fundamental character of level 1 is simply the mod ℓ cyclotomic character cycℓ, and that the product of all fundamental characters of a given level is the cyclotomic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6 (Serre [Ser72], Raynaud [Ray74], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' [Die02][Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ℓ be a semistable prime for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V /Fℓ be an n-dimensional Jordan–H¨older factor of the Iℓ-module A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then V admits a 1-dimensional Fℓn-vector space structure such that ρA,ℓ∣Iℓ acts on V via the character ψd1 n,1⋯ψdn n,n with each di equal to either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' On the other hand, the following fundamental result of Grothendieck constrains the action of inertia at semistable primes p ≠ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7 (Grothendieck [GRR72, Expos´e IX, Prop 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be an abelian variety over a number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then A has semistable reduction at p ≠ ℓ if and only if the action of Ip ⊂ GK on TℓA is unipotent of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Combining these two results allows one fine control of the determinant of a subquotient of A[ℓ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' this will be used in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A/Q be an abelian surface, and let Xℓ be a Jordan–H¨older factor of the Fℓ[GQ]- module A[ℓ] ⊗ Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ is a semistable prime, then detXℓ ≃ ϵ ⋅ cycx ℓ for some character ϵ∶GQ → Fℓ that is unramified at ℓ and some 0 ≤ x ≤ dimXℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, ϵ120 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The first part follows immediately from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For the fact that ϵ120 = 1, every abelian surface attains semistable reduction over an extension K/Q with [K ∶ Q] dividing 120 by [LV14a, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2], and so this follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7 since there are no nontrivial unramified characters of GQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ We can now state Dieulefait’s classification of maximal subgroups of GSp4(Fℓ) that can occur as the image ρA,ℓ(GQ) for a semistable prime ℓ > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9 ([Die02]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be the Jacobian of a genus 2 curve defined over Q with Weil pairing ω on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ > 7 is a semistable prime, then ρA,ℓ(GQ) is either all of GSp(A[ℓ],ω) or it is contained in one of the maximal subgroups of Types (1) or (2) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See also [Lom16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='15] for an expanded exposition of why the image of GQ cannot be contained in maximal subgroup of Type (3) for a semistable prime ℓ > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' However, if ℓ is a prime of additive reduction, or if ℓ ≤ 7, then the image of GQ may also be contained in any of the four types of maximal subgroups described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Nevertheless, by [LV22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6], for any prime ℓ > 24, we have that the exponent of the projective image is bounded exp(PρA,ℓ) ≥ (ℓ−1)/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since exp(G1920) = 2exp(S6) = 120 and exp(G720) = exp(S5) = 60, the exceptional maximal subgroups cannot occur as ρA,ℓ(GQ) for ℓ > 1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A consequence of the Chebotarev density theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let K/Q be a finite Galois exten- sion with Galois group G = Gal(K/Q) and absolute discriminant dK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let S ⊆ G be a nonempty subset that is closed under conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By the Chebotarev density theorem, we know that (3) lim x→∞ ∣{p ≤ x ∶ p is unramified in K and Frobp ∈ S}∣ ∣{p ≤ x}∣ = ∣S∣ ∣G∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be the least prime such that p is unramified in K and Frobp ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' There are effective versions of the Chebotarev density theorem that give bounds on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The best known unconditional bounds are polynomial in dK [LMO79, AK19, KW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Under GRH, the best known bounds are polynomial in log dK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular Bach and Sorenson [BS96] showed that under GRH, (4) p ≤ (4log dK + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5[K ∶ Q] + 5)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The present goal is to give an effective version of the Chebotarev density theorem in the context of abelian surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will use a corollary of (4) that is noted in [MW21] which allows for the avoidance of a prescribed set of primes by taking a quadratic extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We do this because we will take K = Q(A[ℓ]), and p being unramified in K is not sufficient to imply that p is a prime of good reduction for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lastly, we will use that by [Ser81, Proposition 6], if K/Q is finite Galois, then (5) log dK ≤ ([K ∶ Q] − 1)log rad(dK) + [K ∶ Q]log([K ∶ Q]), where radn = ∏p∣n p denotes the radical of an integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A/Q be a typical principally polarized abelian surface with conductor NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let q be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let S ⊆ ρA,q(GQ) be a nonempty subset that is closed under conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be the least prime of good reduction for A such that p ≠ q and ρA,q(Frobp) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, we have p ≤ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let K = Q(A[q]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then K/Q is Galois and [K ∶ Q] ≤ ∣GSp4(Fq)∣ = q4(q4 − 1)(q2 − 1)(q − 1) ≤ q11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As raddK is the product of primes that ramify in Q(A[q]), the criterion of N´eron-Ogg-Shafarevich for abelian varieties [ST68, Theorem 1] implies that rad(dK) divides rad(qNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ˜K ∶= K(√m) where m ∶= rad(2NA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Note that the primes that ramify in ˜K are precisely 2, q, and the primes of bad reduction for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Thus rad(d ˜ K) = rad(2qNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover [ ˜K ∶ Q] ≤ 2q11 and by (5), log(d ˜ K) ≤ (2q11 − 1)log rad(2qNA) + 22q11 log(2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Applying [MW21, Corollary 6] to the field ˜K, we get that (under GRH) there exists a prime p satisfying the claimed bound, that does not divide m, and for which ρA,q(Frobp) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Finding a finite set containing all nonsurjective primes In this section we describe Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 referenced in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This algorithm produces a finite list PossiblyNonsurjectivePrimes that provably includes all nonsurjective primes ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We also prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since our goal is to produce a finite list (from which we will later remove extraneous primes) it is harmless to include the finitely many bad primes as well as 2,3,5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9, it suffices to find conditions on ℓ > 7 for which ρA,ℓ(GQ) could be contained in one of the maximal subgroups of type (1) and (2) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We first find primes ℓ for which ρA,ℓ has (geometrically) reducible image (and hence is contained in a maximal subgroup in case (1) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 or in a subgroup Mℓ in case (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' To treat the geometrically irreducible cases, we then make use of the observation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 1a that every element outside of an index 2 subgroup has trace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 12 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 curve C/Q with conductor N and Jacobian A, compute a finite list PossiblyNonsurjectivePrimes of primes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Initialize PossiblyNonsurjectivePrimes = [2,3,5,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) Add to PossiblyNonsurjectivePrimes all primes dividing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗ Fℓ could be reducible via Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (4) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗Fℓ could be irreducible but nonsurjective via Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (5) Return PossiblyNonsurjectivePrimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' At a very high-level, each of the subalgorithms of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 makes use of a set of auxiliary good primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We compute the integral characteristic polynomial of Frobenius Pp(t) and use it to constrain those ℓ ≠ p for which the image could have a particular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Even though robust methods to compute the conductor N of a genus 2 curve are not implemented at the time of writing, the odd-part Nodd of N can be computed via genus2red function of PARI and the genus2reduction module of SageMath, both based on an algorithm of Liu [Liu94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, [BK94, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2] bounds the 2-exponent of N above by 20 and hence N can be bounded above by 220Nodd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' While these algorithms can be run only with the bound 220Nodd, it will substantially increase the run-time of the limiting Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now explain each of these steps in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Good primes that are not geometrically irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this section we describe the conditions that ℓ must satisfy for the base-extension A[ℓ] ∶= A[ℓ] ⊗Fℓ Fℓ to be reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this case, the representation A[ℓ] is an extension (6) 0 → Xℓ → A[ℓ] → Yℓ → 0 of a (quotient) representation Yℓ by a (sub) representation Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Recall that Nsq denotes the largest square divisor of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ℓ be a prime of good reduction for A and suppose that A[ℓ] sits in sequence (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p ≠ ℓ be a good prime for A and let f denote the order of p in (Z/NsqZ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then there exists 0 ≤ x ≤ dimXℓ and 0 ≤ y ≤ dimYℓ such that Frobgcd(f,120) p acts on detXℓ by pgcd(f,120)x, respectively on detYℓ by pgcd(f,120)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since ℓ is a good prime and Xℓ is composed of Jordan–H¨older factors of A[ℓ], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8 constrains its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We have detXℓ = ϵcycx ℓ for some character ϵ∶GQ → Fℓ unramified at ℓ, and 0 ≤ x ≤ dimXℓ, and ϵ120 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence Frob120 p acts on detXℓ by cycℓ(Frobp)120x = p120x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In fact, we can do slightly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since detA[ℓ] ≃ cyc2 ℓ, we have detYℓ ≃ ϵ−1 cyc2−x ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the conductor is multiplicative in extensions, we conclude that cond(ϵ)2 ∣ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By class field theory, the character ϵ factors through (Z/cond(ϵ)Z)×, and hence through (Z/NsqZ)×, sending Frobp to p (mod Nsq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since pf ≡ 1 (mod Nsq), we have that ϵ(Frobp)gcd(f,120) = 1, and we see that Frobgcd(f,120) p acts on detXℓ by pgcd(f,120)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Exchanging the roles of Xℓ and Yℓ, we deduce the analogous statement for Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ This is often enough information to find all ℓ for which A[ℓ] has a nontrivial subquotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Namely, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1, every root of Pp(t) has complex absolute value p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Thus the gcd(f,120)-th power of each root has complex absolute value pgcd(f,120)/2, and hence is never integrally equal to 1 or pgcd(f,120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 guarantees that this equality must hold modulo ℓ for any good prime ℓ for which A[ℓ] is reducible with a 1-dimensional subquotient, we always get a nontrivial condition on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Some care must be taken to rule out ℓ for which A[ℓ] only has 2-dimensional subquotient(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Odd-dimensional subquotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be a good prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a polynomial P(t) and an integer f, write P (f)(t) for the polynomial whose roots are the fth powers of roots of P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Universal formulas for such polynomials in terms of the coefficients of P(t) are easy to compute, and are implemented in our code in the case where P is a degree 4 polynomial whose roots multiply in pairs to pα, and f ∣ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of p in (Z/NsqZ)× and write f′ = gcd(f,120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Compute an integer Modd as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Choose a nonempty finite set T of auxiliary good primes p ∤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each p, compute Rp ∶= P (f′) p (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Let Modd = gcdp∈T (pRp) over all auxiliary primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Modd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] has an odd-dimensional subrepresentation is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since A[ℓ] is 4-dimensional and has an odd-dimensional subrepresentation, it has a 1- dimensional subquotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For any p ∈ T , Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 shows that Frobf′ p acts on detXℓ by either pf′ or by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Thus, the action of Frobf′ p on A[ℓ] has an eigenvalue that is congruent to pf′ or 1 modulo ℓ, and so P (f′) p (t) has a root that is congruent to 1 or pf′ modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the roots of P (f′)(t) multiply in pairs to pf′, we have P (f′) p (pf′) = p2f′P (f′) p (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence ℓ divides p ⋅ P (f′) p (1) = pRp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1, we can give a theoretical bound on the “worst case” of this step of the algorithm using only one auxiliary prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Of course, taking the greatest common divisor over multiple auxiliary primes will likely remove extraneous factors, and in practice this step of the algorithm runs substantially faster than other steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if p is any good prime for A, then 0 ≠ ∣Modd∣ ≪ p240 where the implied constant is absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This follows from the fact that the coefficient of ti in P (f′) p (t) has magnitude on the order of p(2−i)f′ and f′ ≤ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Two-dimensional subquotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now assume that A[ℓ] is reducible, but does not have any odd-dimensional subquotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, it has an irreducible subrepresentation Xℓ of dimension 2, with irreducible quotient Yℓ of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If A[ℓ] is reducible but indecomposable, then Xℓ is the unique subrepresentation of A[ℓ] and Y ∨ ℓ ⊗ cycℓ is the unique subrepresentation of A[ℓ] ∨ ⊗ cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The isomorphism TℓA ≃ (TℓA)∨ ⊗ cycℓ from (1) yields an isomorphism A[ℓ] ≃ (A[ℓ])∨ ⊗ cycℓ and hence Xℓ ≃ Y ∨ ℓ ⊗ cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Otherwise, A[ℓ] ≃ Xℓ ⊕ Yℓ and so the nondegeneracy of the Weil pairing gives Xℓ ⊕ Yℓ ≃ (X∨ ℓ ⊗ cycℓ) ⊕ (Y ∨ ℓ ⊗ cycℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Therefore either: (a) Xℓ ≃ Y ∨ ℓ ⊗ cycℓ and Yℓ ≃ X∨ ℓ ⊗ cycℓ, or (b) Xℓ ≃ X∨ ℓ ⊗ cycℓ and Yℓ ≃ Y ∨ ℓ ⊗ cycℓ and A[ℓ] ≃ Xℓ ⊕ Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We call the first case related 2-dimensional subquotients and the second case self-dual 2-dimensional subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will see that the ideas of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 easily extend to treat the related subquotient case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' we will use the validity of Serre’s conjecture to treat the self-dual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the 14 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT case that A[ℓ] is decomposable, the above two cases correspond respectively to the index 2 subgroup Mℓ in cases (2a) (the isotropic case) and (2b) (the nondegenerate case) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Related two-dimensional subquotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be a good prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let Pp(t) ∶= t4−at3+bt2−pat+p2 be the characteristic polynomial of Frobp acting on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Suppose that α and β are the eigenvalues of Frobp acting on the subrepresentation Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then, since Xℓ ≃ Y ∨ ℓ ⊗ cycℓ, the eigenvalues of the action of Frobp on Yℓ are p/α and p/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The action of Frobp on detXℓ is therefore by a product of two of the roots of Pp(t) that do not multiply to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Note that there are four such pairs of roots of Pp(t) that do not multiply to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let Qp(t) be the quartic polynomial whose roots are the products of pairs of roots of Pp(t) that do not multiply to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By design, the roots of Qp(t) have complex absolute value p, but are not equal to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (It is elementary to work out that Qp(t) = t4 − (b − 2p)t3 + p(a2 − 2b + 2p)t2 − p2(b − 2p)t + p4 and is a quartic whose roots multiply in pairs to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=') Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of p in (Z/NsqZ)× and write f′ = gcd(f,120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Compute an integer Mrelated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Choose a finite set T of auxiliary good primes p ∤ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each p, compute the product Rp ∶= Q(f′) p (1)Q(f′) p (pf′) (3) Let Mrelated = gcdp∈T (pRp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Mrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] has related two-dimensional subquotients is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proceed similarly as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 — in particular, ℓ divides Q(f′) p (1), Q(f′) p (pf′), or Q(f′) p (p2f′) and hence ℓ divides pRp since Q(f′) p (p2f′) = p4f′Q(f′) p (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ A theoretical “worst case” analysis yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if q is the smallest surjective prime for A, then a good prime p for which Rp is nonzero is bounded by a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, p ≪ q22 log2(qN), where the implied constants are absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, for such a prime p, ∣Mrelated∣ ≪ p961 ≪ q21142 log1922(qN), where the implied constants are absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By Serre’s open image theorem for genus 2 curves, such a prime q exists, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10, the prime p can be chosen such that Rp is nonzero modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Finally, Mrelated ≤ pRp = pQ(f′)(1)Q(f′)(pf′) ≪ p8f′+1 ≪ p961, since the coefficient of ti in Q(f′)(t) has magnitude on the order of p(4−i)f′ and f′ ≤ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Self-dual two-dimensional subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this case, both subrepresentations Xℓ and Yℓ are absolutely irreducible 2-dimensional Galois representations with determinant the cyclotomic character cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It follows that the representations are odd (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', the determinant of complex con- jugation is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=') Therefore, by the Khare–Wintenberger theorem (formerly Serre’s conjecture on the modularity of mod-ℓ Galois representations) [Kha06, KW09a, KW09b], both Xℓ and Yℓ are modular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, for i = 1,2, there exist newforms fi ∈ Snew ki (Γ1(Ni),ϵi) such that Xℓ ≅ ρf1,ℓ and Yℓ ≅ ρf2,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Furthermore, by the multiplicativity of Artin conductors, we obtain the divisibility N1N2 ∣ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Both f1 and f2 have weight two and trivial Nebentypus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, k1 = k2 = 2, and ϵ1 = ϵ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6, we have that Xℓ∣Iℓ and Yℓ∣Iℓ must each be conjugate to either of the following subgroups of GL2(Fℓ): (1 ∗ 0 cycℓ ) or (ψ2 0 0 ψℓ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The assertion of weight 2 now follows from [Ser87, Proposition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (Alternatively, one may use Proposition 4 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', observing that Xℓ and Yℓ are finite and flat as group schemes over Zℓ because ℓ is a prime of good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=') From Section 1 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', the Nebentypus ϵi of fi satisfies, for all p ∤ ℓN, detXℓ(Frobp) = p ⋅ ϵi(p), where this equality is viewed inside F × ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The triviality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ We therefore have newforms fi ∈ Snew 2 (Γ0(Ni)) such that (7) A[ℓ] ≃ ρf1,ℓ ⊕ ρf2,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We may assume without loss of generality that N1 ≤ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p ∤ N be an auxiliary prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We obtain from equation (7) that the integral characteristic polynomial of Frobenius factors: Pp(t) ≡ (t2 − ap(f1)t + p)(t2 − ap(f2)t + p) mod ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' here we use the standard property that, for f a normalised eigenform with trivial Nebentypus, ρf,ℓ(Frobp) satisfies the polynomial equation t2 − ap(f)t + p for p ≠ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, we have Res(Pp(t),t2 − ap(f1)t + p) ≡ 0 mod ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This serves as the basis of the algorithm to find all primes ℓ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mself-dual as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Compute the set S of divisors d of N with d ≤ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each d ∈ S: (a) compute the Hecke L-polynomial Qd(t) ∶= ∏ f (t2 − ap(f)t + p), where the product is taken over the finitely many newforms in Snew 2 (Γ0(d));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) choose a finite set T of auxiliary primes p ∤ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (c) for each auxiliary prime p, compute the resultant Rp(d) ∶= Res(Pp(t),Qd(t));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 16 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT (d) Take the greatest common divisor M(d) ∶= gcd p∈T (pRp(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Let Mself-dual ∶= ∏d∈S M(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Mself-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] has self-dual two-dimensional subrepresenta- tions is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ is in T for any d ∈ S, then ℓ is in the output because Mself-dual is a multiple of M(d) which in turn is a multiple of any element of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Otherwise, as explained before Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10, there is some N1 ∈ S and some newform f1 ∈ Snew 2 (Γ0(N1)) such that Res(Pp(t),t2 − apf1t + p) ≡ 0 (mod ℓ) for every p ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, Rp(N1) ≡ 0 (mod ℓ), so ℓ divides M(N1) and Mself-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ We can again do a “worst case” theoretical analysis of this algorithm to conclude the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As this indicates, this is by far the limiting step of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if q is the smallest surjective prime for A, then a good prime p for which Rp(d) is nonzero is bounded by a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, p ≪ q22 log2(qN), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, for such a prime p, we have ∣Rp(d)∣ ≪ (2p1/2)8 dim Snew 2 (Γ0(d)) ≪ (4p)(d+1)/3, and so all together ∣Mself-dual∣ ≪ (4q)N1/2+ϵ, where the implied constants are absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8, we use Serre’s open image theorem and the Effective Chebotarev Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If Rp(d) is zero integrally, then in particular Rp(d) ≡ 0 (mod q) and Pp(t) is reducible modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since GSp4(Fq) contains elements that do not have reducible characteristic polynomial, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10 implies that such elements are the image of Frobp for p bounded as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The resultant Rp(d) is the product of the pairwise differences of the roots of Pp(t) and Qd(t), which all have complex absolute value p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence the pairwise differences have absolute value at most 2p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover dimSnew 2 (Γ0(d)) ≤ (d + 1)/12 by [Mar05, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since there are 8dimSnew 2 (Γ0(d)) such terms multiplied to give Rp(d), the bound for Rp(d) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since Mself-dual = ∏ d∣N d≤ √ N pRp(d), it suffices to bound ∑ d∣N d≤ √ N d + 4 3 ≤ ∑ d∣N d≤ √ N √ N + 4 3 ≤ σ0(N) √ N + 4 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since σ0(N) ≪ Nϵ by [Apo76, (31) on page 296], we obtain the claimed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The polynomial Qd(t) in step (2) of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10 is closely related to the charac- teristic polynomial Hd(t) of the Hecke operator Tp acting on the space S2(Γ0(d)), which may be computed via modular symbols computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' One may recover Qd(t) from Hd(t) by first homoge- nizing H with an auxiliary variable z (say) to obtain Hd(t,z), and setting t = 1+pz2 (an observation we made in conjunction with Joseph Wetherell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In our computation of nonsurjective primes for the database of genus 2 curves with conductor at most 220 (including those in the LMFDB), we only needed to use polynomials Qd(t) for level up to 210 (since step (1) of the Algorithm has a √ N term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We are grateful to Andrew Sutherland for providing us with a precomputed dataset for these levels resulting from the creation of an extensive database of modular forms going well beyond what was previously available [BBB+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 17 Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Our Sage implementation uses two auxiliary primes in Step 2(b) of the above algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Increasing the number of such primes yields smaller supersets at the expense of longer runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Good primes that are geometrically irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let φ be any quadratic Dirichlet char- acter φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Our goal in this subsection is to find all good primes ℓ governed by φ, by which we mean that tr(ρA,ℓ(Frobp)) ≡ ap ≡ 0 mod ℓ whenever φ(p) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will consider the set of all quadratic Dirichlet character φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using the struc- ture theorem for finite abelian groups and the fact that φ factors through (Z/NZ)×/((Z/NZ)×)2, this set has the structure of an F2-vector space of dimension d(N) ∶= ω(N) + ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ 0 ∶ v2(N) = 0 −1 ∶ v2(N) = 1 0 ∶ v2(N) = 2 1 ∶ v2(N) ≥ 3, where ω(m) denotes the number of prime factors of m and v2(m) is the 2-adic valuation of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, d(N) ≤ ω(N) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mquad as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Compute the set S of quadratic Dirichlet characters φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each φ ∈ S: (a) Choose a nonempty finite set T of “auxiliary” primes p ∤ N for which ap ≠ 0 and φ(p) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) Take the greatest common divisor Mφ ∶= gcd p∈T (pap), over all auxiliary primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Let Mquad ∶= ∏φ∈S Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Mquad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] is governed by a quadratic character is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Suppose that A[ℓ] is governed by the quadratic character φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then for every good prime p ≠ ℓ for which φ(p) = −1, the prime ℓ must divide the integral trace of Frobenius ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence ℓ divides Mφ and Mquad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if q is the smallest surjective prime for A, then a good prime p for which φ(p) = −1 and ap is nonzero is bounded by a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, p ≪ 22d(N)q22 log2(qN), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, we have ∏ φ∈S ∏ ℓ governed by φ ℓ ≪ (23d(N)q33 log3(qN))2−21−d(N) ≪ 26ω(N)q66 log6(qN), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We imitate the proof of [LV14b, Lemma 21] in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V be the d-dimensional F2-vector space of quadratic Dirichlet characters of modulus N (equivalently, quadratic Galois characters unramified outside of N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ρV ∶GK → V ∨ denote the representation sending Frobp to the linear functional φ ↦ φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the character for PGSp4(Fq)/PSp4(Fq) is the abelianization 18 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT of PρA,q, we conclude in the same way as [LV14b, Proof of Lemma 21] that for any α ∈ V ∨, there exists an Xα ∈ GSp4(Fq) with tr(Xα) ≠ 0 such that (α,Xα) is in the image of ρV × ρA,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Apply the effective Chebotarev density theorem to the Galois extension corresponding to ρV × ρA,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This has degree at most 2d(N)∣GSp4(Fq)∣ and is unramified outside of qN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Therefore, assum- ing GRH and combining (4) and (5), there exists a prime pα ≪ 22d(N)q22 log2(qN) for which (α,Xα) = (ρV (Frobpα),ρA,q(Frobpα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let φ be a character not in the kernel of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any exceptional prime ℓ governed by φ must divide pαapα, which is nonzero because it is nonzero modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This proves that the algorithm terminates, since every φ is not in the kernel of precisely half of all α ∈ V ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now bound the size of the product of all ℓ governed by a character in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ is governed by φ, then ℓ divides the quantity p∣ap∣ ≤ p3/2 ≪ 23d(N)q33 log3(qN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Taking the product over all nonzero α in V (of which there are 2d(N) − 1), each ℓ will show up half the time, so we obtain: ⎛ ⎜⎜⎜ ⎝ ∏ ℓ governed by φ ∈ S ℓ ⎞ ⎟⎟⎟ ⎠ 2d(N)−1 ≪ (23d(N)q33 log3(qN)) 2d(N)−1 , which implies the result by taking the (2d(N)−1)th root of both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Putting all of these pieces together, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ρA,ℓ is nonsurjective, ℓ > 7, and ℓ ∤ N, then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9 implies that ρA,ℓ(GQ) must be in one of the maximal subgroups of Type (1) or (2) listed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If it is contained in one of the reducible subgroups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' the subgroups of Type (1), then ρA,ℓ(GQ) (and, hence, ρA,ℓ(GQ) ⊗ Fℓ) is reducible, and so ℓ is added to PossiblyNonsurjectivePrimes in Step (3) by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ρA,ℓ(GQ) is contained in one of the index 2 subgroups Mℓ of an irreducible subgroup of Type (2) listed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, then again ℓ is added to PossiblyNonsurjectivePrimes in Step (3), since Mℓ ⊗ Fℓ is always reducible by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4(1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence we may assume that ρA,ℓ(GQ) is contained in one of the irreducible maximal subgroups Gℓ of Type (2) listed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, but not in the index 2 subgroup Mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The normalizer character GQ ρA,ℓ ��→ Gℓ → Gℓ/Mℓ = {±1} is nontrivial and unramified outside of N, and so it corresponds to a quadratic Dirichlet character φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4(1a) shows that tr(g) = 0 in Fℓ for any g ∈ Gℓ ∖ Mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Consequently, ℓ is governed by φ (in the language of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2), so ℓ is added to PossiblyNonsurjectivePrimes in Step (4) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Bounds on Serre’s open image theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this section we combine the theoretical worst case bounds in the Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13 to give a bound on the smallest surjective good prime q, and the product of all nonsurjective primes, thereby establishing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A/Q be a typical genus 2 Jacobian of conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, we have ∏ ℓ nonsurjective ℓ ≪ exp(N1/2+ϵ), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let q be the smallest surjective good prime for A, which is finite by Serre’s open image theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Multiplying the bounds in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='12, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='15 by the conductor N, the product of all nonsurjective primes is bounded by a function of q and N of the following shape (8) ∏ ℓ nonsurjective ℓ ≪ qN1/2+ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' On the other hand, since q is the smallest surjective prime by definition, the product of all primes less than q divides the product of all nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using [Ser81, Lemme 11], we have exp(q) ≪ ∏ ℓ 1 and vice versa. In fact, how much the fixation probability of the mutant type +increases as one increases r depends on the network structure. Some networks are amplifiers of selection; +on these networks, a single mutant has a higher probability of fixation than in the well-mixed population, +corresponding to the Moran process, at any r > 1 and a smaller fixation probability than in the Moran +process at any r < 1. Other networks are suppressors of selection; on these networks, a single mutant has a +lower fixation probability than in the Moran process at any r > 1 and a higher fixation probability than in +the Moran process at any r < 1. Under the so-called birth-death processes, which we focus on, most networks +are known to be amplifiers, at least when the initial mutant is located on a node selected uniformly at random +[20–22]. Research has discovered various classes of amplifiers [5, 23–27] while few for suppressors [28]. We +show that, contrary to these results, most hypergraphs are suppressors of natural selection even under the +birth-death process and uniform initial condition. We reach this conclusion by theoretical investigations for +hypergraphs with high symmetry and numerical simulations on empirical hypergraphs. The Python codes +for generating the numerical results in this article are available at GitHub [29]. +2. Models +We introduce two models of evolutionary dynamics on undirected hypergraphs. Let H be an undirected +hypergraph with node set V = {1, . . . , N}, where N is the number of nodes, and hyperedge set E. Each +e ∈ E, where e ⊂ V and e ̸= ∅, is a hyperedge, intuitively representing group interaction among the nodes +belonging to e. If each e ∈ E is a set containing exactly two nodes, H is a conventional undirected network. +We assume that H is connected in the sense that there is a hyperedge intersecting both W and V − W for +every non-empty proper subset W of V . +We define model 1 by the following discrete-time evolutionary dynamics on hypergraph H, which extends +the birth-death process for conventional networks and the Moran process in well-mixed populations. We +assume that there are two types of individuals, referred to as A and B, and that A and B have fitness r and +1, respectively. We refer to A as the mutant type and B as the resident type. In each time step, we select one +node for reproduction with the probability proportional to its fitness. We call this node the parent. Then, +the parent node selects one of the hyperedges to which it belongs, denoted by e, with the equal probability. +Finally, the parent converts the type (i.e., A or B) of all other nodes belonging to e into the parent’s type. +We repeat this process until all nodes in V have the same type. Once this unanimity configuration is reached, +which is the fixation, no node will change its type even if one further runs the evolutionary dynamics. We +examine the probability that every node eventually has type A, which we call the fixation probability of type +A. +Model 2 is the same as model 1 in that there are two types of individuals with fitness r and 1, that we +select a parent node in each time step with the probability proportional to its fitness, and that the parent +selects one of its hyperedges, e. Differently from the model 1, the parent node converts all the other nodes +belonging to e into the parent’s type if and only if the parent’s type is the majority in e, i.e., if more than half +of the nodes in e including the parent have the same type as the parent’s type. Model 2 is an evolutionary +dynamics variant of opinion formation models under the majority rule [9, 30, 31]. +When the network is the complete graph, which is a conventional undirected network and therefore a +hypergraph, model 1 is called the Moran process. For the Moran process, the fixation probability of type A +2 + +when there are initially i individuals of type A, denoted by xi, is given by [1] +xi = 1 − 1/ri +1 − 1/rN . +(2.1) +3. Results for synthetic hypergraphs +In this section, for three model hypergraphs that are mathematically convenient, we calculate the fixation +probability for mutant type A when there are initially i nodes of type A and N − i nodes of type B, for both +models 1 and 2. The fixation probability depends on the initial condition. We select the i mutant nodes from +the N nodes uniformly at random, which is called the uniform initialization [32]. +For any hypergraph of size N, at any time step, either type A or B inhabits each node. Therefore, there +are 2N states in total. The fixation probability for type A depends on each state, not just on i. To know +the fixation probability for type A for the initial states with i mutants, we need to solve a linear system of +2N − 2 unknowns, where each unknown is the fixation probability for an initial state. Note that we safely +excluded the two unknowns corresponding to the two initial states in which all nodes unanimously have type +A or B, in which case the fixation probability for type A is trivially equal to 1 and 0, respectively. To solve +a linear system with 2N − 2 unknowns is daunting except when N is small. Therefore, we analyze three +types of symmetric hypergraphs in which all or most nodes are structurally equivalent to each other. For +these hypergraphs, we only need to track the number of nodes with type A among the structurally equivalent +nodes, which drastically reduces the dimension of the linear system to be solved. In this section, we denote +the number of nodes of type A in the entire hypergraph by i ∈ {0, . . . , N}. The fixation of type A and B +corresponds to i = N and i = 0, respectively. +3.1. Model 1 +3.1.1. Complete 3-uniform hypergraph +We have mentioned that our model 1 on the complete graph is equivalent to the Moran process. To +investigate whether model 1 on counterparts of the complete graph for hypergraphs is equivalent to the +Moran process, we consider the complete 3-uniform hypergraph [33]. A complete 3-uniform hypergraph on +node set V is defined by hyperedge set E, which is the set of all subsets of V containing just three nodes. +In other words, E = {{v1, v2, v3}; v1, v2, v3 ∈ V, v1 ̸= v2, v1 ̸= v3, v2 ̸= v3}. We show the complete 3-uniform +hypergraph on four nodes in Fig. 1(a) as an example. +Figure 1: Examples of 3-uniform hypergraphs. (a) Complete 3-uniform hypergraph with N = 4 nodes. (b) Cyclic +3-uniform hypergraph with N = 8 nodes. +(c) Star 3-uniform hypergraph with N = 5 nodes. +Each colored oval +represents a hyperedge. +In this section, we refer to i, i.e., the number of nodes of type A, as the state of the evolutionary dynamics. +Note that knowing the dynamics of i is enough for completely understanding the evolutionary dynamics on +3 + +(a) +(b) +(c)the complete 3-uniform hypergraph owing to its symmetry. Under model 1, state i either remains unchanged +or moves to i−2, i−1, i+1, or i+2 in a single time step of the evolutionary dynamics. This is because every +hyperedge of the complete 3-uniform hypergraph has three nodes such that there are at most two nodes that +flip their type in a time step. +We denote the (N + 1) × (N + 1) transition probability matrix by P = [pi,j], where pi,j is the probability +that the state moves from i to j in a time step. At state i, the probability that a node of type A and B is +selected as parent is equal to ri/(ri + N − i) and (N − i)/(ri + N − i), respectively. If the selected parent +node is of type A, a hyperedge containing the parent and two nodes of type B is used for reproduction +with probability +�N−i +2 +� +/ +�N−1 +2 +� +, where +�� +represents the binomial coefficient. In this case, i increases by 2. +Alternatively, a hyperedge containing the parent, a different node of type A, and a node of type B is used +for reproduction with probability +�i−1 +1 +��N−i +1 +� +/ +�N−1 +2 +� +. In this case, i increases by 1. Otherwise, a hyperedge +containing the parent and two other nodes of type A is used for reproduction. In this case, i does not change. +If the parent is of type B, a hyperedge containing the parent and two nodes of type A is used for reproduction +with probability +�i +2 +� +/ +�N−1 +2 +� +. In this case, i decreases by 2. Alternatively, a hyperedge containing the parent, +a node of type A, and a different node of type B is used for reproduction with probability +�i +1 +��N−i−1 +1 +� +/ +�N−1 +2 +� +. +In this case, i decreases by 1. Otherwise, a hyperedge containing the parent and two other nodes of type B +is selected. In this case, i does not change. Therefore, the transition probabilities are given by +p0,0 = 1, +(3.1) +pN,N = 1, +(3.2) +pi,i−2 = +N − i +ri + N − i · +�i +2 +� +�N−1 +2 +� = +N − i +ri + N − i · +i(i − 1) +(N − 1)(N − 2), +i ∈ {2, . . . , N}, +(3.3) +pi,i−1 = +N − i +ri + N − i · +�i +1 +��N−i−1 +1 +� +�N−1 +2 +� += +N − i +ri + N − i · +2i(N − i − 1) +(N − 1)(N − 2), +i ∈ {1, . . . , N}, +(3.4) +pi,i+1 = +ri +ri + N − i · +�i−1 +1 +��N−i +1 +� +�N−1 +2 +� += +ri +ri + N − i · 2(i − 1)(N − i) +(N − 1)(N − 2), +i ∈ {2, . . . , N − 1}, +(3.5) +pi,i+2 = +ri +ri + N − i · +�N−i +2 +� +�N−1 +2 +� = +ri +ri + N − i · (N − i)(N − i − 1) +(N − 1)(N − 2) +, +i ∈ {1, . . . , N − 2}, +(3.6) +p1,1 = 1 − p1,0 − p1,2 − p1,3, +(3.7) +pi,i = 1 − pi,i−2 − pi,i−1 − pi,i+1 − pi,i+2, +i ∈ {2, . . . , N − 2}, +(3.8) +pN−1,N−1 = 1 − pN−1,N−3 − pN−1,N−2 − pN−1,N. +(3.9) +All the other entries of P are equal to zero. Therefore, P is a pentadiagonal matrix. States i = 0 and i = N +are absorbing states. +Denote by xi the probability of ending up in state N, i.e., fixation probability of the mutant type A, when +the initial state is i. We obtain +x0 = 0, +(3.10) +x1 = p1,0x0 + p1,1x1 + p1,2x2 + p1,3x3, +(3.11) +xi = pi,i−2xi−2 + pi,i−1xi−1 + pi,ixi + pi,i+1xi+1 + pi,i+2xi+2, +i ∈ {2, . . . , N − 2}, +(3.12) +xN−1 = pN−1,N−3xN−3 + pN−1,N−2xN−2 + pN−1,N−1xN−1 + pN−1,NxN, +(3.13) +xN = 1. +(3.14) +In vector notation, we can concisely write Eqs. (3.10) to (3.14) as +x = Px, +(3.15) +where x = (x0, x1, x2, . . . , xN)⊤, and ⊤ represents the transposition. Equation (3.15) is equivalent to +(P − I)x = 0, +(3.16) +4 + +where I is the identity matrix. Using Eqs. (3.10), (3.14), and (3.16), we obtain +Mx = b, +(3.17) +where b = (0, 0, . . . , 0, 1)⊤, and +M = +� +� +� +� +� +� +� +� +� +1 +0 +0 +0 +0 +· · · +0 +0 +0 +0 +p1,0 +p1,1 − 1 +p1,2 +p1,3 +0 +· · · +0 +0 +0 +0 +p2,0 +p2,1 +p2,2 − 1 +p2,3 +p2,4 +· · · +0 +0 +0 +0 +... +... +... +... +... +· · · +... +... +... +... +0 +0 +0 +0 +0 +· · · +pN−1,N−3 +pN−1,N−2 +pN−1,N−1 − 1 +pN−1,N +0 +0 +0 +0 +0 +· · · +0 +0 +0 +1 +� +� +� +� +� +� +� +� +� +. +(3.18) +Like P, the (N + 1) × (N + 1) matrix M is a pentadiagonal matrix. +PTRANS-I and PTRANS-II are numerical algorithms for efficiently solving pentadiagonal linear systems +[34]. Note that we need to calculate x although we only need x1. These two algorithms run in O(log n) time, +where n is the number of unknowns, and they are about ten times faster than the SciPy algorithm for banded +matrices, scipy.linalg.solve banded [35]. We use the PTRANS-II built in a Python package pentapy [36] to +calculate the fixation probability as a function of r unless N is small. For a given value of r, the calculation +requires O(log N) time, which is much faster than solving a full linear system of 2N − 2 unknowns. +For small complete 3-uniform hypergraphs, one can analytically calculate x1 by directly solving Eq. (3.17). +We obtain +x1 = +r2 +r2 + 2r + 1 +(3.19) +for N = 4 and +x1 = +r2(8r2 + 12r + 1) +8r4 + 28r3 + 33r2 + 28r + 8 +(3.20) +for N = 5. We compare Eqs. (3.19) and (3.20) with x1 for the Moran process (i.e., Eq. (2.1) with i = 1) with +N = 4 and N = 5 as a function of r in Figs. 2(a) and 2(b), respectively. Although the results are shown +by blue lines, they are hidden behind the orange lines. The figure indicates that these complete 3-uniform +hypergraphs are suppressors because their x1 is smaller than that for the Moran process for r > 1 and larger +for r < 1. We also obtained x1 by numerically solving Eq. (3.17) for N = 20 and N = 200. The results +shown in Figs. 2(c) and 2(d) for N = 20 and N = 200, respectively, indicate that these larger complete 3- +uniform hypergraphs are also suppressors. Therefore, we conclude that the complete 3-uniform hypergraphs +are suppressors under the evolutionary dynamics described by model 1. +3.1.2. Cyclic 3-uniform hypergraph +The fixation probability as a function of r for undirected cycle graphs is the same as that for the Moran +process because the cycle graphs are so-called isothermal graphs [1, 5, 37]. To examine whether the same +equivalence result holds true for hypergraphs, we consider an extension of the cycle graph to the hypergraph, +which we call the cyclic 3-uniform hypergraph. A cyclic 3-uniform hypergraph consists of a node set V = +{1, . . . , N} and a hyperedge set E = {{1, 2, 3}, {2, 3, 4}, . . . , {N − 2, N − 1, N}, {N − 1, N, 1}, {N, 1, 2}}, i.e., +any three consecutively numbered nodes with a periodic boundary condition form a hyperedge. The cyclic +3-uniform hypergraph with N = 3 nodes is the complete 3-uniform hypergraph. Therefore, we consider cyclic +3-uniform hypergrapys with N ≥ 4. We show the cyclic 3-uniform hypergraph on 8 nodes in Fig. 1(b). +We assume that there is initially one individual of type A. Then, under model 1, all the nodes of type A +are consecutively numbered at any time, without being interrupted by nodes of type B. Therefore, similarly +to the analysis of evolutionary dynamics on cycles [38, 39], it suffices to track the number of nodes of type +A, which we again denote by i, to understand the evolutionary dynamics on this hypergraph. +5 + +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +Moran +complete +cyclic +star +0 +1 +2 +3 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0 +1 +2 +3 +0.9 +0.95 +1.0 +r +0.01 +0.03 +0.05 +0.99 +0.995 +1.0 +r +0.001 +0.003 +0.005 +(a) +N = 4 +(b) +N = 5 +(c) +N = 20 +(d) +N = 200 +r +Fixation probability +Figure 2: Fixation probability for different hypergraph models under model 1. We compare the Moran process, which +is the baseline, complete 3-uniform hypergraphs, cyclic 3-uniform hypergraphs, and star 3-uniform hypergraphs. (a) +N = 4. (b) N = 5. (c) N = 20. (d) N = 200. The insets in (c) and (d) magnify the results for r values smaller than +and close to r = 1. In these insets and main panel (b), the results for the complete 3-uniform hypergraph are close +to those for the cyclic hypergraph such that the blue lines are almost hidden behind the orange lines. In (a), the two +results are exactly the same such that the blue line is completely hidden behind the orange line. +Figure 3: State transitions in the cyclic 3-uniform hypergraph. (a) A state with just one node of type A (i.e., i = 1). +(b) A state with just one node of type B (i.e., i = N − 1). (c) A state with more than one nodes of each type (i.e., +2 ≤ i ≤ N − 2). +6 + +(a) +(b) +Vt-2 +Vt-1 +Vt +V++1 +Vt+2 +Vt-2 +Ve-1 +Vt +Vt+1 +Vt+2 +B +B +(c) +Vt-2 +Vt-1 +Vt +Vt+1 +Vt+i-2 +Vt+i-1 +V+i +V+i+1 +AWhen i = 1, there are three types of events that can occur next. Without loss of generality, we assume +that the ℓth node is the only node of type A (see Fig. 3(a) for a schematic). The state moves from i = 1 +to i = 0 in one time step such that B fixates in the following two cases. In the first case, either node vℓ−2 +or vℓ+2, which is of type B, is selected as parent. If N ≥ 5, these two nodes are distinct. Therefore, this +event occurs with probability 2/(r + N − 1). Then, the hyperedge that contains the parent and vℓ is used for +reproduction, which occurs with probability 1/3. For example, if vℓ−2 is selected as parent and hyperedge +{ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from 1 to 0. In the second case, either vℓ−1 +or vℓ+1 is selected as parent, which occurs with probability 2/(r + N − 1). Then, one of the two hyperedges +that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3. For example, if +vℓ−1 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the +state moves from 1 to 0. By summing up these probabilities, we obtain +p1,0 = +2 +r + N − 1. +(3.21) +In fact, Eq. (3.21) also holds true for N = 4 although vℓ−2 and vℓ+2 are identical nodes when N = 4; +see Appendix A for the proof. Alternatively, the state moves from i = 1 to i = 3 whenever vℓ is selected as +parent, which occurs with probability r/(r + N − 1). Therefore, we obtain +p1,3 = +r +r + N − 1. +(3.22) +If any other event occurs, then i = 1 remains unchanged. Therefore, we obtain +p1,1 = 1 − p1,0 − p1,3 = +N − 3 +r + N − 1, +(3.23) +p1,j = 0 if j ̸= 0, 1, 3. +(3.24) +When i = N − 1, there are similarly three types of events that can occur in a time step. Without loss +of generality, we assume that the ℓth node is the only node of type B (see Fig. 3(b) for a schematic). The +state moves from i = N − 1 to i = N − 3 whenever vℓ is selected as parent, which occurs with probability +1/[r(N − 1) + 1]. Therefore, we obtain +pN−1,N−3 = +1 +r(N − 1) + 1. +(3.25) +Alternatively, the state moves from i = N − 1 to i = N such that type A fixates in the following two cases. +In the first case, either node vℓ−2 or vℓ+2, which is of type A, is selected as parent. If N ≥ 5, these two +nodes are distinct. Therefore, this event occurs with probability 2r/[r(N −1)+1]. Then, the hyperedge that +contains the parent and vℓ is used for reproduction, which occurs with probability 1/3. In the second case, +either vℓ−1 or vℓ+1 is selected as parent, which occurs with probability 2r/[r(N − 1) + 1]. Then, one of the +two hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3. +By summing up these probabilities, we obtain +pN−1,N = +2r +r(N − 1) + 1. +(3.26) +In fact, Eq. (3.26) also holds true for N = 4; see Appendix B for the proof. If any other event occurs, then +i = N − 1 remains unchanged. Therefore, we obtain +pN−1,N−1 = 1 − pN−1,N−3 − pN−1,N = +r(N − 3) +r(N − 1) + 1, +(3.27) +pN−1,j = 0 if j ̸= N − 3, N − 1, N. +(3.28) +When i ∈ {2, . . . , N − 2}, there are five types of possible events. Without loss of generality, we assume +that the ℓth to the (ℓ + i − 1)th nodes are of type A and that all other nodes are of type B (see Fig. 3(c) +for a schematic). If ℓ + i − 1 is larger than N, then we interpret ℓ + i − 1 as the number modulo N (i.e., +7 + +ℓ + i − 1 − N); the same convention applies in the following text. In the first type of event, either node vℓ−1 +or vℓ+i, which is of type B, is selected as parent; this event occurs with probability 2/(ri + N − i). Then, +the hyperedge that contains the parent and two nodes of type A is used for reproduction, which occurs with +probability 1/3. In this case, the state i decreases by two. For example, if vℓ−1 is selected as parent and +hyperedge {ℓ − 1, ℓ, ℓ + 1} is selected, then the state moves from i to i − 2. Therefore, we obtain +pi,i−2 = +2 +ri + N − i · 1 +3. +(3.29) +In the second type of event, either vℓ−2, vℓ−1, vℓ+i, or vℓ+i+1, which is of type B, is selected as parent. If +i ≤ N − 4, these four nodes are distinct. Therefore, this event occurs with probability 4/(ri + N − i). Then, +the hyperedge that contains the parent, a node of type B, and a node of type A is used for reproduction, +which occurs with probability 1/3. In this case, the state i decreases by one. For example, if vℓ−2 is selected +as parent and hyperedge {ℓ−2, ℓ−1, ℓ} is used for reproduction, then the state moves from i to i−1 because +vℓ turns from A to B. Therefore, we obtain +pi,i−1 = +4 +ri + N − i · 1 +3. +(3.30) +In fact, Eq. (3.30) also holds true for i = N − 3 and i = N − 2 although some of vℓ−2, vℓ−1, vℓ+i, and +vℓ+i+1 are identical nodes when i = N − 3 or i = N − 2; see Appendix C for the proof. In the third type +of event, either vℓ, vℓ+1, vℓ+i−2, or vℓ+i−1, which is of type A, is selected as parent. If i ≥ 4, these four +nodes are distinct. Therefore, this event occurs with probability 4r/(ri + N − i). Then, the hyperedge that +contains the parent node, a node of type A, and a node of type B is used for reproduction, which occurs with +probability 1/3. In this case, state i increases by one. For example, if vℓ is selected as parent and hyperedge +{ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from i to i + 1 because vℓ−1 turns from B to +A. Therefore, we obtain +pi,i+1 = +4r +ri + N − i · 1 +3. +(3.31) +In fact, Eq. (3.31) also holds true for i = 2 and i = 3 although some of vℓ, vℓ+1, vℓ+i−2, and vℓ+i−1 are +identical nodes when i = 2 or i = 3; see Appendix D for the proof. In the fourth type of event, either node vℓ +or vℓ+i−1, which is of type A, is selected as parent; this event occurs with probability 2r/(ri + N − i). Then, +the hyperedge that contains the parent and two nodes of type B is used for reproduction, which occurs with +probability 1/3. In this case, state i increases by two. For example, if vℓ is selected as parent and hyperedge +{ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from i to i + 2. Therefore, we obtain +pi,i+2 = +2r +ri + N − i · 1 +3. +(3.32) +If any other event occurs, then i remains unchanged. Therefore, we obtain +pi,i = 1 − pi,i−2 − pi,i−1 − pi,i+1 − pi,i+2. +(3.33) +All the other entries of transition probability matrix P are equal to zero. +By solving Eq. (3.17), we obtain +x1 = +r2 +r2 + 2r + 1 +(3.34) +for N = 4 and +x1 = +r2(6r2 + 8r + 1) +6r4 + 20r3 + 23r2 + 20r + 6 +(3.35) +for N = 5. We compare Eqs. (3.34) and (3.35) with x1 for the Moran process with N = 4 and N = 5 in +Figs. 2(a) and 2(b), respectively. We find that these cyclic 3-uniform hypergraphs are suppressors. The result +for N = 4 given by Eq. (3.34) coincides with that for the complete 3-uniform hypergraph given by Eq. (3.19). +For larger N, we use the same numerical method for solving Eq. (3.17) as that for the complete 3-uniform +hypergraph because P is a pentadiagonal matrix. We show the thus calculated x1 for N = 20 and N = 200 +in Figs. 2(c) and 2(d), respectively. These figures confirm that these larger cyclic 3-uniform hypergraphs are +8 + +also suppressors. Therefore, we conclude that the cyclic 3-uniform hypergraph is a suppressor. +3.1.3. Star 3-uniform hypergraph +The conventional star graphs are a strong amplifier [5, 26]. To examine whether counterparts of the star +graph for hypergraphs are also amplifiers, we define the star 3-uniform hypergraph as follows and examine +the fixation probability of the mutant on it. A star 3-uniform hypergraph consists of a node set V with a +single hub node and N − 1 leaf nodes, and hyperedges each of which is of size three and consists of the hub +node and a pair of N − 1 leaf nodes. We use all the +�N−1 +2 +� +pairs of leaf nodes to form hyperedges such that +the hub node belongs to +�N−1 +2 +� +hyperedges. Each leaf node belongs to N − 1 hyperedges and are structurally +equivalent to each other. The star 3-uniform hypergraph with N = 3 is the complete 3-uniform hypergraph. +Therefore, we consider star 3-uniform hypergraphs with N ≥ 4. We show the 3-uniform star hypergraph on +5 nodes in Fig. 1(c) as an example. +Owing to the structural equivalence among the N − 1 leaf nodes, the state of the evolutionary dynamics +on the star 3-uniform hypergraph is completely determined by the type on the hub (i.e., either A or B) and +the number of leaf nodes of type A, which ranges between 0 and N − 1. Therefore, we denote the state by +(i1, i2), where i1 = 1 or 0 if the hub node is of type A or B, respectively, and i2 ∈ {0, 1, . . . , N −1} represents +the number of the leaf nodes of type A. The total number of nodes of type A is given by i = i1 + i2. The +fixation of type A and B corresponds to (i1, i2) = (1, N − 1) and (0, 0), respectively. +Assume that there are currently i nodes of type A and that the state is (i1, i2) = (1, i − 1) with i ∈ +{1, . . . , N − 1}. There are five types of events that can occur next. +In the first type of event, a leaf node of type B is selected as parent, which occurs with probability +(N − i)/(ri + N − i). Then, a hyperedge that contains the parent, the hub node, and a different leaf node +of type B is used for reproduction, which occurs with probability (N − i − 1)/(N − 2). The state after this +entire event is (i1, i2) = (0, i − 1) with i ∈ {1, . . . , N − 2}. Therefore, we obtain +p(1,i−1)→(0,i−1) = +N − i +ri + N − i +N − i − 1 +N − 2 +. +(3.36) +In the second type of event, a leaf node of type B is selected as parent with probability (N − i)/(ri + N − i). +Then, a hyperedge that contains the parent, the hub node, and a leaf node of type A is used for reproduction, +which occurs with probability (i − 1)/(N − 2). The state after this event is (0, i − 2) with i ∈ {2, . . . , N − 1}. +Therefore, we obtain +p(1,i−1)→(0,i−2) = +N − i +ri + N − i +i − 1 +N − 2. +(3.37) +In the third type of event, the hub node, which is of type A, is selected as parent, which occurs with probability +r/(ri+N −i). Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B is +used for reproduction, which occurs with probability (i − 1)(N − i)/ +�N−1 +2 +� +. Alternatively, a leaf node of type +A is selected as parent, which occurs with probability r(i − 1)/(ri + N − i). Then, a hyperedge that contains +the parent, the hub node, and a leaf node of type B is used for reproduction, which occurs with probability +(N − i)/(N − 2). In both cases, the state after the event is (1, i) with i ∈ {2, . . . , N − 1}. Therefore, we +obtain +p(1,i−1)→(1,i) = +r +ri + N − i +(i − 1)(N − i) +�N−1 +2 +� ++ +r(i − 1) +ri + N − i +N − i +N − 2. +(3.38) +In the fourth type of event, the hub node is selected as parent with probability r/(ri + N − i). Then, a +hyperedge that contains the parent and two leaf nodes of type B is used for reproduction, which occurs with +probability +�N−i +2 +� +/ +�N−1 +2 +� +. The state after this event is (1, i+1) with i ∈ {1, . . . , N −2}. Therefore, we obtain +p(1,i−1)→(1,i+1) = +r +ri + N − i +�N−i +2 +� +�N−1 +2 +�. +(3.39) +If any other event occurs, then the state remains unchanged. Therefore, we obtain +p(1,i−1)→(1,i−1) = 1 − p(1,i−1)→(0,i−1) − p(1,i−1)→(0,i−2) − p(1,i−1)→(1,i) − p(1,i−1)→(1,i+1). +(3.40) +9 + +We denote the probability that type A fixates starting with state (i1, i2) by ˜x(i1,i2). We obtain +˜x(1,i−1) =p(1,i−1)→(0,i−1)˜x(0,i−1) + p(1,i−1)→(0,i−2)˜x(0,i−2) + p(1,i−1)→(1,i)˜x(1,i) ++ p(1,i−1)→(1,i+1)˜x(1,i+1) + p(1,i−1)→(1,i−1)˜x(1,i−1). +(3.41) +Assume that the current state is (i1, i2) = (0, i) with i ∈ {1, . . . , N − 1}. There are five types of events +that can occur next. +In the first type of event, a leaf node of type A is selected as parent with probability ri/(ri + N − i). +Then, a hyperedge that contains the parent, the hub node, and a different leaf node of type A is used for +reproduction with probability (i − 1)/(N − 2). +The state after this entire event is (i1, i2) = (1, i) with +i ∈ {2, . . . , N − 1}. Therefore, we obtain +p(0,i)→(1,i) = +ri +ri + N − i +i − 1 +N − 2. +(3.42) +In the second type of event, a leaf node of type A is selected as parent with probability ri/(ri+N −i). Then, +a hyperedge that contains the parent, the hub node, and a leaf node of type B is used for reproduction, which +occurs with probability (N − i − 1)/(N − 2). The state after this event is (1, i + 1) with i ∈ {1, . . . , N − 2}. +Therefore, we obtain +p(0,i)→(1,i+1) = +ri +ri + N − i +N − i − 1 +N − 2 +. +(3.43) +In the third type of event, the hub node, which is of type B, is selected as parent, with probability 1/(ri+N−i). +Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B, is used for +reproduction, which occurs with probability i(N − i − 1)/ +�N−1 +2 +� +. Alternatively, a leaf node of type B is +selected as parent with probability (N − i − 1)/(ri + N − i). Then, a hyperedge that contains the parent, +the hub node, and a leaf node of type A is used for reproduction, which occurs with probability i/(N − 2). +In both cases, the state after the event is (0, i − 1) with i ∈ {1, . . . , N − 2}. Therefore, we obtain +p(0,i)→(0,i−1) = +1 +ri + N − i +i(N − i − 1) +�N−1 +2 +� ++ N − i − 1 +ri + N − i +i +N − 2. +(3.44) +In the fourth type of event, the hub node is selected as parent with probability 1/(ri + N − i). Then, the +hyperedge that contains the parent and two leaf nodes of type A is used for reproduction, which occurs with +probability +�i +2 +� +/ +�N−1 +2 +� +. The state after this event is (0, i − 2) with i ∈ {2, . . . , N − 1}. Therefore, we obtain +p(0,i)→(0,i−2) = +1 +ri + N − i +�i +2 +� +�N−1 +2 +�. +(3.45) +If any other event occurs, then the state remains unchanged. Therefore, we obtain +p(0,i)→(0,i) = 1 − p(0,i)→(1,i) − p(0,i)→(1,i+1) − p(0,i)→(0,i−1) − p(0,i)→(0,i−2). +(3.46) +Using these transition probabilities, we obtain +˜x(0,i) =p(0,i)→(1,i)˜x(1,i) + p(0,i)→(1,i+1)˜x(1,i+1) + p(0,i)→(0,i−1)˜x(0,i−1) ++ p(0,i)→(0,i−2)˜x(0,i−2) + p(0,i)→(0,i)˜x(0,i). +(3.47) +We rewrite Eqs. (3.41) and (3.47) as +˜x = P ˜x, +(3.48) +where ˜x = (˜x(0,0), ˜x(0,1), . . . , ˜x(0,N−1), ˜x(1,0), ˜x(1,1), . . . , ˜x(1,N−1))⊤. The 2N ×2N stochastic matrix P is given +by +P = +� C +D +E +F +� +, +(3.49) +10 + +where C, D, E, and F are N × N matrices given by +C = +� +� +� +� +� +� +� +1 +0 +0 +· · · +0 +0 +0 +p(0,1)→(0,0) +p(0,1)→(0,1) +0 +· · · +0 +0 +0 +p(0,2)→(0,0) +p(0,2)→(0,1) +p(0,2)→(0,2) +· · · +0 +0 +0 +... +... +... +· · · +... +... +0 +0 +0 +· · · +p(0,N−1)→(0,N−3) +p(0,N−1)→(0,N−2) +p(0,N−1)→(0,N−1) +� +� +� +� +� +� +� +, +(3.50) +D = +� +� +� +� +� +� +� +� +� +0 +0 +0 +0 +· · · +0 +0 +0 +0 +p(0,1)→(1,1) +p(0,1)→(1,2) +0 +· · · +0 +0 +0 +0 +0 +p(0,2)→(1,2) +p(0,2)→(1,3) +· · · +0 +0 +0 +... +... +... +... +· · · +... +... +0 +0 +0 +0 +· · · +0 +p(0,N−2)→(1,N−2) +p(0,N−2)→(1,N−1) +0 +0 +0 +0 +· · · +0 +0 +p(0,N−1)→(1,N−1) +� +� +� +� +� +� +� +� +� +, +(3.51) +E = +� +� +� +� +� +� +� +� +� +p(1,0)→(0,0) +0 +0 +· · · +0 +0 +0 +p(1,1)→(0,0) +p(1,1)→(0,1) +0 +· · · +0 +0 +0 +0 +p(1,2)→(0,1) +p(1,2)→(0,2) +· · · +0 +0 +0 +... +... +... +· · · +... +... +... +0 +0 +0 +· · · +p(1,N−2)→(0,N−3) +p(1,N−2)→(0,N−2) +0 +0 +0 +0 +· · · +0 +p(1,N−1)→(0,N−2) +p(1,N−1)→(0,N−1) +� +� +� +� +� +� +� +� +� +, +(3.52) +and +F = +� +� +� +� +� +� +� +p(1,0)→(1,0) +p(1,0)→(1,1) +p(1,0)→(1,2) +0 +· · · +0 +0 +0 +0 +p(1,1)→(1,1) +p(1,1)→(1,2) +p(1,1)→(1,3) +· · · +0 +0 +0 +... +... +... +... +· · · +... +... +... +0 +0 +0 +0 +· · · +0 +p(1,N−2)→(1,N−2) +p(1,N−2)→(1,N−1) +0 +0 +0 +0 +· · · +0 +0 +1 +� +� +� +� +� +� +� +. +(3.53) +The transition matrix P is a sparse matrix. We use the scipy implementation of the DGESV routine of +LAPACK, scipy.linalg.solve, to numerically solve Eq. (3.48) to obtain ˜x(0,i) and ˜x(1,i−1). +We remind that xi is the probability that type A fixates when there are initially i nodes of type A that +are selected uniformly at random. There are +�N−1 +i−1 +� +states in which i − 1 nodes have type A and the hub has +type A. There are +�N−1 +i +� +states in which i nodes have type A and the hub has type B. Therefore, we obtain +xi = +�N−1 +i−1 +� +�N−1 +i−1 +� ++ +�N−1 +i +� ˜x(1,i−1) + +�N−1 +i +� +�N−1 +i−1 +� ++ +�N−1 +i +� ˜x(0,i) += i +N ˜x(1,i−1) + N − i +N +˜x(0,i). +(3.54) +We analytically solve Eqs. (3.48) and (3.54) to obtain +x1 = +r2(3r + 5) +4(3r3 + 14r2 + 18r + 9) + +9r2 +4(3r2 + 5r + 3) +(3.55) +for N = 4 and +x1 = +r2(72r3 + 202r2 + 145r + 6) +5(72r5 + 490r4 + 1025r3 + 1070r2 + 880r + 288) +11 + ++ +4r2(72r2 + 94r + 4) +5(72r4 + 202r3 + 217r2 + 202r + 72) +(3.56) +for N = 5. We compare Eqs. (3.55) and (3.56) with x1 for the Moran process with N = 4 and N = 5 in +Figs. 2(a) and 2(b), respectively. We find that these star 3-uniform hypergraphs are suppressors. We then +computed x1 by numerically solving Eq. (3.48) for N = 20 and N = 200. Figures 2(c) and 2(d) compare +the obtained x1 values with those for the Moran process with N = 20 and N = 200, respectively. These +figures indicate that these larger star 3-uniform hypergraphs are also suppressors although the degree of +suppression is smaller for larger star 3-uniform hypergraphs; Fig. 2(d) shows that the difference between the +star 3-uniform hypergraph and the Moran process in terms of x1 is tiny when N = 200. In sum, we conclude +that the star 3-uniform hypergraph is a suppressor. +3.1.4. Comparison with conventional networks obtained by one-mode projection +A lossy representation of a hypergraph as a conventional network is the one-mode projection, in which +two nodes are adjacent by an edge if and only if they belong to at least one common hyperedge. A weighted +network version of the one-mode projection defines the edge weight by the number of hyperedges that the +two nodes share. The one-mode projection is a common approach for analyzing dynamics on hypergraphs +including evolutionary dynamics [4]. The one-mode projections of the complete and cyclic 3-uniform hyper- +graphs are regular networks (i.e., networks in which all the nodes have the same degree), in the case of both +unweighted and weighted variants of the one-mode projection. Then, the isothermal theorem [5] guarantees +that the birth-death process on the one-mode projections of the complete and cyclic 3-uniform hypergraphs +is equivalent to the Moran process. Therefore, our result that the complete and cyclic 3-uniform hypergraphs +are suppressors is not an artifact of one-mode projection. +The same argument does not apply for the star 3-uniform hypergraph because the one-mode projection +of the star 3-uniform hypergraph is not a regular network. Furthermore, the star 3-uniform hypergraph are +only analogously similar to the conventional star graph. In fact, leaf nodes are adjacent to each other in the +star 3-uniform hypergraph, whereas they are not in the conventional star graph. Then, the direct connection +between leaf nodes might be a reason why the star 3-uniform hypergraph is a suppressor. To exclude this +possibility, using similar analytical techniques, we investigated the fixation probability for the weighted one- +mode projection of the star 3-uniform hypergraph, which is a weighted complete graph. +Note that the +unweighted one-mode projection of the star 3-uniform hypergraph is the unweighted complete graph, which +is trivially equivalent to the Moran process. As we show in Appendix E, we have found that the obtained +weighted complete graph is an amplifier although the amplifying effect is weak. Therefore, our result that +the star 3-uniform hypergraph is a suppressor is not expected from the one-mode projection. +3.2. Model 2 +In this section, we analyze the fixation probability for the birth-death process governed by model 2 on +the complete, cyclic, and star 3-uniform hypergraphs. +3.2.1. Complete 3-uniform hypergraph +On the complete 3-uniform hypergraph, the state i either remains unchanged or moves to i − 1 or i + 1 +in a single time step because all the hyperedges are composed of three nodes such that there are at most one +node that changes the state under the majority rule given by model 2. At state i, the probability that a node +of type A and B is selected as parent is equal to ri/(ri + N − i) and (N − i)/(ri + N − i), respectively. If the +parent is of type A, then the following three types of events are possible. First, if a hyperedge containing the +parent and two nodes of type B is used for reproduction with probability +�N−i +2 +� +/ +�N−1 +2 +� +, then the state does +not change. Second, if a hyperedge containing the parent, a different node of type A, and a node of type B +is used for reproduction with probability +�i−1 +1 +��N−i +1 +� +/ +�N−1 +2 +� +, then the state moves from i to i + 1. Third, if a +hyperedge containing the parent and two other nodes of type A is used for reproduction with the remaining +probability, then the state does not change. If the parent is of type B, then the following three types of events +are possible. First, if a hyperedge containing the parent and two nodes of type A is used for reproduction +with probability +�i +2 +� +/ +�N−1 +2 +� +, then the state does not change. If a hyperedge containing the parent, a node +of type A, and a different node of type B is used for reproduction with probability +�i +1 +��N−i−1 +1 +� +/ +�N−1 +2 +� +, then +the state moves from i to i − 1. Third, if a hyperedge containing the parent and two other nodes of type +12 + +B is used for reproduction with the remaining probability, then the state does not change. Therefore, the +transition probabilities are given by +p0,0 = 1, +(3.57) +pN,N = 1, +(3.58) +pi,i−1 = +N − i +ri + N − i · +�i +1 +��N−i−1 +1 +� +�N−1 +2 +� += +N − i +ri + N − i · +2i(N − i − 1) +(N − 1)(N − 2), +i ∈ {1, . . . , N}, +(3.59) +pi,i+1 = +ri +ri + N − i · +�i−1 +1 +��N−i +1 +� +�N−1 +2 +� += +ri +ri + N − i · 2(i − 1)(N − i) +(N − 1)(N − 2), +i ∈ {0, . . . , N − 1}, +(3.60) +pi,i = 1 − pi,i−1 − pi,i+1, +i ∈ {1, . . . , N − 1}. +(3.61) +All the other entries of P are equal to zero. Therefore, P is a tridiagonal matrix. We note that the states +i = 0 and i = N are absorbing states. +The fixation probability of type A starting from state i, i.e., xi, satisfies +x0 = x1 = 0, +(3.62) +xi = pi,i−1xi−1 + pi,ixi + pi,i+1xi+1, +i ∈ {2, . . . , N − 2}, +(3.63) +xN−1 = xN = 1. +(3.64) +Similar to the analysis of the fixation probability for the Moran process [1], we set +yi ≡ xi − xi−1, +i ∈ {1, . . . , N}. +(3.65) +Note that �N +i=1 yi = xN − x0 = 1. +Let γi = pi,i−1/pi,i+1. +Equation (3.63) leads to yi+1 = yiγi with +i ∈ {2, . . . , N − 2}. Therefore, we obtain y1 = 0, y2 = x2, y3 = x2γ2, y4 = x2γ2γ3, . . ., yN−1 = x2 +�N−2 +k=2 γk. +By summing all these expressions and using yN = 0, which Eq. (3.64) implies, we obtain +1 = +N +� +i=1 +yi = x2 +� +1 + γ2 + γ2γ3 + · · · + +N−2 +� +k=2 +γk +� += x2 +� +1 + N − 3 +r ++ N − 4 +2r +N − 3 +r ++ · · · + +1 +r(N − 3) +2 +r(N − 4) · · · N − 4 +2r +N − 3 +r +� += x2 +N−3 +� +i=0 +r−i +�N − 3 +i +� += x2 +� +1 + 1 +r +�N−3 +. +(3.66) +Therefore, we obtain +x2 = +� +1 + 1 +r +�3−N +(3.67) +and +xi =x2 +� +�1 + +i−1 +� +j=2 +j� +k=2 +γk +� +� += +� +1 + 1 +r +�3−N +� +�1 + +i−1 +� +j=2 +j� +k=2 +γk +� +� , +i ∈ {3, 4, . . . , N − 2}. +(3.68) +For model 2, it always holds true that x1 = 0 because the evolutionary dynamics is driven by a majority +rule. Therefore, we compare x2, instead of x1, as a function of r with x2 for the Moran process, to examine +13 + +whether a given hypergraph is an amplifier, suppressor, equivalent to the Moran process, or neither. We +compare x2 calculated from Eq. (3.67) with that for the Moran process at four values of N in Fig. 4. The +figure indicates that the complete 3-uniform hypergraph with 4 nodes is a suppressor. However, the complete +3-uniform hypergraph with N > 4 is neither amplifier nor suppressor. This is because Eq. (3.67) implies that +x2 = 23−N when r = 1, which is different from the result for the Moran process, i.e., x2 = 2/N, when N > 4. +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +Moran +complete +cyclic +star +0 +1 +2 +3 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +0 +1 +2 +3 +(a) +N = 4 +(b) +N = 5 +(c) +N = 20 +(d) +N = 200 +r +Fixation probability +Figure 4: Fixation probability for different hypergraph models under model 2. We compare the Moran process, which +is the baseline, complete 3-uniform hypergraphs, cyclic 3-uniform hypergraphs, and star 3-uniform hypergraphs. (a) +N = 4. (b) N = 5. (c) N = 20. (d) N = 200. In (a), the result for the complete 3-uniform hypergraph is exactly +the same as that for the cyclic 3-uniform hypergraph such that the blue line is completely hidden behind the orange +line. In (c) and (d), the results for the complete 3-uniform hypergraph (shown by the blue lines) are not identical but +close to those for the star hypergraph (shown by the green lines) such that the former are hidden behind the latter. +3.2.2. Cyclic 3-uniform hypergraph +Consider the evolutionary dynamics under model 2 on the cyclic 3-uniform hypergraph. Assume that +there are initially two nodes of type A, which are distributed uniformly at random, and N − 2 nodes of type +B. We derived in Appendix F the fixation probability for type A, x2, using techniques similar to those used +for the complete 3-uniform hypergraph (section 3.2.1). We obtain +x2 = +� +� +� +2 +N−1 +� +1−r−1 +1−r−(N−2) + +r−r−1 +(r+4)[1−r−(N−2)] +� +(N ≥ 5), +r +1+r +(N = 4). +(3.69) +We compare x2 given by Eqs. (3.69) with x2 for the Moran process at four values of N in Fig. 4. The +figure indicates that the cyclic 3-uniform hypergraph with N = 4 is a suppressor. +However, the cyclic +14 + +3-uniform hypergraph with N > 4 is neither amplifier nor suppressor under model 2. +This is because +x2 = 14/ [5(N − 1)(N − 2)] when r = 1 for cyclic 3-uniform hypergraphs, which is different from the value +for the Moran process, i.e., x2 = 2/N. +3.2.3. Star 3-uniform hypergraph +We calculate the fixation probability for model 2 on the star 3-uniform hypergraph. As in the case of +model 1, we exploit the fact that the combination of the type of the hub (i.e., either A or B) and the number +of leaf nodes of type A, which ranges between 0 and N − 1, completely specifies the state of the evolutionary +dynamics on this hypergraph. We show the derivation of the fixation probability in Appendix G. +We obtain +x2 = +5r +2(5r + 3) + +15r2 + 9r +2(15r2 + 34r + 15) +(3.70) +for N = 4 and +x2 = +6(r3 + 3r2) +5(3r3 + 13r2 + 15r + 4) + +3r2 +5(r2 + 3r + 1) +(3.71) +for N = 5. We compare Eqs. (3.70) and (3.71) with x2 for the Moran process with N = 4 and N = 5 in +Figs. 4(a) and 4(b), respectively. We find that the star 3-uniform hypergraph with N = 4 is a suppressor. In +contrast, the star 3-uniform hypergraph with N = 5 is neither an amplifier nor suppressor because x2 = 9/35 +at r = 1, which is different from the corresponding result for the Moran process, i.e., x2 = 2/5. We also +obtained x2 for N = 20 and N = 200 by numerically solving a system of linear equations with 2N − 2 +unknowns (see Appendix G). Figures 4(c) and 4(d), which compare the obtained x2 values with x2 for the +Moran process for N = 20 and N = 200, respectively, indicate that these larger star 3-uniform hypergraphs +are neither an amplifier nor suppressor. Therefore, we conclude that star 3-uniform hypergraphs are neither +an amplifier nor suppressor for N ≥ 5 under the evolutionary dynamics described by model 2. +3.3. Neutral drift +The discussion of amplifier and suppressor of natural selection requires x1 = 1/N (or x2 = 2/N in the case +of our model 2) when r = 1. In this section, we prove the following theorem, which justifies such discussion +for model 1 and not for model 2. +Theorem 1. Consider the neutral drift, i.e., r = 1. When there are initially i mutants (i.e., type A) that +are distributed uniformly at random over the N nodes, the fixation probability for the mutant is equal to i/N. +Proof. Consider model 1’, which is defined by the same evolutionary dynamics as that for model 1 with the +initial condition in which each node i is occupied by a distinct neutral type ˜Ai. Therefore, there are initially +N types, all of which have the constant fitness equal to 1. The evolutionary dynamics terminates when a +single type fixates. We denote by qi the fixation probability for type ˜Ai under the aforementioned initial +condition. It should be noted that �N +i=1 qi = 1. +Now we consider the original model 1 with the initial condition in which there is just one mutant located +at node i. Then, the fixation probability for the mutant is equal to qi. This is because model 1’ is reduced +to model 1 with the present initial condition if we regard type ˜Ai as the mutant type and all the N − 1 types +˜Aj, where j ̸= i, as the resident type. We recall that x1 is the average of the fixation probability over the N +initial conditions in which there is just one mutant. Therefore, we obtain x1 = �N +i=1 qi/N = 1/N. +Next we consider model 1 with the initial condition in which there are two mutants whose locations are +selected uniformly at random. Assume that the two mutants are initially located at nodes 1 and 2. The +mutant type fixates with probability q1 + q2 because model 1’ is reduced to model 1 with the present initial +condition if we regard type ˜A1 and ˜A2 as the mutant type and all the other N − 2 types ˜Aj, where j ̸= 1, 2, +as the resident type. Therefore, we obtain x2 = �N +i=1 +�i−1 +j=1(qi + qj)/ +�N +2 +� += (N − 1) �N +i=1 qi/ +�N +2 +� += 2/N. +Similarly, we obtain x3 = �N +i=1 +�i−1 +j=1 +�j−1 +k=1(qi + qj + qk)/ +�N +3 +� += +�N−1 +2 +� �N +i=1 qi/ +�N +3 +� += 3/N. +It is +straightforward to verify xi = i/N for the other i values. +Remark 2. This theorem also holds true for the birth-death process on conventional networks with the proof +being unchanged. +15 + +Remark 3. The theorem does not hold true for model 2. This is because, in model 2, whether the parent +node u propagates its type to the other nodes in the selected hyperedge e containing u depends on the other +nodes belonging to e. As an example, consider the case in which e contains three nodes, u, v1, and v2. Parent +u imposes its type on v1 and v2 only when either v1 or v2 has the same type as u. Otherwise, i.e., if both v1 +and v2 are of the opposite type as u’s, then no state change occurs after u is selected as parent and hyperedge +e is selected. Model 1’ cannot handle this situation. In model 1’, the parent disseminates its type to all the +other nodes in the selected hyperedge e, as in model 1, regardless of the type of the other nodes belonging +to e. Therefore, we cannot map model 1’ to model 2. In fact, Fig. 4 shows that x2 ̸= 2/N in most cases, +verifying that Theorem 1 does not hold true in general. +4. Numerical results for empirical hypergraphs +In this section, we carry out numerical simulations for empirical hypergraphs. +The present study is +motivated by the result that most networks are amplifiers under the birth-death process [20–22]. In section 3.1, +we showed that three model hypergraphs are suppressors under model 1. In section 3.2, we argued that one +cannot discuss whether the same hypergraphs are amplifier or suppressor under model 2 except for the star +3-uniform hypergraph with N = 4 nodes. This is because model 2 does not respect x2 = 2/N in general (see +Remark 3). Therefore, we focus on model 1 in this section and examine whether empirical hypergraphs tend +to be suppressors, as is the case for the complete, cyclic, and star 3-uniform hypergraphs. +Empirical, or general, hypergraphs are distinct from the complete, cyclic, and star 3-uniform hypergraphs +in two main aspects. First, in general, empirical hypergraphs do not have much symmetry that we can exploit +to simplify the probability transition matrix P. Therefore, pursuing analytical solutions, which would involve +the solution of a linear system with 2N − 2 unknowns, is formidable unless N is small. Second, empirical +hypergraphs contain hyperedges of different sizes, whereas the 3-uniform hypergraphs only have hyperedges of +size 3. Therefore, in this section, we numerically examine the stochastic evolutionary dynamics on empirical +hypergraphs. In each time step, we select a parent node with the probability proportional to its fitness from +all the N nodes. Then, the parent propagates its type to all the other nodes in a hyperedge to which the +parent belongs, which we select uniformly at random from all the hyperedges to which the parent belongs +regardless of the size of the hyperedge. +To calculate the fixation probability of a single mutant of type A for an arbitrary hypergraph and for +each value of r, we run the birth-death process until type A or B fixates for each node v initially occupied by +type A. Note that all the N − 1 nodes except v are initially of type B. For each v, we run 3 × 103 simulations +for r ≥ 1 and 4 × 104 simulations for r < 1. We use a substantially larger number of simulations for r < 1 +than r ≥ 1 because the fixation probability is small when r is small and therefore it can be estimated with a +higher accuracy with more simulations. For a given value of r, we estimate the fixation probability of type +A as a fraction of the runs in which type A has fixated among the 3 × 103 × N or 4 × 104 × N simulations; +the factor N is due to the N different choices of v. +We examine the fixation probability on four empirical hypergraphs. +The corporate club membership +hypergraph (corporate hypergraph for short) contains 25 nodes and 15 hyperedges [40, 41]. +Each node +represents a corporate executive officer. Each hyperedge represents a social organization such as clubs and +boards of which some officers are members. +The Enron hypergraph is an email communication network +and has 143 nodes and 10,885 hyperedges [42, 43]. Each node represents an email address at Enron. Each +hyperedge is comprised of all recipient addresses of an email. The Senate committee hypergraph (Senate +hypergraph for short) has 282 nodes and 315 hyperedges. Each node is a member of the US Senate. Each +hyperedge represents committee memberships [44, 45]. The high-school hypergraph is a social contact network +with 327 nodes and 7,818 hyperedges. Each node is a student. Each hyperedge represents to a group of +students that were in proximity of one another [45, 46]. All the four hypergraphs are connected hypergraphs. +In Fig. 5, we show the relationships between the fixation probability for a single mutant as a function +of r, for the four empirical hypergraphs, one per panel. We find that all the hypergraphs are suppressors. +We also simulate the birth-death process on the weighted one-mode projection of each empirical hypergraph. +We find that the fixation probability for the obtained weighted network is almost the same as that for the +Moran process for the corporate (Fig. 5(a)), Senate (Fig. 5(c)), and high-school (Fig. 5(d)) hypergraphs. The +one-mode projection of the Enron hypergraph is a suppressor (Fig. 5(b)). However, the fixation probability +for the one-mode projection of the Enron hypergraph as a function of r is close to that for the Moran process. +16 + +In contrast, the original Enron hypergraph is a much stronger suppressor. Therefore, our result that the four +empirical hypergraphs are suppressors is not expected from the one-mode projection. +To further examine the result that the empirical hypergraphs are suppressors, we now simulate the same +evolutionary dynamics on the randomized hypergraph. We obtained the randomized hypergraph for each +empirical hypergraph by randomly shuffling the hyperedges of the original hypergraph. +In the random +shuffling, we preserved the degree of each node and the size of each hyperedge [47, 48]. We show the fixation +probability for the randomized hypergraphs by the green circles in Fig. 5. We find that the randomized +hypergraph is also a suppressor for all the four hypergraphs. The randomized hypergraph is less suppressing +than the original hypergraph for three empirical hypergraphs (see Figs. 5(a), (b), and (c)) and vice versa for +the other one empirical hypergraph (see Fig. 5(d)). Therefore, how the features of empirical hypergraphs +except for the distribution of the node’s degree and hyperedge’s size affects the fixation probability is non- +equivocal. However, these results for the randomized hypergraphs further strengthen our main claim that +hypergraphs are suppressors of evolutionary dynamics under model 1 in most cases. +0 +1 +2 +3 +0 +0.4 +0.8 +Moran +empirical +one-mode projection +randomized +0 +1 +2 +3 +0 +1 +2 +3 +0 +0.4 +0.8 +0 +1 +2 +3 +0.98 +0.99 +1 +r +0 +0.005 +0.01 +0.98 +0.99 +1 +r +0 +0.002 +0.004 +0.98 +0.99 +1 +r +0 +0.001 +0.002 +(a) +corporate +(b) +Enron +(c) +Senate +(d) +high-school +r +Fixation probability +Figure 5: Fixation probability for empirical hypergraphs, their one-mode projection, and the randomized hyper- +graphs. (a) Corporate. (b) Enron. (c) Senate. (d) High-school. The insets of (b), (c), and (d) magnify the results for +r values smaller than and close to r = 1. +5. Discussion +We have proposed two models of evolutionary dynamics on hypergraphs that are extensions of the birth- +death process on networks. +For both models of evolutionary dynamics, we semi-analytically derived the +17 + +fixation probability for the mutant type under the constant selection for three synthetic hypergraphs with +high symmetry, which generalize the complete graph, cycle graph, and star graph. For model 1, which is +appropriate for discussing the strength of natural selection, we showed that these synthetic hypergraphs are +suppressors of natural selection with few exceptions. Furthermore, by numerical simulations, we have shown +that four empirical hypergraphs of our arbitrary choices are also suppressors. Our results are in stark contrast +to the known result that most networks are amplifiers under the birth-death updating rule and the uniform +initialization [20–22], which we also assumed. It is often the case that interaction among individuals is often +of higher order, and hypergraphs rather than conventional networks are more direct representation of many +empirical data of social and ecological interactions [13, 14]. Therefore, amplification of natural selection by +birth-death processes on networks may not be as universal as it has been suggested once we expand the class +of network models to be considered. +For conventional networks, finding suppressors under the birth-death process is difficult [28]. In contrast, +under the death-birth process, most conventional networks are suppressors [20], and amplification of natural +selection is bounded and transient [49]. +Our main result that most hypergraphs are suppressors under +model 1 begs various related research questions. Is there a theoretical bound on amplification of natural +selection in birth-death processes on hypergraphs? Is there a systematically constructed class of amplifying +hypergraphs even if they are rare? Are hypergraphs suppressors under the death-birth process? If it is the +case, hypergraphs are even more suppressing under the death-birth than birth-death processes? Can we find +optimal amplifier [24, 26, 27] or suppressor for hypergraphs? We save these questions for the future. +Evolutionary set theory is a mathematical framework with which to analyze coevolution of the strategy +(i.e., type) of the node and the membership of the node to sets (i.e., groups) [50–52]. It assumes that each +node belongs to different groups and play games with other nodes belonging to the same group. A node v with +a large fitness obtained from playing the game disseminates v’s group membership as well as v’s type to other +nodes with a high probability. Therefore, evolutionary set theory is a dynamic graph theory. Evolutionary +set theory is distinct from evolutionary dynamics on hypergraphs studied in the present study in the following +aspects. First, in evolutionary set theory, interaction between players is pairwise by default. In contrast, in +evolutionary dynamics on hypergraphs, the outcome of an interaction in a group (i.e., hyperedge) may not be +decomposed into the summation of pairwise interaction in the group. For example, we assumed that a parent +node v simultaneously imposes its type on all the other nodes in the selected hyperedge. Second, the group +membership evolves in evolutionary set theory, whereas it is fixed in the evolutionary dynamics considered +in this study, which is a simplification assumption. Third, reproduction in evolutionary set theory occurs +globally, not limited to between nodes in the same group. In contrast, we have assumed that reproduction +occurs between nodes in the same hyperedge. These differences create opportunities for future work. For +example, extending the present model to the case of dynamic hypergraphs may be interesting. +For conventional networks, the time to fixation has been shown to be in a tradeoff relationship with the +fixation probability [53–57]. In other words, a strongly amplifying network tends to accompany a large mean +fixation time. Our mathematical framework is readily adaptable to the examination of fixation times. The +fixation time for hypergraphs including the comparison with the case of conventional networks warrants future +work. Other topics for further investigations include the effects of different initial conditions [7, 32, 58, 59], +mathematical analyses of other symmetric hypergraphs, weak selection expansion of fixation probability +[22] in the case of hypergraphs, amplification and suppression of natural selection in the mutation-selection +equilibrium under a small mutation rate [60], and fixation probability of cooperation in evolutionary game +dynamics on hypergraphs [15, 16]. Focusing on fixation of a mutant (or resident) type in general allows +us to use Markov chain theory to reach various mathematical insights. We believe that the present study +is an important step toward deploying this powerful mathematical machinery to evolutionary dynamics on +higher-order networks. +Acknowledgments +N.M. acknowledges support from AFOSR European Office (under Grant No. FA9550-19-1-7024), the +Japan Science and Technology Agency (JST) Moonshot R&D (under Grant No. JPMJMS2021), and the +National Science Foundation (under Grant No. 2052720 and 2204936). +18 + +Appendix A. Proof of Eq. (3.21) for N = 4 +If N = 4, then vℓ−2 and vℓ+2 are identical. In this case, there are two sequences of events through which +the state moves from i = 1 to i = 0 in one time step. In the first sequence, vℓ−2, which is of type B, is +selected as parent with probability 1/(r + 3). Then, the hyperedge that contains the parent and vℓ, i.e., +{ℓ − 2, ℓ − 1, ℓ} or {ℓ, ℓ + 1, ℓ − 2}, is used for reproduction, which occurs with probability 2/3. In the second +sequence, either vℓ−1 or vℓ+1 is selected as parent, which occurs with probability 2/(r + 3). Then, one of +the two hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability +2/3. For example, if vℓ−1 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 1, ℓ, ℓ + 1} is used for +reproduction, then the state moves from 1 to 0. By summing up these probabilities, we obtain Eq. (3.21). +Appendix B. Proof of Eq. (3.26) for N = 4 +If N = 4, then vℓ−2 and vℓ+2 are identical. In this case, there are two sequences of events through which +the state moves from i = N −1 to i = N. In the first sequence, vℓ−2, which is of type A, is selected as parent +with probability r/(3r + 1). Then, the hyperedge that contains the parent and vℓ, i.e., {ℓ − 2, ℓ − 1, ℓ} or +{ℓ, ℓ + 1, ℓ − 2}, is used for reproduction, which occurs with probability 2/3. In the second sequence, either +vℓ−1 or vℓ+1, which is of type A, is selected as parent with probability 2r/(3r + 1). Then, one of the two +hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3. By +summing up these probabilities, we obtain Eq. (3.26). +Appendix C. Proof of Eq. (3.30) for i = N − 3 and i = N − 2 +If i = N − 3, then vℓ−2 and vℓ+i+1 are identical. In this case, there are two sequences of events through +which the state decreases from i to i − 1. In the first sequence, either vℓ−1 or vℓ−3 is selected as parent, +which occurs with probability 2/(rN − 3r + 3). Then, the hyperedge containing two nodes of type B (i.e., +hyperedge {ℓ − 2, ℓ − 1, ℓ} if vℓ−1 is the parent and hyperedge {ℓ − 4, ℓ − 3, ℓ − 2} if vℓ−3 is the parent) is used +for reproduction, which occurs with probability 1/3. In the second sequence, vℓ−2 is selected as parent with +probability 1/(rN − 3r + 3). Then, hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 4, ℓ − 3, ℓ − 2} is used for reproduction, +which occurs with probability 2/3. By summing up these probabilities, we obtain Eq. (3.30). +If i = N − 2, either of the two nodes of type B, i.e., vℓ−1 or vℓ−2, must be selected as parent for the +state to move from i to i − 1. This event occurs with probability 2/(rN − 2r + 2). Then, either hyperedge +{ℓ − 2, ℓ − 1, ℓ} or {ℓ − 3, ℓ − 2, ℓ − 1} must be used for reproduction, which occurs with probability 2/3. The +product of these two probabilities yields Eq. (3.30). +Appendix D. Proof of Eq. (3.31) for i = 2 and i = 3 +If i = 2, for the state to move from i to i + 1, either vℓ or vℓ+1, which is of type A, must be selected as +parent. This event occurs with probability 2r/(2r + N − 2). Then, a hyperedge containing both vℓ and vℓ+1 +must be selected, which occurs with probability 2/3. The product of these two probabilities yields Eq. (3.31). +If i = 3, then vℓ+1 and vℓ+i−2 are identical. In this case, there are two sequences of events with which the +state increases from i to i + 1. In the first sequence, either vℓ or vℓ+2 is selected as parent with probability +2r/(3r + N − 3). Then, the hyperedge composed of the parent, vℓ+1, which is of type A, and a node of +type B (i.e., vℓ−1 if the parent is vℓ, and vℓ+3 if the parent is vℓ+2) is used for reproduction with probability +1/3. In the second sequence, vℓ+1 is selected as parent with probability r/(3r + N − 3). Then, hyperedge +{ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ + 3} is used for reproduction with probability 2/3. By summing up these +probabilities, we obtain Eq. (3.31). +Appendix E. Fixation probability for the birth-death process on the weighted one-mode pro- +jection of the star 3-uniform hypergraph +In this section, we consider the weighted one-mode projection of the star 3-uniform hypergraph and +examine the fixation probability on the obtained weighted network. +19 + +We denote the weighted one-mode projection of the star 3-uniform hypergraph by G. Note that G is +a weighted complete graph; the edge between the hub node and any leaf node has weight N − 2, and the +edge between any pair of leaf nodes has weight 1. In the birth-death process on G, in each time step, we +select one node as parent with the probability proportional to its fitness. Then, the parent selects one of its +neighbors with the probability proportional to the edge weight and converts the neighbor into the parent’s +type. As is the case for the star 3-uniform hypergraph, the symmetry in G allows us to specify the state of +the birth-death process by tuple (i1, i2), where i1 ∈ {0, 1} specifies whether the hub is of type A (i.e., i1 = 1) +or B (i.e., i1 = 0), and i2 ∈ {0, 1, . . . , N − 1} is the number of leaf nodes of type A. The total number of +nodes of type A is equal to i = i1 + i2. The fixation of type A and B corresponds to (i1, i2) = (1, N − 1) and +(0, 0), respectively. +Assume that the current state is (i1, i2) = (1, i − 1) with i − 1 ∈ {0, 1, . . . , N − 2}. There are four types +of events that can occur in the next time step. In the first type of event, a leaf node of type B is selected as +parent with probability (N − i)/(ri + N − i). Then, the edge between the parent and the hub node is used +for reproduction with probability 1/2. The state after this entire event is (i1, i2) = (0, i − 1). Therefore, we +obtain +p(1,i−1)→(0,i−1) = +N − i +ri + N − i · 1 +2. +(E.1) +In the second type of event, a leaf node of type B is selected as parent with probability (N − i)/(ri + N − i). +Then, the edge between the parent and a leaf node of type A is used for reproduction with probability +(i − 1)/2(N − 2). The state after this event is (1, i − 2). Therefore, we obtain +p(1,i−1)→(1,i−2) = +N − i +ri + N − i · +i − 1 +2(N − 2). +(E.2) +In the third type of event, the hub node, which is of type A, is selected as parent with probability r/(ri+N−i). +Then, the edge between the parent and a leaf node of type B is used for reproduction with probability +(N −i)/(N −1). Alternatively, a leaf node of type A is selected as parent with probability r(i−1)/(ri+N −i). +Then, the edge between the parent and a leaf node of type B is used for reproduction with probability +(N − i)/2(N − 2). In both cases, the state after the event is (1, i). Therefore, we obtain +p(1,i−1)→(1,i) = +r +ri + N − i · N − i +N − 1 + +r(i − 1) +ri + N − i · +N − i +2(N − 2). +(E.3) +If any other event occurs, then the state remains unchanged. Therefore, we obtain +p(1,i−1)→(1,i−1) = 1 − p(1,i−1)→(0,i−1) − p(1,i−1)→(1,i−2) − p(1,i−1)→(1,i). +(E.4) +We remind that ˜x(i1,i2) represents the probability that A fixates starting with state (i1, i2). We obtain +˜x(1,i−1) = p(1,i−1)→(0,i−1)˜x(0,i−1) + p(1,i−1)→(1,i−2)˜x(1,i−2) + p(1,i−1)→(1,i)˜x(1,i) + p(1,i−1)→(1,i−1)˜x(1,i−1). +(E.5) +Now we assume that the current state is (i1, i2) = (0, i) with i ∈ {1, 2, . . . , N − 1}. There are four types +of events that can occur in the next time step. In the first type of event, a leaf node of type A is selected +as parent with probability ri/(ri + N − i). Then, the edge between the parent and the hub node is used for +reproduction with probability 1/2. The state after this event is (i1, i2) = (1, i). Therefore, we obtain +p(0,i)→(1,i) = +ri +ri + N − i · 1 +2. +(E.6) +In the second type of event, a leaf node of type A is selected as parent with probability ri/(ri + N − i). +Then, the edge between the parent and a leaf node of type B is used for reproduction with probability +(N − i − 1)/2(N − 2). The state after this event is (0, i + 1). Therefore, we obtain +p(0,i)→(0,i+1) = +ri +ri + N − i · N − i − 1 +2(N − 2) . +(E.7) +20 + +In the third type of event, the hub node, which is of type B, is selected as parent with probability 1/(ri+N−i). +Then, the edge between the parent and a leaf node of type A is used for reproduction with probability +i/(N − 1). Alternatively, a leaf node of type B is selected as parent with probability (N − i − 1)/(ri + N − i). +Then, the edge between the parent and a leaf node of type A is used for reproduction with probability +i/2(N − 2). In both cases, the state after the event is (0, i − 1). Therefore, we obtain +p(0,i)→(0,i−1) = +1 +ri + N − i · +i +N − 1 + N − i − 1 +ri + N − i · +i +2(N − 2). +(E.8) +If any other event occurs, then the state remains unchanged. Therefore, we obtain +p(0,i)→(0,i) = 1 − p(0,i)→(1,i) − p(0,i)→(0,i+1) − p(0,i)→(0,i−1). +(E.9) +Using these transition probabilities, we obtain +˜x(0,i) = p(0,i)→(1,i)˜x(1,i) + p(0,i)→(0,i+1)˜x(0,i+1) + p(0,i)→(0,i−1)˜x(0,i−1) + p(0,i)→(0,i)˜x(0,i). +(E.10) +Equation (3.48) also holds true for the one-mode projection. We use the scipy DGESV algorithm to numer- +ically solve Eq. (3.48) to obtain ˜x(0,i) and ˜x(1,i−1). Then, we obtain xi using Eq. (3.54). +We computed x1 by numerically solving Eq. (3.48) for N = 4, 5, 20, and 200. Figure E.6 compares the +obtained x1 values with those for the Moran process and the star 3-uniform hypergraph under model 1. We +find that the one-mode projection of the star 3-uniform hypergraph is weak amplifier. Note that, for N = 4, +5, and 200, the results for the one-mode projection almost overlap those for the Moran process such that the +orange lines are hidden behind the black lines in Figs. E.6(a), (b), and (d). +21 + +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +Moran +star hypergraph +one-mode projection +0 +1 +2 +3 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0 +1 +2 +3 +0.9 +0.95 +1.0 +r +0.01 +0.03 +0.05 +0.99 +0.995 +1.0 +r +0.001 +0.003 +0.005 +(a) +N = 4 +(b) +N = 5 +(c) +N = 20 +(d) +N = 200 +r +Fixation probability +Figure E.6: Fixation probability for the weighted one-mode projection of star 3-uniform hypergraphs. We compare +it with the fixation probability for the Moran process and star 3-uniform hypergraphs. (a) N = 4. (b) N = 5. (c) +N = 20. (d) N = 200. The insets in (c) and (d) magnify the results for r values smaller than and close to r = 1. In +the inset in (d), the results for the star 3-uniform hypergraph (shown by the blue line) and the Moran process (shown +by the black line) are close to that for the one-mode projection (shown by the orange line) such that the blue and +the black lines are almost hidden behind the orange line. In (a), (b), and the main panel of (d), the results for the +Moran process (shown by the black lines) are not identical but close to those for the one-mode projection (shown by +the orange lines) such that the former are hidden behind the latter. +Appendix F. Derivation of the fixation probability for the cyclic 3-uniform hypergraph under +model 2 +In this section, we derive the fixation probability for the cyclic 3-uniform hypergraph under model 2. We +assume that there are initially just two mutants that are uniformly distributed. +The fixation of type A can occur only when the two nodes that are initially of type A share at least one +hyperedge. Once such a hyperedge is selected for reproduction, all the nodes of type A are consecutive along +the cycle without being interrupted by nodes of type B. Note that the two nodes that initially have type A +may be next to each other on the cycle already in the initial condition. Therefore, to calculate the fixation +probability on the cyclic 3-uniform hypergraph, it suffices to track the number of consecutive nodes having +type A, which we denote by i. For N ≥ 5, the initial condition is either of the following three types. +22 + +Figure F.7: Initial position of the two nodes of type A on the cyclic 3-uniform hypergraph. (a) The two nodes of +type A do not share any hyperedge. (b) The two nodes of type A share two hyperedges. (c) The two nodes of type +A share one hyperedge. +First type of initial condition +In the first type of initial condition, the two nodes of type A do not share any hyperedge (see Fig. F.7(a) +for a schematic), which occurs with probability N(N −5)/ +� +2 +�N +2 +�� +. The probability that type A fixates under +this initial condition, denoted by x′ +2, is given by +x′ +2 = 0. +(F.1) +Second type of initial condition +In the second type of initial condition, the two nodes of type A share two hyperedges, i.e., these two +nodes are next to each other on the cycle (see Fig. F.7(b)). This initial condition occurs with probability +N/ +�N +2 +� +. Let x′′ +2 be the fixation probability for type A under this initial condition. In general, let x′′ +i with +i ∈ {1, 2, . . . , N − 1} be the fixation probability for type A when there are i consecutive nodes of type A and +all the other nodes are of type B. We calculate x′′ +2 by tracking the number of consecutive nodes with type A, +i.e., i, as follows. +Move of the state from i to i − 1: If i ∈ {2, . . . , N −2}, there are three types of events that can occur +next. Without loss of generality, we assume that the ℓth to the (ℓ + i − 1)th nodes are of type A and that +all the other nodes are of type B (see Fig. 3(c)). In the first type of event, the state moves from i to i − 1. +If i ≤ N − 4, either vℓ−2, vℓ−1, vℓ+i, and vℓ+i+1, which is of type B, is selected as parent with probability +4/(ri + N − i). Then, the hyperedge that contains the parent node, a node of type B, and a node of type +A is used for reproduction with probability 1/3. In this case, i decreases by one. For example, if vℓ−2 is +selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from i to +i − 1. Therefore, we obtain +pi,i−1 = +4 +ri + N − i · 1 +3. +(F.2) +If i = N − 3, there are two sequences of events through which the state decreases from i to i − 1. In the first +sequence, either vℓ−1 or vℓ−3 is selected as parent, which occurs with probability 2/(rN − 3r + 3). Then, the +hyperedge containing two nodes of type B (i.e., hyperedge {ℓ − 2, ℓ− 1, ℓ} if vℓ−1 is the parent and hyperedge +{ℓ − 4, ℓ − 3, ℓ − 2} if vℓ−3 is the parent) is used for reproduction, which occurs with probability 1/3. In the +second sequence, vℓ−2 is selected as parent with probability 1/(rN −3r +3). Then, hyperedge {ℓ−2, ℓ−1, ℓ} +or {ℓ − 4, ℓ − 3, ℓ − 2} is used for reproduction, which occurs with probability 2/3. By summing up these +probabilities, we obtain Eq. (F.2). If i = N − 2, either of the two nodes of type B, i.e., vℓ−1 or vℓ−2, must be +selected as parent for the state to move from i to i − 1. This event occurs with probability 2/(rN − 2r + 2). +Then, either hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 3, ℓ − 2, ℓ − 1} must be used for reproduction, which occurs +with probability 2/3. The product of these two probabilities coincides with Eq. (F.2). Therefore, Eq. (F.2) +holds true for any i ∈ {2, . . . , N − 2}. +23 + +(a) +(b) +V-2 +Vt-1 +Vt +Vt+1 +Ve+2 +Vt+3 +Ve+4Move of the state from i to i + 1: In the second type of event, the state moves from i to i + 1. If +i ≥ 4, either vℓ, vℓ+1, vℓ+i−2, or vℓ+i−1, which is of type A, has to be selected as parent with probability +4r/(ri + N − i). Then, the hyperedge that contains the parent node, a node of type A, and a node of type B +has to be used for reproduction, which occurs with probability 1/3. For example, if vℓ is selected as parent +and hyperedge {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from i to i + 1. Therefore, we +obtain +pi,i+1 = +4r +ri + N − i · 1 +3. +(F.3) +If i = 3, there are two sequences of events through which the state increases from i to i + 1. In the first +sequence, either vℓ or vℓ+2 is selected as parent, which occurs with probability 2r/(3r + N − 3). Then, the +hyperedge containing two nodes of type A (i.e., hyperedge {ℓ − 1, ℓ, ℓ + 1} if vℓ is the parent and hyperedge +{ℓ + 1, ℓ + 2, ℓ + 3} if vℓ+2 is the parent) is used for reproduction, which occurs with probability 1/3. In +the second sequence, vℓ+1 is selected as parent with probability r/(3r + N − 3). Then, hyperedge {ℓ − 1, ℓ, +ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ + 3} is used for reproduction, which occurs with probability 2/3. If we sum these +probabilities, we obtain Eq. (F.3). If i = 2, either of the two nodes of type A, i.e., vℓ or vℓ+1, must be +selected as parent for the state to move from i to i + 1. This event occurs with probability 2r/(2r + N − 2). +Then, either hyperedge {ℓ − 1, ℓ, ℓ + 1} or {ℓ, ℓ + 1, ℓ + 2} must be used for reproduction, which occurs with +probability 2/3. The product of these two probabilities coincides with Eq. (F.3). Therefore, Eq. (F.3) holds +true for any i ∈ {2, . . . , N − 2}. +No move of the state from i: Because i remains unchanged if any other event occurs, we obtain +pi,i = 1 − pi,i−1 − pi,i+1. +(F.4) +Derivation of x′′ +2: Therefore, the fixation probability of type A starting from i consecutive nodes of type +A, i.e., x′′ +i , satisfies +x0 = x1 = 0, +(F.5) +x′′ +i = pi,i−1x′′ +i−1 + pi,ix′′ +i + pi,i+1x′′ +i+1, +i ∈ {2, . . . , N − 2}, +(F.6) +xN−1 = xN = 1. +(F.7) +Note that x′′ +1 = x1 and x′′ +N−1 = xN−1. Similar to the analysis of the fixation probability for the complete +3-uniform hypergraph, we set +yi ≡ x′′ +i − x′′ +i−1, +i ∈ {2, . . . , N − 1}. +(F.8) +Note that �N−1 +i=2 yi = x′′ +N−1 − x′′ +1 = 1. Let +γi = pi,i−1/pi,i+1. +(F.9) +By combining Eqs. (F.4), (F.6), (F.8), and (F.9), we obtain +yi+1 = yiγi, +(F.10) +which leads to +yi = y2 +i−1 +� +k=2 +γk = x′′ +2 +i−1 +� +k=2 +γk. +(F.11) +Using Eq. (F.11), we obtain +1 = +N−1 +� +i=2 +yi = x′′ +2 +� +1 + γ2 + γ2γ3 + · · · + +N−2 +� +k=2 +γk +� += x′′ +2 +� +1 + r−1 + r−2 + · · · + r−(N−3)� += x′′ +2 +1 − r−(N−2) +1 − r−1 +. +(F.12) +24 + +Therefore, we obtain +x′′ +2 = +1 − r−1 +1 − r−(N−2) . +(F.13) +Third type of initial condition +In the third type of initial condition, the two nodes of type A share one hyperedge, implying that there +is a node of type B between the two nodes of type A. Without loss of generality, we assume that the ℓth and +the (ℓ + 2)th nodes are of type A and that all the other nodes are of type B (see Fig. F.7(c)). This initial +condition, which we denote by 2∗, occurs with probability N/ +�N +2 +� +. Now we calculate the fixation probability +for type A starting from state 2∗, which we denote by x +′′′ +2 . To ease the discussion, in the remainder of this +section, we denote by i the state in which consecutive i nodes on the cycle are of type A and the other N − i +nodes are of type B. +Move of the state from 2∗ to i = 1: If N ≥ 7, the state moves from 2∗ to i = 1 in one time step if +either of the following two types of events occurs. In the first type of event, either node vℓ−2 or vℓ+4, which +is of type B, is selected as parent. This event occurs with probability 2/(2r + N − 2). Then, the hyperedge +that contains the parent, a node of type B, and a node of type A, is used for reproduction, which occurs with +probability 1/3. For example, if vℓ−2 is the parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, +then the state moves from 2∗ to i = 1. In the second type of event, one of the nodes vℓ−1, vℓ+1, and vℓ+3, +which is of type B, is selected as parent, which occurs with probability 3/(2r + N − 2). Then, one of the +two hyperedges that contains the parent, a node of type B, and a node of type A, is used for reproduction, +which occurs with probability 2/3. For example, if vℓ−1 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} +or {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from 2∗ to i = 1. By summing up these +probabilities, we obtain the probability that the state moves from 2∗ to i = 1 as +p2∗,1 = +1 +2r + N − 2 · 8 +3. +(F.14) +If N = 6, the state moves from 2∗ to i = 1 in one time step if the following event occurs. Either vℓ−2, +vℓ−1, vℓ+1, and vℓ+3, which is of type B, is selected as parent with probability 4/(2r + 4). Then, one of the +two hyperedges that contains the parent, a node of type B, and a node of type A, is used for reproduction, +which occurs with probability 2/3. For example, if vℓ−2 is selected as parent and hyperedge {ℓ−4, ℓ−3, ℓ−2} +or {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from 2∗ to i = 1. The product of these two +probabilities coincides with Eq. (F.14). +If N = 5, the state moves from 2∗ to i = 1 in one time step if either of the following two types of events +occurs. In the first type of event, either node vℓ−1 or vℓ+3, which is of type B, is selected as parent with +probability 2/(2r + 3). Then, the hyperedge that contains the parent, a node of type B, and a node of type +A, is used for reproduction, which occurs with probability 1. For example, if vℓ−1 is the parent and any +hyperedge that contains vℓ−1 is used for reproduction, then the state moves from 2∗ to i = 1. In the second +type of event, the node vℓ+1, which is of type B, is selected as parent with probability 1/(2r + 3). Then, +one of the two hyperedges {ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ + 3} is used for reproduction, which occurs with +probability 2/3. By summing up these probabilities, we obtain Eq. (F.14). Therefore, Eq. (F.14) holds true +for any N ≥ 5. +Move of the state from 2∗ to i = 3: The state moves from 2∗ to i = 3 if either vℓ or vℓ+2, which is +of type A, is selected as parent with probability 2r/(2r + N − 2), and then, the hyperedge that contains vℓ, +vℓ+1, and vℓ+2 is used for reproduction with probability 1/3. Therefore, we obtain +p2∗,3 = +2r +2r + N − 2 · 1 +3. +(F.15) +No move of the state from 2∗: If any other event occurs at state 2∗, the state remains unchanged. +Therefore, we obtain +p2∗,2∗ = 1 − p2∗,1 − p2∗,3 = 4r + 3N − 14 +3(2r + N − 2), +(F.16) +p2∗,j = 0 if j ̸= 1, 2∗, 3. +(F.17) +25 + +Derivation of x′′′ +2 : If the state moves from 2∗ to either i = 1 or i = 3, all the nodes of type A are +consecutively numbered without being interrupted by nodes of type B afterwards. Therefore, we obtain +x +′′′ +2 = p2∗,1x1 + p2∗,2∗x +′′′ +2 + p2∗,3x′′ +3. +(F.18) +By substituting Eqs. (F.5) and (F.13) in Eq. (F.6) for i = 2, we obtain +x′′ +3 = +1 − r−2 +1 − r−(N−2) . +(F.19) +By substituting Eqs. (F.5) and (F.19) in Eq. (F.18), we obtain +x +′′′ +2 = +r − r−1 +(r + 4) +� +1 − r−(N−2)�. +(F.20) +Weighted sum to obtain x2 +By combining Eqs. (F.1), (F.13), and (F.20) with the respective probability, we obtain the fixation prob- +ability for type A when there are initially two uniformly randomly distributed mutants, given in Eq. (3.69), +as follows: +x2 = N − 5 +N − 1x′ +2 + +2 +N − 1x +′′ +2 + +2 +N − 1x +′′′ +2 += +2 +N − 1 +� +1 − r−1 +1 − r−(N−2) + +r − r−1 +(r + 4) +� +1 − r−(N−2)� +� +, +(F.21) +where N ≥ 5. +Derivation of x2 for N = 4 +For N = 4, the first type of initial condition occurs with probability 0. +The second type of initial condition occurs with the same probability as in the case of N ≥ 5, i.e., with +probability 4/ +�4 +2 +� += 2/3. In this case, the state moves from 2 to 1 if the following event occurs. Either +vℓ−1 or vℓ+2, which is of type B, is selected as parent with probability 2/(2r + 2). Then, either hyperedge +{ℓ + 2, ℓ − 1, ℓ} or {ℓ + 1, ℓ + 2, ℓ − 1} must be used for reproduction, which occurs with probability 2/3. Note +that the node indices ℓ − 1 and ℓ are equivalent to ℓ + 3 and ℓ + 4 because we interpret the node index with +modulo N. The product of these two probabilities coincides with Eq. (F.2). Therefore, Eq. (F.2) also holds +true for N = 4. Similarly, Eq. (F.3) holds true for N = 4. Therefore, using Eq. (F.12), we obtain +x′′ +2 = +r +r + 1. +(F.22) +Under the third type of initial condition, which occurs with probability 2/ +�4 +2 +� += 1/3, the state moves +from 2∗ to 1 if the following event occurs. Either vℓ−1 or vℓ+1, which is of type B, is selected as parent +with probability 2/(2r + 2). Then, either hyperedge {ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ − 1} must be used for +reproduction, which occurs with probability 2/3. Therefore, we obtain +p2∗,1 = +2 +3r + 3. +(F.23) +The state moves from 2∗ to 3 if the following event occurs. Either vℓ or vℓ+2, which is of type A, is selected +as parent with probability 2r/(2r + 2). Then, either hyperedge {ℓ, ℓ + 1, ℓ + 2} or {ℓ + 2, ℓ − 1, ℓ} must be +used for reproduction, which occurs with probability 2/3. Therefore, we obtain +p2∗,3 = +2r +3r + 3. +(F.24) +26 + +If any other event occurs at state 2∗, the state remains unchanged. Therefore, we obtain +p2∗,2∗ = 1 − p2∗,1 − p2∗,3 = r + 1 +3r + 3, +(F.25) +p2∗,j = 0 if j ̸= 1, 2∗, 3. +(F.26) +By substituting Eqs. (F.5) and (F.7) in Eq. (F.18), we obtain +x +′′′ +2 = +r +r + 1. +(F.27) +Therefore, we obtain +x2 = 2 +3x +′′ +2 + 1 +3x +′′′ +2 = +r +r + 1 +(F.28) +for N = 4. +Appendix G. Derivation of the fixation probability for the star 3-uniform hypergraph under +model 2 +We derive the fixation probability for the star 3-uniform hypergraph under model 2 in this section. We +use same notations as those in section 3.1.3. +Assume that the current state is (i1, i2) = (1, i − 1) with i − 1 ∈ {0, 1, . . . , N − 2}. There are three types +of events that can occur in the next time step. In the first type of event, a leaf node of type B is selected +as parent with probability (N − i)/(ri + N − i). Then, a hyperedge that contains the parent, the hub node, +and a different leaf node of type B, is used for reproduction with probability (N − i − 1)/(N − 2). The state +after this entire event is (i1, i2) = (0, i − 1). Therefore, we obtain +p(1,i−1)→(0,i−1) = +N − i +ri + N − i · N − i − 1 +N − 2 +. +(G.1) +In the second type of event, the hub node, which is of type A, is selected as parent with probability r/(ri + +N − i). Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B, is used +for reproduction with probability (i − 1)(N − i)/ +�N−1 +2 +� +. Alternatively, a leaf node of type A is selected as +parent with probability r(i − 1)/(ri + N − i). Then, a hyperedge that contains the parent, the hub node, and +a leaf node of type B, is used for reproduction with probability (N − i)/(N − 2). In both cases, the state +after the event is (1, i). Therefore, we obtain +p(1,i−1)→(1,i) = +r +ri + N − i · (i − 1)(N − i) +�N−1 +2 +� ++ +r(i − 1) +ri + N − i · N − i +N − 2. +(G.2) +If any other event occurs, then the state remains unchanged. Therefore, we obtain +p(1,i−1)→(1,i−1) = 1 − p(1,i−1)→(0,i−1) − p(1,i−1)→(1,i). +(G.3) +We remind that ˜x(i1,i2) is the probability that type A fixates when the initial state is (i1, i2). We obtain +˜x(1,i−1) = p(1,i−1)→(0,i−1)˜x(0,i−1) + p(1,i−1)→(1,i)˜x(1,i) + p(1,i−1)→(1,i−1)˜x(1,i−1). +(G.4) +Now we assume that the current state is (i1, i2) = (0, i) with i ∈ {1, . . . , N − 1}. Then, there are three +types of events that can occur in the next time step. In the first type of event, a leaf node of type A is +selected as parent with probability ri/(ri + N − i). Then, a hyperedge that contains the parent, the hub +node, and a different leaf node of type A, is used for reproduction with probability (i−1)/(N −2). The state +after this event is (i1, i2) = (1, i). Therefore, we obtain +p(0,i)→(1,i) = +ri +ri + N − i · i − 1 +N − 2. +(G.5) +27 + +In the second type of event, the hub node, which is of type B, is selected as parent with probability 1/(ri + +N − i). Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B, is +used for reproduction with probability i(N − i − 1)/ +�N−1 +2 +� +. Alternatively, a leaf node of type B is selected as +parent with probability (N − i − 1)/(ri + N − i). Then, a hyperedge that contains the parent, the hub node, +and a leaf node of type A, is used for reproduction with probability i/(N − 2). In both cases, the state after +the event is (0, i − 1). Therefore, we obtain +p(0,i)→(0,i−1) = +1 +ri + N − i · i(N − i − 1) +�N−1 +2 +� ++ N − i − 1 +ri + N − i · +i +N − 2. +(G.6) +If any other event occurs, then the state remains unchanged. Therefore, we obtain +p(0,i)→(0,i) = 1 − p(0,i)→(1,i) − p(0,i)→(0,i−1). +(G.7) +Using these transition probabilities, we obtain +˜x(0,i) = p(0,i)→(1,i)˜x(1,i) + p(0,i)→(0,i−1)˜x(0,i−1) + p(0,i)→(0,i)˜x(0,i). +(G.8) +Equations (G.4) and (G.8) lead to +˜x(1,i) = αi˜x(1,i−1) + (1 − αi)˜x(0,i−1) +(G.9) +and +˜x(0,i) = βi˜x(1,i) + (1 − βi)˜x(0,i−1), +(G.10) +respectively, where +αi = 1 + (N − i − 1)(N − 1) +r(i − 1)(N + 1) +, +(G.11) +βi = +r(i − 1)(N − 1) +r(i − 1)(N − 1) + (N − i − 1)(N + 1). +(G.12) +We rewrite Eqs. (G.9) and (G.10) as +ϱi = Aiϱi−1, +(G.13) +where ϱi = (˜x(1,i), ˜x(0,i))⊤, and Ai is the 2 × 2 matrix given by +Ai = +� αi +1 − αi +αiβi +1 − αiβi +� +. +(G.14) +Equation (G.13) yields +ϱi = AiAi−1 · · · A2ϱ1. +(G.15) +Therefore, +� +1 +1 +� += ϱN−1 = AN−1AN−2 · · · A2ϱ1. +(G.16) +Equation (3.54) also holds true for model 2. +Therefore, we obtain ˜x(1,1) from Eq. (G.16), ˜x(0,2) from +Eq. (G.13), and finally x2 from Eq. (3.54). +By analytically solving Eq. 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Wu, Close spatial arrangement of mutants favors and disfavors fixation, PLoS Comput. +Biol., 15, e1007212 (2019). +[60] N. Sharma and A. Traulsen, Suppressors of fixation can increase average fitness beyond amplifiers of +selection, Proc. Natl. Acad. Sci. USA, 119, e2205424119 (2022). +31 + diff --git a/KtE4T4oBgHgl3EQf7w7r/content/tmp_files/load_file.txt b/KtE4T4oBgHgl3EQf7w7r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca04aab2c3a1ea84e5ee3f992eba421ebcf6e715 --- /dev/null +++ b/KtE4T4oBgHgl3EQf7w7r/content/tmp_files/load_file.txt @@ -0,0 +1,1899 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf,len=1898 +page_content='Fixation dynamics on hypergraphs Ruodan Liua, Naoki Masudaa,b,∗ aDepartment of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA bComputational and Data-Enabled Sciences and Engineering Program, State University of New York at Buffalo, Buffalo, NY 14260-5030, USA Abstract Hypergraphs have been a useful tool for analyzing population dynamics such as opinion formation and the public goods game occurring in overlapping groups of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the present study, we propose and analyze evolutionary dynamics on hypergraphs, in which each node takes one of the two types of different but constant fitness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For the same dynamics on conventional networks, under the birth-death process and uniform initial conditions, most networks are known to be amplifiers of natural selection, which by definition enhances the difference in the strength of the two completing types in terms of the probability that the mutant type fixates in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In contrast, we show that a vast majority of hypergraphs are suppressors of selection under the same conditions by combining theoretical and numerical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We also show that this suppressing effect is not explained by one-mode projection, which is a standard method for expressing hypergraph data as a conventional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Our results suggest that the modeling framework for structured populations in addition to the specific network structure is an important determinant of evolutionary dynamics, paving a way to studying fixation dynamics on higher-order networks including hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Keywords: Evolutionary dynamics, fixation probability, constant selection, hypergraph, amplifier, suppressor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Introduction Populations of individuals, cells, and habitats, on which evolutionary processes take place often have structure that may be described by networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Evolutionary graph theory enables us to mathematically and computationally investigate population dynamics in which multiple types of different fitness compete on networks under a selection pressure [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A minimal evolutionary dynamics model on graphs, or networks, is to assume that there are two types of different fitness values, 1 and r, which are constant over time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', constant selection), and that each node is occupied by either type, which can change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A question then is which type eventually occupies the entire population, which is called the fixation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In physics and mathematics literature, the same model is often called the biased voter model, and the fixation is often called the consensus [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Crucially, the fixation probability of each type, as well as other properties of the evolutionary dynamics such as the average time to fixation, depends on the network structure in addition to the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Application of evolutionary graph theory to social dilemma games has also been successful in giving analytical insights into various conditions under which cooperation is favored over defection [1– 3, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Evolutionary graph theory with conventional networks assumes that interaction between nodes occurs pairwise because each edge in a conventional network represents direct connectivity between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' However, in reality, more than two individuals may simultaneously interact and compete in evolutionary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, an evolutionary dynamics under constant selection can be regarded as a model of opinion dynamics in a population of human or animal individuals, in which they may meet in groups with more than two individuals for opinion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' As another example, the public goods game is naturally ∗Corresponding author Email address: naokimas@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='edu (Naoki Masuda) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='05343v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='soc-ph] 13 Jan 2023 defined for group interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' each individual in the group decides whether or not to contribute to a collective good, and all individuals will receive a share of an augmented amount of the collective good regardless of whether or not they contributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Hypergraphs are a natural extension of conventional networks to the case of group interactions [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In a hypergraph, the concept of edge is extended to the hyperedge, which represents a unit of interaction and may contain more than two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Population dynamics on hypergraphs such as evolutionary dynamics of social dilemma games [15, 16] and opinion formation [17, 18] have been investigated (see [13, 19] for reviews including different types of dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the present study, we study constant-selection evolutionary dynamics on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We propose two such models and formulate the fixation probability of mutants using theory based on Markov chains, which extends the same approach for conventional networks to the case of hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, we ask a key question in the study of constant-selection dynamics on networks: whether the given network is amplifier or suppressor of natural selection [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Suppose that the resident and mutant type have constant fitness values 1 and r, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We anticipate that the mutant type is more likely to fixate than the resident type under the same condition if r > 1 and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In fact, how much the fixation probability of the mutant type increases as one increases r depends on the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Some networks are amplifiers of selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' on these networks, a single mutant has a higher probability of fixation than in the well-mixed population, corresponding to the Moran process, at any r > 1 and a smaller fixation probability than in the Moran process at any r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Other networks are suppressors of selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' on these networks, a single mutant has a lower fixation probability than in the Moran process at any r > 1 and a higher fixation probability than in the Moran process at any r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Under the so-called birth-death processes, which we focus on, most networks are known to be amplifiers, at least when the initial mutant is located on a node selected uniformly at random [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Research has discovered various classes of amplifiers [5, 23–27] while few for suppressors [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show that, contrary to these results, most hypergraphs are suppressors of natural selection even under the birth-death process and uniform initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We reach this conclusion by theoretical investigations for hypergraphs with high symmetry and numerical simulations on empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The Python codes for generating the numerical results in this article are available at GitHub [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Models We introduce two models of evolutionary dynamics on undirected hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Let H be an undirected hypergraph with node set V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N}, where N is the number of nodes, and hyperedge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each e ∈ E, where e ⊂ V and e ̸= ∅, is a hyperedge, intuitively representing group interaction among the nodes belonging to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If each e ∈ E is a set containing exactly two nodes, H is a conventional undirected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We assume that H is connected in the sense that there is a hyperedge intersecting both W and V − W for every non-empty proper subset W of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We define model 1 by the following discrete-time evolutionary dynamics on hypergraph H, which extends the birth-death process for conventional networks and the Moran process in well-mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We assume that there are two types of individuals, referred to as A and B, and that A and B have fitness r and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We refer to A as the mutant type and B as the resident type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In each time step, we select one node for reproduction with the probability proportional to its fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We call this node the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the parent node selects one of the hyperedges to which it belongs, denoted by e, with the equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Finally, the parent converts the type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', A or B) of all other nodes belonging to e into the parent’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We repeat this process until all nodes in V have the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Once this unanimity configuration is reached, which is the fixation, no node will change its type even if one further runs the evolutionary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We examine the probability that every node eventually has type A, which we call the fixation probability of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Model 2 is the same as model 1 in that there are two types of individuals with fitness r and 1, that we select a parent node in each time step with the probability proportional to its fitness, and that the parent selects one of its hyperedges, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Differently from the model 1, the parent node converts all the other nodes belonging to e into the parent’s type if and only if the parent’s type is the majority in e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', if more than half of the nodes in e including the parent have the same type as the parent’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Model 2 is an evolutionary dynamics variant of opinion formation models under the majority rule [9, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' When the network is the complete graph, which is a conventional undirected network and therefore a hypergraph, model 1 is called the Moran process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For the Moran process, the fixation probability of type A 2 when there are initially i individuals of type A, denoted by xi, is given by [1] xi = 1 − 1/ri 1 − 1/rN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Results for synthetic hypergraphs In this section, for three model hypergraphs that are mathematically convenient, we calculate the fixation probability for mutant type A when there are initially i nodes of type A and N − i nodes of type B, for both models 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation probability depends on the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We select the i mutant nodes from the N nodes uniformly at random, which is called the uniform initialization [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For any hypergraph of size N, at any time step, either type A or B inhabits each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, there are 2N states in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation probability for type A depends on each state, not just on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To know the fixation probability for type A for the initial states with i mutants, we need to solve a linear system of 2N − 2 unknowns, where each unknown is the fixation probability for an initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that we safely excluded the two unknowns corresponding to the two initial states in which all nodes unanimously have type A or B, in which case the fixation probability for type A is trivially equal to 1 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To solve a linear system with 2N − 2 unknowns is daunting except when N is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we analyze three types of symmetric hypergraphs in which all or most nodes are structurally equivalent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For these hypergraphs, we only need to track the number of nodes with type A among the structurally equivalent nodes, which drastically reduces the dimension of the linear system to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this section, we denote the number of nodes of type A in the entire hypergraph by i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation of type A and B corresponds to i = N and i = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Model 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Complete 3-uniform hypergraph We have mentioned that our model 1 on the complete graph is equivalent to the Moran process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To investigate whether model 1 on counterparts of the complete graph for hypergraphs is equivalent to the Moran process, we consider the complete 3-uniform hypergraph [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A complete 3-uniform hypergraph on node set V is defined by hyperedge set E, which is the set of all subsets of V containing just three nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In other words, E = {{v1, v2, v3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' v1, v2, v3 ∈ V, v1 ̸= v2, v1 ̸= v3, v2 ̸= v3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show the complete 3-uniform hypergraph on four nodes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 1(a) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Figure 1: Examples of 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) Complete 3-uniform hypergraph with N = 4 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) Cyclic 3-uniform hypergraph with N = 8 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) Star 3-uniform hypergraph with N = 5 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each colored oval represents a hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this section, we refer to i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', the number of nodes of type A, as the state of the evolutionary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that knowing the dynamics of i is enough for completely understanding the evolutionary dynamics on 3 (a) (b) (c)the complete 3-uniform hypergraph owing to its symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Under model 1, state i either remains unchanged or moves to i−2, i−1, i+1, or i+2 in a single time step of the evolutionary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This is because every hyperedge of the complete 3-uniform hypergraph has three nodes such that there are at most two nodes that flip their type in a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We denote the (N + 1) × (N + 1) transition probability matrix by P = [pi,j], where pi,j is the probability that the state moves from i to j in a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' At state i, the probability that a node of type A and B is selected as parent is equal to ri/(ri + N − i) and (N − i)/(ri + N − i), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If the selected parent node is of type A, a hyperedge containing the parent and two nodes of type B is used for reproduction with probability �N−i 2 � / �N−1 2 � , where �� represents the binomial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i increases by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a hyperedge containing the parent, a different node of type A, and a node of type B is used for reproduction with probability �i−1 1 ��N−i 1 � / �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i increases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Otherwise, a hyperedge containing the parent and two other nodes of type A is used for reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If the parent is of type B, a hyperedge containing the parent and two nodes of type A is used for reproduction with probability �i 2 � / �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i decreases by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a hyperedge containing the parent, a node of type A, and a different node of type B is used for reproduction with probability �i 1 ��N−i−1 1 � / �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i decreases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Otherwise, a hyperedge containing the parent and two other nodes of type B is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, the transition probabilities are given by p0,0 = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1) pN,N = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2) pi,i−2 = N − i ri + N − i · �i 2 � �N−1 2 � = N − i ri + N − i · i(i − 1) (N − 1)(N − 2), i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3) pi,i−1 = N − i ri + N − i · �i 1 ��N−i−1 1 � �N−1 2 � = N − i ri + N − i · 2i(N − i − 1) (N − 1)(N − 2), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4) pi,i+1 = ri ri + N − i · �i−1 1 ��N−i 1 � �N−1 2 � = ri ri + N − i · 2(i − 1)(N − i) (N − 1)(N − 2), i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) pi,i+2 = ri ri + N − i · �N−i 2 � �N−1 2 � = ri ri + N − i · (N − i)(N − i − 1) (N − 1)(N − 2) , i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6) p1,1 = 1 − p1,0 − p1,2 − p1,3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7) pi,i = 1 − pi,i−2 − pi,i−1 − pi,i+1 − pi,i+2, i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8) pN−1,N−1 = 1 − pN−1,N−3 − pN−1,N−2 − pN−1,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9) All the other entries of P are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, P is a pentadiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' States i = 0 and i = N are absorbing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Denote by xi the probability of ending up in state N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', fixation probability of the mutant type A, when the initial state is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain x0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10) x1 = p1,0x0 + p1,1x1 + p1,2x2 + p1,3x3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='11) xi = pi,i−2xi−2 + pi,i−1xi−1 + pi,ixi + pi,i+1xi+1 + pi,i+2xi+2, i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='12) xN−1 = pN−1,N−3xN−3 + pN−1,N−2xN−2 + pN−1,N−1xN−1 + pN−1,NxN, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13) xN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14) In vector notation, we can concisely write Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14) as x = Px, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='15) where x = (x0, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , xN)⊤, and ⊤ represents the transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='15) is equivalent to (P − I)x = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='16) 4 where I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='16), we obtain Mx = b, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='17) where b = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , 0, 1)⊤, and M = � � � � � � � � � 1 0 0 0 0 · · 0 0 0 0 p1,0 p1,1 − 1 p1,2 p1,3 0 · · 0 0 0 0 p2,0 p2,1 p2,2 − 1 p2,3 p2,4 · · 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 0 0 0 0 · · pN−1,N−3 pN−1,N−2 pN−1,N−1 − 1 pN−1,N 0 0 0 0 0 · · 0 0 0 1 � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='18) Like P, the (N + 1) × (N + 1) matrix M is a pentadiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' PTRANS-I and PTRANS-II are numerical algorithms for efficiently solving pentadiagonal linear systems [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that we need to calculate x although we only need x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' These two algorithms run in O(log n) time, where n is the number of unknowns, and they are about ten times faster than the SciPy algorithm for banded matrices, scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='linalg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='solve banded [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We use the PTRANS-II built in a Python package pentapy [36] to calculate the fixation probability as a function of r unless N is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For a given value of r, the calculation requires O(log N) time, which is much faster than solving a full linear system of 2N − 2 unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For small complete 3-uniform hypergraphs, one can analytically calculate x1 by directly solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain x1 = r2 r2 + 2r + 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='19) for N = 4 and x1 = r2(8r2 + 12r + 1) 8r4 + 28r3 + 33r2 + 28r + 8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='20) for N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='20) with x1 for the Moran process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1) with i = 1) with N = 4 and N = 5 as a function of r in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2(a) and 2(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Although the results are shown by blue lines, they are hidden behind the orange lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The figure indicates that these complete 3-uniform hypergraphs are suppressors because their x1 is smaller than that for the Moran process for r > 1 and larger for r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We also obtained x1 by numerically solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='17) for N = 20 and N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2(c) and 2(d) for N = 20 and N = 200, respectively, indicate that these larger complete 3- uniform hypergraphs are also suppressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we conclude that the complete 3-uniform hypergraphs are suppressors under the evolutionary dynamics described by model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Cyclic 3-uniform hypergraph The fixation probability as a function of r for undirected cycle graphs is the same as that for the Moran process because the cycle graphs are so-called isothermal graphs [1, 5, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To examine whether the same equivalence result holds true for hypergraphs, we consider an extension of the cycle graph to the hypergraph, which we call the cyclic 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A cyclic 3-uniform hypergraph consists of a node set V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N} and a hyperedge set E = {{1, 2, 3}, {2, 3, 4}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , {N − 2, N − 1, N}, {N − 1, N, 1}, {N, 1, 2}}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', any three consecutively numbered nodes with a periodic boundary condition form a hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The cyclic 3-uniform hypergraph with N = 3 nodes is the complete 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we consider cyclic 3-uniform hypergrapys with N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show the cyclic 3-uniform hypergraph on 8 nodes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We assume that there is initially one individual of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, under model 1, all the nodes of type A are consecutively numbered at any time, without being interrupted by nodes of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, similarly to the analysis of evolutionary dynamics on cycles [38, 39], it suffices to track the number of nodes of type A, which we again denote by i, to understand the evolutionary dynamics on this hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 Moran complete cyclic star 0 1 2 3 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='005 (a) N = 4 (b) N = 5 (c) N = 20 (d) N = 200 r Fixation probability Figure 2: Fixation probability for different hypergraph models under model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare the Moran process, which is the baseline, complete 3-uniform hypergraphs, cyclic 3-uniform hypergraphs, and star 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (d) N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The insets in (c) and (d) magnify the results for r values smaller than and close to r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In these insets and main panel (b), the results for the complete 3-uniform hypergraph are close to those for the cyclic hypergraph such that the blue lines are almost hidden behind the orange lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In (a), the two results are exactly the same such that the blue line is completely hidden behind the orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Figure 3: State transitions in the cyclic 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) A state with just one node of type A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', i = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) A state with just one node of type B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', i = N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) A state with more than one nodes of each type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', 2 ≤ i ≤ N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 6 (a) (b) Vt-2 Vt-1 Vt V++1 Vt+2 Vt-2 Ve-1 Vt Vt+1 Vt+2 B B (c) Vt-2 Vt-1 Vt Vt+1 Vt+i-2 Vt+i-1 V+i V+i+1 AWhen i = 1, there are three types of events that can occur next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Without loss of generality, we assume that the ℓth node is the only node of type A (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3(a) for a schematic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state moves from i = 1 to i = 0 in one time step such that B fixates in the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first case, either node vℓ−2 or vℓ+2, which is of type B, is selected as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If N ≥ 5, these two nodes are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, this event occurs with probability 2/(r + N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and vℓ is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−2 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second case, either vℓ−1 or vℓ+1 is selected as parent, which occurs with probability 2/(r + N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−1 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain p1,0 = 2 r + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='21) In fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='21) also holds true for N = 4 although vℓ−2 and vℓ+2 are identical nodes when N = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' see Appendix A for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, the state moves from i = 1 to i = 3 whenever vℓ is selected as parent, which occurs with probability r/(r + N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p1,3 = r r + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='22) If any other event occurs, then i = 1 remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p1,1 = 1 − p1,0 − p1,3 = N − 3 r + N − 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='23) p1,j = 0 if j ̸= 0, 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='24) When i = N − 1, there are similarly three types of events that can occur in a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Without loss of generality, we assume that the ℓth node is the only node of type B (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3(b) for a schematic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state moves from i = N − 1 to i = N − 3 whenever vℓ is selected as parent, which occurs with probability 1/[r(N − 1) + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pN−1,N−3 = 1 r(N − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='25) Alternatively, the state moves from i = N − 1 to i = N such that type A fixates in the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first case, either node vℓ−2 or vℓ+2, which is of type A, is selected as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If N ≥ 5, these two nodes are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, this event occurs with probability 2r/[r(N −1)+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and vℓ is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second case, either vℓ−1 or vℓ+1 is selected as parent, which occurs with probability 2r/[r(N − 1) + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain pN−1,N = 2r r(N − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='26) In fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='26) also holds true for N = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' see Appendix B for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If any other event occurs, then i = N − 1 remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pN−1,N−1 = 1 − pN−1,N−3 − pN−1,N = r(N − 3) r(N − 1) + 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='27) pN−1,j = 0 if j ̸= N − 3, N − 1, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='28) When i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}, there are five types of possible events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Without loss of generality, we assume that the ℓth to the (ℓ + i − 1)th nodes are of type A and that all other nodes are of type B (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3(c) for a schematic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If ℓ + i − 1 is larger than N, then we interpret ℓ + i − 1 as the number modulo N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', 7 ℓ + i − 1 − N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' the same convention applies in the following text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, either node vℓ−1 or vℓ+i, which is of type B, is selected as parent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' this event occurs with probability 2/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and two nodes of type A is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, the state i decreases by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−1 is selected as parent and hyperedge {ℓ − 1, ℓ, ℓ + 1} is selected, then the state moves from i to i − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i−2 = 2 ri + N − i · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='29) In the second type of event, either vℓ−2, vℓ−1, vℓ+i, or vℓ+i+1, which is of type B, is selected as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i ≤ N − 4, these four nodes are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, this event occurs with probability 4/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent, a node of type B, and a node of type A is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, the state i decreases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−2 is selected as parent and hyperedge {ℓ−2, ℓ−1, ℓ} is used for reproduction, then the state moves from i to i−1 because vℓ turns from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i−1 = 4 ri + N − i · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='30) In fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='30) also holds true for i = N − 3 and i = N − 2 although some of vℓ−2, vℓ−1, vℓ+i, and vℓ+i+1 are identical nodes when i = N − 3 or i = N − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' see Appendix C for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the third type of event, either vℓ, vℓ+1, vℓ+i−2, or vℓ+i−1, which is of type A, is selected as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i ≥ 4, these four nodes are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, this event occurs with probability 4r/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent node, a node of type A, and a node of type B is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, state i increases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ is selected as parent and hyperedge {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from i to i + 1 because vℓ−1 turns from B to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i+1 = 4r ri + N − i · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='31) In fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='31) also holds true for i = 2 and i = 3 although some of vℓ, vℓ+1, vℓ+i−2, and vℓ+i−1 are identical nodes when i = 2 or i = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' see Appendix D for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the fourth type of event, either node vℓ or vℓ+i−1, which is of type A, is selected as parent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' this event occurs with probability 2r/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and two nodes of type B is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, state i increases by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from i to i + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i+2 = 2r ri + N − i · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='32) If any other event occurs, then i remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i = 1 − pi,i−2 − pi,i−1 − pi,i+1 − pi,i+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='33) All the other entries of transition probability matrix P are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='17), we obtain x1 = r2 r2 + 2r + 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='34) for N = 4 and x1 = r2(6r2 + 8r + 1) 6r4 + 20r3 + 23r2 + 20r + 6 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='35) for N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='35) with x1 for the Moran process with N = 4 and N = 5 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2(a) and 2(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that these cyclic 3-uniform hypergraphs are suppressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The result for N = 4 given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='34) coincides with that for the complete 3-uniform hypergraph given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For larger N, we use the same numerical method for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='17) as that for the complete 3-uniform hypergraph because P is a pentadiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show the thus calculated x1 for N = 20 and N = 200 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2(c) and 2(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' These figures confirm that these larger cyclic 3-uniform hypergraphs are 8 also suppressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we conclude that the cyclic 3-uniform hypergraph is a suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Star 3-uniform hypergraph The conventional star graphs are a strong amplifier [5, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To examine whether counterparts of the star graph for hypergraphs are also amplifiers, we define the star 3-uniform hypergraph as follows and examine the fixation probability of the mutant on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A star 3-uniform hypergraph consists of a node set V with a single hub node and N − 1 leaf nodes, and hyperedges each of which is of size three and consists of the hub node and a pair of N − 1 leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We use all the �N−1 2 � pairs of leaf nodes to form hyperedges such that the hub node belongs to �N−1 2 � hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each leaf node belongs to N − 1 hyperedges and are structurally equivalent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The star 3-uniform hypergraph with N = 3 is the complete 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we consider star 3-uniform hypergraphs with N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show the 3-uniform star hypergraph on 5 nodes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 1(c) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Owing to the structural equivalence among the N − 1 leaf nodes, the state of the evolutionary dynamics on the star 3-uniform hypergraph is completely determined by the type on the hub (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', either A or B) and the number of leaf nodes of type A, which ranges between 0 and N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we denote the state by (i1, i2), where i1 = 1 or 0 if the hub node is of type A or B, respectively, and i2 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N −1} represents the number of the leaf nodes of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The total number of nodes of type A is given by i = i1 + i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation of type A and B corresponds to (i1, i2) = (1, N − 1) and (0, 0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Assume that there are currently i nodes of type A and that the state is (i1, i2) = (1, i − 1) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are five types of events that can occur next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, a leaf node of type B is selected as parent, which occurs with probability (N − i)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a different leaf node of type B is used for reproduction, which occurs with probability (N − i − 1)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this entire event is (i1, i2) = (0, i − 1) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(0,i−1) = N − i ri + N − i N − i − 1 N − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='36) In the second type of event, a leaf node of type B is selected as parent with probability (N − i)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a leaf node of type A is used for reproduction, which occurs with probability (i − 1)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (0, i − 2) with i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(0,i−2) = N − i ri + N − i i − 1 N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='37) In the third type of event, the hub node, which is of type A, is selected as parent, which occurs with probability r/(ri+N −i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B is used for reproduction, which occurs with probability (i − 1)(N − i)/ �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a leaf node of type A is selected as parent, which occurs with probability r(i − 1)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a leaf node of type B is used for reproduction, which occurs with probability (N − i)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In both cases, the state after the event is (1, i) with i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i) = r ri + N − i (i − 1)(N − i) �N−1 2 � + r(i − 1) ri + N − i N − i N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='38) In the fourth type of event, the hub node is selected as parent with probability r/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent and two leaf nodes of type B is used for reproduction, which occurs with probability �N−i 2 � / �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (1, i+1) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N −2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i+1) = r ri + N − i �N−i 2 � �N−1 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='39) If any other event occurs, then the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i−1) = 1 − p(1,i−1)→(0,i−1) − p(1,i−1)→(0,i−2) − p(1,i−1)→(1,i) − p(1,i−1)→(1,i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='40) 9 We denote the probability that type A fixates starting with state (i1, i2) by ˜x(i1,i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain ˜x(1,i−1) =p(1,i−1)→(0,i−1)˜x(0,i−1) + p(1,i−1)→(0,i−2)˜x(0,i−2) + p(1,i−1)→(1,i)˜x(1,i) + p(1,i−1)→(1,i+1)˜x(1,i+1) + p(1,i−1)→(1,i−1)˜x(1,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='41) Assume that the current state is (i1, i2) = (0, i) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are five types of events that can occur next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, a leaf node of type A is selected as parent with probability ri/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a different leaf node of type A is used for reproduction with probability (i − 1)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this entire event is (i1, i2) = (1, i) with i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(1,i) = ri ri + N − i i − 1 N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='42) In the second type of event, a leaf node of type A is selected as parent with probability ri/(ri+N −i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a leaf node of type B is used for reproduction, which occurs with probability (N − i − 1)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (1, i + 1) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(1,i+1) = ri ri + N − i N − i − 1 N − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='43) In the third type of event, the hub node, which is of type B, is selected as parent, with probability 1/(ri+N−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B, is used for reproduction, which occurs with probability i(N − i − 1)/ �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a leaf node of type B is selected as parent with probability (N − i − 1)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a leaf node of type A is used for reproduction, which occurs with probability i/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In both cases, the state after the event is (0, i − 1) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i−1) = 1 ri + N − i i(N − i − 1) �N−1 2 � + N − i − 1 ri + N − i i N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='44) In the fourth type of event, the hub node is selected as parent with probability 1/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and two leaf nodes of type A is used for reproduction, which occurs with probability �i 2 � / �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (0, i − 2) with i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i−2) = 1 ri + N − i �i 2 � �N−1 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='45) If any other event occurs, then the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i) = 1 − p(0,i)→(1,i) − p(0,i)→(1,i+1) − p(0,i)→(0,i−1) − p(0,i)→(0,i−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='46) Using these transition probabilities, we obtain ˜x(0,i) =p(0,i)→(1,i)˜x(1,i) + p(0,i)→(1,i+1)˜x(1,i+1) + p(0,i)→(0,i−1)˜x(0,i−1) + p(0,i)→(0,i−2)˜x(0,i−2) + p(0,i)→(0,i)˜x(0,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='47) We rewrite Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='41) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='47) as ˜x = P ˜x, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) where ˜x = (˜x(0,0), ˜x(0,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , ˜x(0,N−1), ˜x(1,0), ˜x(1,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , ˜x(1,N−1))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The 2N ×2N stochastic matrix P is given by P = � C D E F � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='49) 10 where C, D, E, and F are N × N matrices given by C = � � � � � � � 1 0 0 · · 0 0 0 p(0,1)→(0,0) p(0,1)→(0,1) 0 · · 0 0 0 p(0,2)→(0,0) p(0,2)→(0,1) p(0,2)→(0,2) · · 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 0 0 · · p(0,N−1)→(0,N−3) p(0,N−1)→(0,N−2) p(0,N−1)→(0,N−1) � � � � � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='50) D = � � � � � � � � � 0 0 0 0 · · 0 0 0 0 p(0,1)→(1,1) p(0,1)→(1,2) 0 · · 0 0 0 0 0 p(0,2)→(1,2) p(0,2)→(1,3) · · 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 0 0 0 · · 0 p(0,N−2)→(1,N−2) p(0,N−2)→(1,N−1) 0 0 0 0 · · 0 0 p(0,N−1)→(1,N−1) � � � � � � � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='51) E = � � � � � � � � � p(1,0)→(0,0) 0 0 · · 0 0 0 p(1,1)→(0,0) p(1,1)→(0,1) 0 · · 0 0 0 0 p(1,2)→(0,1) p(1,2)→(0,2) · · 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 0 0 · · p(1,N−2)→(0,N−3) p(1,N−2)→(0,N−2) 0 0 0 0 · · 0 p(1,N−1)→(0,N−2) p(1,N−1)→(0,N−1) � � � � � � � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='52) and F = � � � � � � � p(1,0)→(1,0) p(1,0)→(1,1) p(1,0)→(1,2) 0 · · 0 0 0 0 p(1,1)→(1,1) p(1,1)→(1,2) p(1,1)→(1,3) · · 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 0 0 0 · · 0 p(1,N−2)→(1,N−2) p(1,N−2)→(1,N−1) 0 0 0 0 · · 0 0 1 � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='53) The transition matrix P is a sparse matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We use the scipy implementation of the DGESV routine of LAPACK, scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='linalg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='solve, to numerically solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) to obtain ˜x(0,i) and ˜x(1,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We remind that xi is the probability that type A fixates when there are initially i nodes of type A that are selected uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are �N−1 i−1 � states in which i − 1 nodes have type A and the hub has type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are �N−1 i � states in which i nodes have type A and the hub has type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain xi = �N−1 i−1 � �N−1 i−1 � + �N−1 i � ˜x(1,i−1) + �N−1 i � �N−1 i−1 � + �N−1 i � ˜x(0,i) = i N ˜x(1,i−1) + N − i N ˜x(0,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='54) We analytically solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='54) to obtain x1 = r2(3r + 5) 4(3r3 + 14r2 + 18r + 9) + 9r2 4(3r2 + 5r + 3) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='55) for N = 4 and x1 = r2(72r3 + 202r2 + 145r + 6) 5(72r5 + 490r4 + 1025r3 + 1070r2 + 880r + 288) 11 + 4r2(72r2 + 94r + 4) 5(72r4 + 202r3 + 217r2 + 202r + 72) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='56) for N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='55) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='56) with x1 for the Moran process with N = 4 and N = 5 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2(a) and 2(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that these star 3-uniform hypergraphs are suppressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We then computed x1 by numerically solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) for N = 20 and N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Figures 2(c) and 2(d) compare the obtained x1 values with those for the Moran process with N = 20 and N = 200, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' These figures indicate that these larger star 3-uniform hypergraphs are also suppressors although the degree of suppression is smaller for larger star 3-uniform hypergraphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2(d) shows that the difference between the star 3-uniform hypergraph and the Moran process in terms of x1 is tiny when N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In sum, we conclude that the star 3-uniform hypergraph is a suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Comparison with conventional networks obtained by one-mode projection A lossy representation of a hypergraph as a conventional network is the one-mode projection, in which two nodes are adjacent by an edge if and only if they belong to at least one common hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A weighted network version of the one-mode projection defines the edge weight by the number of hyperedges that the two nodes share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The one-mode projection is a common approach for analyzing dynamics on hypergraphs including evolutionary dynamics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The one-mode projections of the complete and cyclic 3-uniform hyper- graphs are regular networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', networks in which all the nodes have the same degree), in the case of both unweighted and weighted variants of the one-mode projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the isothermal theorem [5] guarantees that the birth-death process on the one-mode projections of the complete and cyclic 3-uniform hypergraphs is equivalent to the Moran process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, our result that the complete and cyclic 3-uniform hypergraphs are suppressors is not an artifact of one-mode projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The same argument does not apply for the star 3-uniform hypergraph because the one-mode projection of the star 3-uniform hypergraph is not a regular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Furthermore, the star 3-uniform hypergraph are only analogously similar to the conventional star graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In fact, leaf nodes are adjacent to each other in the star 3-uniform hypergraph, whereas they are not in the conventional star graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the direct connection between leaf nodes might be a reason why the star 3-uniform hypergraph is a suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To exclude this possibility, using similar analytical techniques, we investigated the fixation probability for the weighted one- mode projection of the star 3-uniform hypergraph, which is a weighted complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that the unweighted one-mode projection of the star 3-uniform hypergraph is the unweighted complete graph, which is trivially equivalent to the Moran process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' As we show in Appendix E, we have found that the obtained weighted complete graph is an amplifier although the amplifying effect is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, our result that the star 3-uniform hypergraph is a suppressor is not expected from the one-mode projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Model 2 In this section, we analyze the fixation probability for the birth-death process governed by model 2 on the complete, cyclic, and star 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Complete 3-uniform hypergraph On the complete 3-uniform hypergraph, the state i either remains unchanged or moves to i − 1 or i + 1 in a single time step because all the hyperedges are composed of three nodes such that there are at most one node that changes the state under the majority rule given by model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' At state i, the probability that a node of type A and B is selected as parent is equal to ri/(ri + N − i) and (N − i)/(ri + N − i), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If the parent is of type A, then the following three types of events are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' First, if a hyperedge containing the parent and two nodes of type B is used for reproduction with probability �N−i 2 � / �N−1 2 � , then the state does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Second, if a hyperedge containing the parent, a different node of type A, and a node of type B is used for reproduction with probability �i−1 1 ��N−i 1 � / �N−1 2 � , then the state moves from i to i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Third, if a hyperedge containing the parent and two other nodes of type A is used for reproduction with the remaining probability, then the state does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If the parent is of type B, then the following three types of events are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' First, if a hyperedge containing the parent and two nodes of type A is used for reproduction with probability �i 2 � / �N−1 2 � , then the state does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If a hyperedge containing the parent, a node of type A, and a different node of type B is used for reproduction with probability �i 1 ��N−i−1 1 � / �N−1 2 � , then the state moves from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Third, if a hyperedge containing the parent and two other nodes of type 12 B is used for reproduction with the remaining probability, then the state does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, the transition probabilities are given by p0,0 = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='57) pN,N = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='58) pi,i−1 = N − i ri + N − i · �i 1 ��N−i−1 1 � �N−1 2 � = N − i ri + N − i · 2i(N − i − 1) (N − 1)(N − 2), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='59) pi,i+1 = ri ri + N − i · �i−1 1 ��N−i 1 � �N−1 2 � = ri ri + N − i · 2(i − 1)(N − i) (N − 1)(N − 2), i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='60) pi,i = 1 − pi,i−1 − pi,i+1, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='61) All the other entries of P are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, P is a tridiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We note that the states i = 0 and i = N are absorbing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation probability of type A starting from state i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', xi, satisfies x0 = x1 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='62) xi = pi,i−1xi−1 + pi,ixi + pi,i+1xi+1, i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='63) xN−1 = xN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='64) Similar to the analysis of the fixation probability for the Moran process [1], we set yi ≡ xi − xi−1, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='65) Note that �N i=1 yi = xN − x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Let γi = pi,i−1/pi,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='63) leads to yi+1 = yiγi with i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain y1 = 0, y2 = x2, y3 = x2γ2, y4 = x2γ2γ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', yN−1 = x2 �N−2 k=2 γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing all these expressions and using yN = 0, which Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='64) implies, we obtain 1 = N � i=1 yi = x2 � 1 + γ2 + γ2γ3 + · · · + N−2 � k=2 γk � = x2 � 1 + N − 3 r + N − 4 2r N − 3 r + · · · + 1 r(N − 3) 2 r(N − 4) · · · N − 4 2r N − 3 r � = x2 N−3 � i=0 r−i �N − 3 i � = x2 � 1 + 1 r �N−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='66) Therefore, we obtain x2 = � 1 + 1 r �3−N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='67) and xi =x2 � �1 + i−1 � j=2 j� k=2 γk � � = � 1 + 1 r �3−N � �1 + i−1 � j=2 j� k=2 γk � � , i ∈ {3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='68) For model 2, it always holds true that x1 = 0 because the evolutionary dynamics is driven by a majority rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we compare x2, instead of x1, as a function of r with x2 for the Moran process, to examine 13 whether a given hypergraph is an amplifier, suppressor, equivalent to the Moran process, or neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare x2 calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='67) with that for the Moran process at four values of N in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The figure indicates that the complete 3-uniform hypergraph with 4 nodes is a suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' However, the complete 3-uniform hypergraph with N > 4 is neither amplifier nor suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This is because Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='67) implies that x2 = 23−N when r = 1, which is different from the result for the Moran process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', x2 = 2/N, when N > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8 Moran complete cyclic star 0 1 2 3 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8 0 1 2 3 (a) N = 4 (b) N = 5 (c) N = 20 (d) N = 200 r Fixation probability Figure 4: Fixation probability for different hypergraph models under model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare the Moran process, which is the baseline, complete 3-uniform hypergraphs, cyclic 3-uniform hypergraphs, and star 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (d) N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In (a), the result for the complete 3-uniform hypergraph is exactly the same as that for the cyclic 3-uniform hypergraph such that the blue line is completely hidden behind the orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In (c) and (d), the results for the complete 3-uniform hypergraph (shown by the blue lines) are not identical but close to those for the star hypergraph (shown by the green lines) such that the former are hidden behind the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Cyclic 3-uniform hypergraph Consider the evolutionary dynamics under model 2 on the cyclic 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Assume that there are initially two nodes of type A, which are distributed uniformly at random, and N − 2 nodes of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We derived in Appendix F the fixation probability for type A, x2, using techniques similar to those used for the complete 3-uniform hypergraph (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain x2 = � � � 2 N−1 � 1−r−1 1−r−(N−2) + r−r−1 (r+4)[1−r−(N−2)] � (N ≥ 5), r 1+r (N = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='69) We compare x2 given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='69) with x2 for the Moran process at four values of N in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The figure indicates that the cyclic 3-uniform hypergraph with N = 4 is a suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' However, the cyclic 14 3-uniform hypergraph with N > 4 is neither amplifier nor suppressor under model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This is because x2 = 14/ [5(N − 1)(N − 2)] when r = 1 for cyclic 3-uniform hypergraphs, which is different from the value for the Moran process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', x2 = 2/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Star 3-uniform hypergraph We calculate the fixation probability for model 2 on the star 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' As in the case of model 1, we exploit the fact that the combination of the type of the hub (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', either A or B) and the number of leaf nodes of type A, which ranges between 0 and N − 1, completely specifies the state of the evolutionary dynamics on this hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show the derivation of the fixation probability in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain x2 = 5r 2(5r + 3) + 15r2 + 9r 2(15r2 + 34r + 15) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='70) for N = 4 and x2 = 6(r3 + 3r2) 5(3r3 + 13r2 + 15r + 4) + 3r2 5(r2 + 3r + 1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='71) for N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='70) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='71) with x2 for the Moran process with N = 4 and N = 5 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 4(a) and 4(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that the star 3-uniform hypergraph with N = 4 is a suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In contrast, the star 3-uniform hypergraph with N = 5 is neither an amplifier nor suppressor because x2 = 9/35 at r = 1, which is different from the corresponding result for the Moran process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', x2 = 2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We also obtained x2 for N = 20 and N = 200 by numerically solving a system of linear equations with 2N − 2 unknowns (see Appendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Figures 4(c) and 4(d), which compare the obtained x2 values with x2 for the Moran process for N = 20 and N = 200, respectively, indicate that these larger star 3-uniform hypergraphs are neither an amplifier nor suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we conclude that star 3-uniform hypergraphs are neither an amplifier nor suppressor for N ≥ 5 under the evolutionary dynamics described by model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Neutral drift The discussion of amplifier and suppressor of natural selection requires x1 = 1/N (or x2 = 2/N in the case of our model 2) when r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this section, we prove the following theorem, which justifies such discussion for model 1 and not for model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Consider the neutral drift, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' When there are initially i mutants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', type A) that are distributed uniformly at random over the N nodes, the fixation probability for the mutant is equal to i/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Consider model 1’, which is defined by the same evolutionary dynamics as that for model 1 with the initial condition in which each node i is occupied by a distinct neutral type ˜Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, there are initially N types, all of which have the constant fitness equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The evolutionary dynamics terminates when a single type fixates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We denote by qi the fixation probability for type ˜Ai under the aforementioned initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' It should be noted that �N i=1 qi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Now we consider the original model 1 with the initial condition in which there is just one mutant located at node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the fixation probability for the mutant is equal to qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This is because model 1’ is reduced to model 1 with the present initial condition if we regard type ˜Ai as the mutant type and all the N − 1 types ˜Aj, where j ̸= i, as the resident type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We recall that x1 is the average of the fixation probability over the N initial conditions in which there is just one mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain x1 = �N i=1 qi/N = 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Next we consider model 1 with the initial condition in which there are two mutants whose locations are selected uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Assume that the two mutants are initially located at nodes 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The mutant type fixates with probability q1 + q2 because model 1’ is reduced to model 1 with the present initial condition if we regard type ˜A1 and ˜A2 as the mutant type and all the other N − 2 types ˜Aj, where j ̸= 1, 2, as the resident type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain x2 = �N i=1 �i−1 j=1(qi + qj)/ �N 2 � = (N − 1) �N i=1 qi/ �N 2 � = 2/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Similarly, we obtain x3 = �N i=1 �i−1 j=1 �j−1 k=1(qi + qj + qk)/ �N 3 � = �N−1 2 � �N i=1 qi/ �N 3 � = 3/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' It is straightforward to verify xi = i/N for the other i values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This theorem also holds true for the birth-death process on conventional networks with the proof being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 15 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The theorem does not hold true for model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This is because, in model 2, whether the parent node u propagates its type to the other nodes in the selected hyperedge e containing u depends on the other nodes belonging to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' As an example, consider the case in which e contains three nodes, u, v1, and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Parent u imposes its type on v1 and v2 only when either v1 or v2 has the same type as u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', if both v1 and v2 are of the opposite type as u’s, then no state change occurs after u is selected as parent and hyperedge e is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Model 1’ cannot handle this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In model 1’, the parent disseminates its type to all the other nodes in the selected hyperedge e, as in model 1, regardless of the type of the other nodes belonging to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we cannot map model 1’ to model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In fact, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 4 shows that x2 ̸= 2/N in most cases, verifying that Theorem 1 does not hold true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Numerical results for empirical hypergraphs In this section, we carry out numerical simulations for empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The present study is motivated by the result that most networks are amplifiers under the birth-death process [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1, we showed that three model hypergraphs are suppressors under model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2, we argued that one cannot discuss whether the same hypergraphs are amplifier or suppressor under model 2 except for the star 3-uniform hypergraph with N = 4 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This is because model 2 does not respect x2 = 2/N in general (see Remark 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we focus on model 1 in this section and examine whether empirical hypergraphs tend to be suppressors, as is the case for the complete, cyclic, and star 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Empirical, or general, hypergraphs are distinct from the complete, cyclic, and star 3-uniform hypergraphs in two main aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' First, in general, empirical hypergraphs do not have much symmetry that we can exploit to simplify the probability transition matrix P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, pursuing analytical solutions, which would involve the solution of a linear system with 2N − 2 unknowns, is formidable unless N is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Second, empirical hypergraphs contain hyperedges of different sizes, whereas the 3-uniform hypergraphs only have hyperedges of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, in this section, we numerically examine the stochastic evolutionary dynamics on empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In each time step, we select a parent node with the probability proportional to its fitness from all the N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the parent propagates its type to all the other nodes in a hyperedge to which the parent belongs, which we select uniformly at random from all the hyperedges to which the parent belongs regardless of the size of the hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To calculate the fixation probability of a single mutant of type A for an arbitrary hypergraph and for each value of r, we run the birth-death process until type A or B fixates for each node v initially occupied by type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that all the N − 1 nodes except v are initially of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For each v, we run 3 × 103 simulations for r ≥ 1 and 4 × 104 simulations for r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We use a substantially larger number of simulations for r < 1 than r ≥ 1 because the fixation probability is small when r is small and therefore it can be estimated with a higher accuracy with more simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For a given value of r, we estimate the fixation probability of type A as a fraction of the runs in which type A has fixated among the 3 × 103 × N or 4 × 104 × N simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' the factor N is due to the N different choices of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We examine the fixation probability on four empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The corporate club membership hypergraph (corporate hypergraph for short) contains 25 nodes and 15 hyperedges [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each node represents a corporate executive officer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each hyperedge represents a social organization such as clubs and boards of which some officers are members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The Enron hypergraph is an email communication network and has 143 nodes and 10,885 hyperedges [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each node represents an email address at Enron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each hyperedge is comprised of all recipient addresses of an email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The Senate committee hypergraph (Senate hypergraph for short) has 282 nodes and 315 hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each node is a member of the US Senate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each hyperedge represents committee memberships [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The high-school hypergraph is a social contact network with 327 nodes and 7,818 hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each node is a student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Each hyperedge represents to a group of students that were in proximity of one another [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' All the four hypergraphs are connected hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5, we show the relationships between the fixation probability for a single mutant as a function of r, for the four empirical hypergraphs, one per panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that all the hypergraphs are suppressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We also simulate the birth-death process on the weighted one-mode projection of each empirical hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that the fixation probability for the obtained weighted network is almost the same as that for the Moran process for the corporate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5(a)), Senate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5(c)), and high-school (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5(d)) hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The one-mode projection of the Enron hypergraph is a suppressor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' However, the fixation probability for the one-mode projection of the Enron hypergraph as a function of r is close to that for the Moran process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 16 In contrast, the original Enron hypergraph is a much stronger suppressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, our result that the four empirical hypergraphs are suppressors is not expected from the one-mode projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To further examine the result that the empirical hypergraphs are suppressors, we now simulate the same evolutionary dynamics on the randomized hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtained the randomized hypergraph for each empirical hypergraph by randomly shuffling the hyperedges of the original hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the random shuffling, we preserved the degree of each node and the size of each hyperedge [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We show the fixation probability for the randomized hypergraphs by the green circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that the randomized hypergraph is also a suppressor for all the four hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The randomized hypergraph is less suppressing than the original hypergraph for three empirical hypergraphs (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5(a), (b), and (c)) and vice versa for the other one empirical hypergraph (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, how the features of empirical hypergraphs except for the distribution of the node’s degree and hyperedge’s size affects the fixation probability is non- equivocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' However, these results for the randomized hypergraphs further strengthen our main claim that hypergraphs are suppressors of evolutionary dynamics under model 1 in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 0 1 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8 Moran empirical one-mode projection randomized 0 1 2 3 0 1 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='99 1 r 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='99 1 r 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='99 1 r 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='002 (a) corporate (b) Enron (c) Senate (d) high-school r Fixation probability Figure 5: Fixation probability for empirical hypergraphs, their one-mode projection, and the randomized hyper- graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) Corporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) Enron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) Senate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (d) High-school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The insets of (b), (c), and (d) magnify the results for r values smaller than and close to r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Discussion We have proposed two models of evolutionary dynamics on hypergraphs that are extensions of the birth- death process on networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For both models of evolutionary dynamics, we semi-analytically derived the 17 fixation probability for the mutant type under the constant selection for three synthetic hypergraphs with high symmetry, which generalize the complete graph, cycle graph, and star graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For model 1, which is appropriate for discussing the strength of natural selection, we showed that these synthetic hypergraphs are suppressors of natural selection with few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Furthermore, by numerical simulations, we have shown that four empirical hypergraphs of our arbitrary choices are also suppressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Our results are in stark contrast to the known result that most networks are amplifiers under the birth-death updating rule and the uniform initialization [20–22], which we also assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' It is often the case that interaction among individuals is often of higher order, and hypergraphs rather than conventional networks are more direct representation of many empirical data of social and ecological interactions [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, amplification of natural selection by birth-death processes on networks may not be as universal as it has been suggested once we expand the class of network models to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For conventional networks, finding suppressors under the birth-death process is difficult [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In contrast, under the death-birth process, most conventional networks are suppressors [20], and amplification of natural selection is bounded and transient [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Our main result that most hypergraphs are suppressors under model 1 begs various related research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Is there a theoretical bound on amplification of natural selection in birth-death processes on hypergraphs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Is there a systematically constructed class of amplifying hypergraphs even if they are rare?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Are hypergraphs suppressors under the death-birth process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If it is the case, hypergraphs are even more suppressing under the death-birth than birth-death processes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Can we find optimal amplifier [24, 26, 27] or suppressor for hypergraphs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We save these questions for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Evolutionary set theory is a mathematical framework with which to analyze coevolution of the strategy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', type) of the node and the membership of the node to sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', groups) [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' It assumes that each node belongs to different groups and play games with other nodes belonging to the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A node v with a large fitness obtained from playing the game disseminates v’s group membership as well as v’s type to other nodes with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, evolutionary set theory is a dynamic graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Evolutionary set theory is distinct from evolutionary dynamics on hypergraphs studied in the present study in the following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' First, in evolutionary set theory, interaction between players is pairwise by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In contrast, in evolutionary dynamics on hypergraphs, the outcome of an interaction in a group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', hyperedge) may not be decomposed into the summation of pairwise interaction in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, we assumed that a parent node v simultaneously imposes its type on all the other nodes in the selected hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Second, the group membership evolves in evolutionary set theory, whereas it is fixed in the evolutionary dynamics considered in this study, which is a simplification assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Third, reproduction in evolutionary set theory occurs globally, not limited to between nodes in the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In contrast, we have assumed that reproduction occurs between nodes in the same hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' These differences create opportunities for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, extending the present model to the case of dynamic hypergraphs may be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For conventional networks, the time to fixation has been shown to be in a tradeoff relationship with the fixation probability [53–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In other words, a strongly amplifying network tends to accompany a large mean fixation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Our mathematical framework is readily adaptable to the examination of fixation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation time for hypergraphs including the comparison with the case of conventional networks warrants future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Other topics for further investigations include the effects of different initial conditions [7, 32, 58, 59], mathematical analyses of other symmetric hypergraphs, weak selection expansion of fixation probability [22] in the case of hypergraphs, amplification and suppression of natural selection in the mutation-selection equilibrium under a small mutation rate [60], and fixation probability of cooperation in evolutionary game dynamics on hypergraphs [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Focusing on fixation of a mutant (or resident) type in general allows us to use Markov chain theory to reach various mathematical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We believe that the present study is an important step toward deploying this powerful mathematical machinery to evolutionary dynamics on higher-order networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Acknowledgments N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' acknowledges support from AFOSR European Office (under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' FA9550-19-1-7024), the Japan Science and Technology Agency (JST) Moonshot R&D (under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' JPMJMS2021), and the National Science Foundation (under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 2052720 and 2204936).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 18 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='21) for N = 4 If N = 4, then vℓ−2 and vℓ+2 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, there are two sequences of events through which the state moves from i = 1 to i = 0 in one time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first sequence, vℓ−2, which is of type B, is selected as parent with probability 1/(r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and vℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', {ℓ − 2, ℓ − 1, ℓ} or {ℓ, ℓ + 1, ℓ − 2}, is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second sequence, either vℓ−1 or vℓ+1 is selected as parent, which occurs with probability 2/(r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−1 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='26) for N = 4 If N = 4, then vℓ−2 and vℓ+2 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, there are two sequences of events through which the state moves from i = N −1 to i = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first sequence, vℓ−2, which is of type A, is selected as parent with probability r/(3r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent and vℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', {ℓ − 2, ℓ − 1, ℓ} or {ℓ, ℓ + 1, ℓ − 2}, is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second sequence, either vℓ−1 or vℓ+1, which is of type A, is selected as parent with probability 2r/(3r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges that contain the parent and vℓ is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='30) for i = N − 3 and i = N − 2 If i = N − 3, then vℓ−2 and vℓ+i+1 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, there are two sequences of events through which the state decreases from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first sequence, either vℓ−1 or vℓ−3 is selected as parent, which occurs with probability 2/(rN − 3r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge containing two nodes of type B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', hyperedge {ℓ − 2, ℓ − 1, ℓ} if vℓ−1 is the parent and hyperedge {ℓ − 4, ℓ − 3, ℓ − 2} if vℓ−3 is the parent) is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second sequence, vℓ−2 is selected as parent with probability 1/(rN − 3r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 4, ℓ − 3, ℓ − 2} is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i = N − 2, either of the two nodes of type B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', vℓ−1 or vℓ−2, must be selected as parent for the state to move from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This event occurs with probability 2/(rN − 2r + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, either hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 3, ℓ − 2, ℓ − 1} must be used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The product of these two probabilities yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='31) for i = 2 and i = 3 If i = 2, for the state to move from i to i + 1, either vℓ or vℓ+1, which is of type A, must be selected as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This event occurs with probability 2r/(2r + N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge containing both vℓ and vℓ+1 must be selected, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The product of these two probabilities yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i = 3, then vℓ+1 and vℓ+i−2 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, there are two sequences of events with which the state increases from i to i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first sequence, either vℓ or vℓ+2 is selected as parent with probability 2r/(3r + N − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge composed of the parent, vℓ+1, which is of type A, and a node of type B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', vℓ−1 if the parent is vℓ, and vℓ+3 if the parent is vℓ+2) is used for reproduction with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second sequence, vℓ+1 is selected as parent with probability r/(3r + N − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, hyperedge {ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ + 3} is used for reproduction with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Fixation probability for the birth-death process on the weighted one-mode pro- jection of the star 3-uniform hypergraph In this section, we consider the weighted one-mode projection of the star 3-uniform hypergraph and examine the fixation probability on the obtained weighted network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 19 We denote the weighted one-mode projection of the star 3-uniform hypergraph by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that G is a weighted complete graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' the edge between the hub node and any leaf node has weight N − 2, and the edge between any pair of leaf nodes has weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the birth-death process on G, in each time step, we select one node as parent with the probability proportional to its fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the parent selects one of its neighbors with the probability proportional to the edge weight and converts the neighbor into the parent’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' As is the case for the star 3-uniform hypergraph, the symmetry in G allows us to specify the state of the birth-death process by tuple (i1, i2), where i1 ∈ {0, 1} specifies whether the hub is of type A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', i1 = 1) or B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', i1 = 0), and i2 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1} is the number of leaf nodes of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The total number of nodes of type A is equal to i = i1 + i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation of type A and B corresponds to (i1, i2) = (1, N − 1) and (0, 0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Assume that the current state is (i1, i2) = (1, i − 1) with i − 1 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are four types of events that can occur in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, a leaf node of type B is selected as parent with probability (N − i)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and the hub node is used for reproduction with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this entire event is (i1, i2) = (0, i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(0,i−1) = N − i ri + N − i · 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1) In the second type of event, a leaf node of type B is selected as parent with probability (N − i)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and a leaf node of type A is used for reproduction with probability (i − 1)/2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (1, i − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i−2) = N − i ri + N − i · i − 1 2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2) In the third type of event, the hub node, which is of type A, is selected as parent with probability r/(ri+N−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and a leaf node of type B is used for reproduction with probability (N −i)/(N −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a leaf node of type A is selected as parent with probability r(i−1)/(ri+N −i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and a leaf node of type B is used for reproduction with probability (N − i)/2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In both cases, the state after the event is (1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i) = r ri + N − i · N − i N − 1 + r(i − 1) ri + N − i · N − i 2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3) If any other event occurs, then the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i−1) = 1 − p(1,i−1)→(0,i−1) − p(1,i−1)→(1,i−2) − p(1,i−1)→(1,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4) We remind that ˜x(i1,i2) represents the probability that A fixates starting with state (i1, i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain ˜x(1,i−1) = p(1,i−1)→(0,i−1)˜x(0,i−1) + p(1,i−1)→(1,i−2)˜x(1,i−2) + p(1,i−1)→(1,i)˜x(1,i) + p(1,i−1)→(1,i−1)˜x(1,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) Now we assume that the current state is (i1, i2) = (0, i) with i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are four types of events that can occur in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, a leaf node of type A is selected as parent with probability ri/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and the hub node is used for reproduction with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (i1, i2) = (1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(1,i) = ri ri + N − i · 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6) In the second type of event, a leaf node of type A is selected as parent with probability ri/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and a leaf node of type B is used for reproduction with probability (N − i − 1)/2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (0, i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i+1) = ri ri + N − i · N − i − 1 2(N − 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7) 20 In the third type of event, the hub node, which is of type B, is selected as parent with probability 1/(ri+N−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and a leaf node of type A is used for reproduction with probability i/(N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a leaf node of type B is selected as parent with probability (N − i − 1)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the edge between the parent and a leaf node of type A is used for reproduction with probability i/2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In both cases, the state after the event is (0, i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i−1) = 1 ri + N − i · i N − 1 + N − i − 1 ri + N − i · i 2(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8) If any other event occurs, then the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i) = 1 − p(0,i)→(1,i) − p(0,i)→(0,i+1) − p(0,i)→(0,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9) Using these transition probabilities, we obtain ˜x(0,i) = p(0,i)→(1,i)˜x(1,i) + p(0,i)→(0,i+1)˜x(0,i+1) + p(0,i)→(0,i−1)˜x(0,i−1) + p(0,i)→(0,i)˜x(0,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) also holds true for the one-mode projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We use the scipy DGESV algorithm to numer- ically solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) to obtain ˜x(0,i) and ˜x(1,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, we obtain xi using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We computed x1 by numerically solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='48) for N = 4, 5, 20, and 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Figure E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 compares the obtained x1 values with those for the Moran process and the star 3-uniform hypergraph under model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We find that the one-mode projection of the star 3-uniform hypergraph is weak amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that, for N = 4, 5, and 200, the results for the one-mode projection almost overlap those for the Moran process such that the orange lines are hidden behind the black lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6(a), (b), and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 21 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 Moran star hypergraph one-mode projection 0 1 2 3 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='0 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='005 (a) N = 4 (b) N = 5 (c) N = 20 (d) N = 200 r Fixation probability Figure E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6: Fixation probability for the weighted one-mode projection of star 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We compare it with the fixation probability for the Moran process and star 3-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (d) N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The insets in (c) and (d) magnify the results for r values smaller than and close to r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the inset in (d), the results for the star 3-uniform hypergraph (shown by the blue line) and the Moran process (shown by the black line) are close to that for the one-mode projection (shown by the orange line) such that the blue and the black lines are almost hidden behind the orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In (a), (b), and the main panel of (d), the results for the Moran process (shown by the black lines) are not identical but close to those for the one-mode projection (shown by the orange lines) such that the former are hidden behind the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Derivation of the fixation probability for the cyclic 3-uniform hypergraph under model 2 In this section, we derive the fixation probability for the cyclic 3-uniform hypergraph under model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We assume that there are initially just two mutants that are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The fixation of type A can occur only when the two nodes that are initially of type A share at least one hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Once such a hyperedge is selected for reproduction, all the nodes of type A are consecutive along the cycle without being interrupted by nodes of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that the two nodes that initially have type A may be next to each other on the cycle already in the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, to calculate the fixation probability on the cyclic 3-uniform hypergraph, it suffices to track the number of consecutive nodes having type A, which we denote by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For N ≥ 5, the initial condition is either of the following three types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 22 Figure F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7: Initial position of the two nodes of type A on the cyclic 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (a) The two nodes of type A do not share any hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (b) The two nodes of type A share two hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (c) The two nodes of type A share one hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' First type of initial condition In the first type of initial condition, the two nodes of type A do not share any hyperedge (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7(a) for a schematic), which occurs with probability N(N −5)/ � 2 �N 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The probability that type A fixates under this initial condition, denoted by x′ 2, is given by x′ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1) Second type of initial condition In the second type of initial condition, the two nodes of type A share two hyperedges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', these two nodes are next to each other on the cycle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This initial condition occurs with probability N/ �N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Let x′′ 2 be the fixation probability for type A under this initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In general, let x′′ i with i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1} be the fixation probability for type A when there are i consecutive nodes of type A and all the other nodes are of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We calculate x′′ 2 by tracking the number of consecutive nodes with type A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', i, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Move of the state from i to i − 1: If i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N −2}, there are three types of events that can occur next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Without loss of generality, we assume that the ℓth to the (ℓ + i − 1)th nodes are of type A and that all the other nodes are of type B (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, the state moves from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i ≤ N − 4, either vℓ−2, vℓ−1, vℓ+i, and vℓ+i+1, which is of type B, is selected as parent with probability 4/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent node, a node of type B, and a node of type A is used for reproduction with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, i decreases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−2 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i−1 = 4 ri + N − i · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2) If i = N − 3, there are two sequences of events through which the state decreases from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first sequence, either vℓ−1 or vℓ−3 is selected as parent, which occurs with probability 2/(rN − 3r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge containing two nodes of type B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', hyperedge {ℓ − 2, ℓ− 1, ℓ} if vℓ−1 is the parent and hyperedge {ℓ − 4, ℓ − 3, ℓ − 2} if vℓ−3 is the parent) is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second sequence, vℓ−2 is selected as parent with probability 1/(rN −3r +3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, hyperedge {ℓ−2, ℓ−1, ℓ} or {ℓ − 4, ℓ − 3, ℓ − 2} is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i = N − 2, either of the two nodes of type B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', vℓ−1 or vℓ−2, must be selected as parent for the state to move from i to i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This event occurs with probability 2/(rN − 2r + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, either hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 3, ℓ − 2, ℓ − 1} must be used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The product of these two probabilities coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2) holds true for any i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' 23 (a) (b) V-2 Vt-1 Vt Vt+1 Ve+2 Vt+3 Ve+4Move of the state from i to i + 1: In the second type of event, the state moves from i to i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i ≥ 4, either vℓ, vℓ+1, vℓ+i−2, or vℓ+i−1, which is of type A, has to be selected as parent with probability 4r/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent node, a node of type A, and a node of type B has to be used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ is selected as parent and hyperedge {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from i to i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain pi,i+1 = 4r ri + N − i · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3) If i = 3, there are two sequences of events through which the state increases from i to i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first sequence, either vℓ or vℓ+2 is selected as parent, which occurs with probability 2r/(3r + N − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge containing two nodes of type A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', hyperedge {ℓ − 1, ℓ, ℓ + 1} if vℓ is the parent and hyperedge {ℓ + 1, ℓ + 2, ℓ + 3} if vℓ+2 is the parent) is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second sequence, vℓ+1 is selected as parent with probability r/(3r + N − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, hyperedge {ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ + 3} is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If we sum these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If i = 2, either of the two nodes of type A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', vℓ or vℓ+1, must be selected as parent for the state to move from i to i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This event occurs with probability 2r/(2r + N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, either hyperedge {ℓ − 1, ℓ, ℓ + 1} or {ℓ, ℓ + 1, ℓ + 2} must be used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The product of these two probabilities coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3) holds true for any i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' No move of the state from i: Because i remains unchanged if any other event occurs, we obtain pi,i = 1 − pi,i−1 − pi,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4) Derivation of x′′ 2: Therefore, the fixation probability of type A starting from i consecutive nodes of type A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', x′′ i , satisfies x0 = x1 = 0, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) x′′ i = pi,i−1x′′ i−1 + pi,ix′′ i + pi,i+1x′′ i+1, i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6) xN−1 = xN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7) Note that x′′ 1 = x1 and x′′ N−1 = xN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Similar to the analysis of the fixation probability for the complete 3-uniform hypergraph, we set yi ≡ x′′ i − x′′ i−1, i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8) Note that �N−1 i=2 yi = x′′ N−1 − x′′ 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Let γi = pi,i−1/pi,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9) By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8), and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9), we obtain yi+1 = yiγi, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10) which leads to yi = y2 i−1 � k=2 γk = x′′ 2 i−1 � k=2 γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='11) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='11), we obtain 1 = N−1 � i=2 yi = x′′ 2 � 1 + γ2 + γ2γ3 + · · · + N−2 � k=2 γk � = x′′ 2 � 1 + r−1 + r−2 + · · · + r−(N−3)� = x′′ 2 1 − r−(N−2) 1 − r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='12) 24 Therefore, we obtain x′′ 2 = 1 − r−1 1 − r−(N−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13) Third type of initial condition In the third type of initial condition, the two nodes of type A share one hyperedge, implying that there is a node of type B between the two nodes of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Without loss of generality, we assume that the ℓth and the (ℓ + 2)th nodes are of type A and that all the other nodes are of type B (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This initial condition, which we denote by 2∗, occurs with probability N/ �N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Now we calculate the fixation probability for type A starting from state 2∗, which we denote by x ′′′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' To ease the discussion, in the remainder of this section, we denote by i the state in which consecutive i nodes on the cycle are of type A and the other N − i nodes are of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Move of the state from 2∗ to i = 1: If N ≥ 7, the state moves from 2∗ to i = 1 in one time step if either of the following two types of events occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, either node vℓ−2 or vℓ+4, which is of type B, is selected as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' This event occurs with probability 2/(2r + N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent, a node of type B, and a node of type A, is used for reproduction, which occurs with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−2 is the parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from 2∗ to i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second type of event, one of the nodes vℓ−1, vℓ+1, and vℓ+3, which is of type B, is selected as parent, which occurs with probability 3/(2r + N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges that contains the parent, a node of type B, and a node of type A, is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−1 is selected as parent and hyperedge {ℓ − 2, ℓ − 1, ℓ} or {ℓ − 1, ℓ, ℓ + 1} is used for reproduction, then the state moves from 2∗ to i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain the probability that the state moves from 2∗ to i = 1 as p2∗,1 = 1 2r + N − 2 · 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14) If N = 6, the state moves from 2∗ to i = 1 in one time step if the following event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Either vℓ−2, vℓ−1, vℓ+1, and vℓ+3, which is of type B, is selected as parent with probability 4/(2r + 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges that contains the parent, a node of type B, and a node of type A, is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−2 is selected as parent and hyperedge {ℓ−4, ℓ−3, ℓ−2} or {ℓ − 2, ℓ − 1, ℓ} is used for reproduction, then the state moves from 2∗ to i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The product of these two probabilities coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' If N = 5, the state moves from 2∗ to i = 1 in one time step if either of the following two types of events occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, either node vℓ−1 or vℓ+3, which is of type B, is selected as parent with probability 2/(2r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, the hyperedge that contains the parent, a node of type B, and a node of type A, is used for reproduction, which occurs with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' For example, if vℓ−1 is the parent and any hyperedge that contains vℓ−1 is used for reproduction, then the state moves from 2∗ to i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the second type of event, the node vℓ+1, which is of type B, is selected as parent with probability 1/(2r + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, one of the two hyperedges {ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ + 3} is used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By summing up these probabilities, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14) holds true for any N ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Move of the state from 2∗ to i = 3: The state moves from 2∗ to i = 3 if either vℓ or vℓ+2, which is of type A, is selected as parent with probability 2r/(2r + N − 2), and then, the hyperedge that contains vℓ, vℓ+1, and vℓ+2 is used for reproduction with probability 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p2∗,3 = 2r 2r + N − 2 · 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='15) No move of the state from 2∗: If any other event occurs at state 2∗, the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p2∗,2∗ = 1 − p2∗,1 − p2∗,3 = 4r + 3N − 14 3(2r + N − 2), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='16) p2∗,j = 0 if j ̸= 1, 2∗, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='17) 25 Derivation of x′′′ 2 : If the state moves from 2∗ to either i = 1 or i = 3, all the nodes of type A are consecutively numbered without being interrupted by nodes of type B afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain x ′′′ 2 = p2∗,1x1 + p2∗,2∗x ′′′ 2 + p2∗,3x′′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='18) By substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6) for i = 2, we obtain x′′ 3 = 1 − r−2 1 − r−(N−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='19) By substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='19) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='18), we obtain x ′′′ 2 = r − r−1 (r + 4) � 1 − r−(N−2)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='20) Weighted sum to obtain x2 By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13), and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='20) with the respective probability, we obtain the fixation prob- ability for type A when there are initially two uniformly randomly distributed mutants, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='69), as follows: x2 = N − 5 N − 1x′ 2 + 2 N − 1x ′′ 2 + 2 N − 1x ′′′ 2 = 2 N − 1 � 1 − r−1 1 − r−(N−2) + r − r−1 (r + 4) � 1 − r−(N−2)� � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='21) where N ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Derivation of x2 for N = 4 For N = 4, the first type of initial condition occurs with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The second type of initial condition occurs with the same probability as in the case of N ≥ 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=', with probability 4/ �4 2 � = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In this case, the state moves from 2 to 1 if the following event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Either vℓ−1 or vℓ+2, which is of type B, is selected as parent with probability 2/(2r + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, either hyperedge {ℓ + 2, ℓ − 1, ℓ} or {ℓ + 1, ℓ + 2, ℓ − 1} must be used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Note that the node indices ℓ − 1 and ℓ are equivalent to ℓ + 3 and ℓ + 4 because we interpret the node index with modulo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The product of these two probabilities coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2) also holds true for N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Similarly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3) holds true for N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='12), we obtain x′′ 2 = r r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='22) Under the third type of initial condition, which occurs with probability 2/ �4 2 � = 1/3, the state moves from 2∗ to 1 if the following event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Either vℓ−1 or vℓ+1, which is of type B, is selected as parent with probability 2/(2r + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, either hyperedge {ℓ − 1, ℓ, ℓ + 1} or {ℓ + 1, ℓ + 2, ℓ − 1} must be used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p2∗,1 = 2 3r + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='23) The state moves from 2∗ to 3 if the following event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Either vℓ or vℓ+2, which is of type A, is selected as parent with probability 2r/(2r + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, either hyperedge {ℓ, ℓ + 1, ℓ + 2} or {ℓ + 2, ℓ − 1, ℓ} must be used for reproduction, which occurs with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p2∗,3 = 2r 3r + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='24) 26 If any other event occurs at state 2∗, the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p2∗,2∗ = 1 − p2∗,1 − p2∗,3 = r + 1 3r + 3, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='25) p2∗,j = 0 if j ̸= 1, 2∗, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='26) By substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='18), we obtain x ′′′ 2 = r r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='27) Therefore, we obtain x2 = 2 3x ′′ 2 + 1 3x ′′′ 2 = r r + 1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='28) for N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Derivation of the fixation probability for the star 3-uniform hypergraph under model 2 We derive the fixation probability for the star 3-uniform hypergraph under model 2 in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We use same notations as those in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Assume that the current state is (i1, i2) = (1, i − 1) with i − 1 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' There are three types of events that can occur in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, a leaf node of type B is selected as parent with probability (N − i)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a different leaf node of type B, is used for reproduction with probability (N − i − 1)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this entire event is (i1, i2) = (0, i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(0,i−1) = N − i ri + N − i · N − i − 1 N − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='1) In the second type of event, the hub node, which is of type A, is selected as parent with probability r/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B, is used for reproduction with probability (i − 1)(N − i)/ �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a leaf node of type A is selected as parent with probability r(i − 1)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a leaf node of type B, is used for reproduction with probability (N − i)/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In both cases, the state after the event is (1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i) = r ri + N − i · (i − 1)(N − i) �N−1 2 � + r(i − 1) ri + N − i · N − i N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='2) If any other event occurs, then the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(1,i−1)→(1,i−1) = 1 − p(1,i−1)→(0,i−1) − p(1,i−1)→(1,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='3) We remind that ˜x(i1,i2) is the probability that type A fixates when the initial state is (i1, i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' We obtain ˜x(1,i−1) = p(1,i−1)→(0,i−1)˜x(0,i−1) + p(1,i−1)→(1,i)˜x(1,i) + p(1,i−1)→(1,i−1)˜x(1,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4) Now we assume that the current state is (i1, i2) = (0, i) with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, there are three types of events that can occur in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In the first type of event, a leaf node of type A is selected as parent with probability ri/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a different leaf node of type A, is used for reproduction with probability (i−1)/(N −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' The state after this event is (i1, i2) = (1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(1,i) = ri ri + N − i · i − 1 N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='5) 27 In the second type of event, the hub node, which is of type B, is selected as parent with probability 1/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, a leaf node of type A, and a leaf node of type B, is used for reproduction with probability i(N − i − 1)/ �N−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Alternatively, a leaf node of type B is selected as parent with probability (N − i − 1)/(ri + N − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Then, a hyperedge that contains the parent, the hub node, and a leaf node of type A, is used for reproduction with probability i/(N − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' In both cases, the state after the event is (0, i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i−1) = 1 ri + N − i · i(N − i − 1) �N−1 2 � + N − i − 1 ri + N − i · i N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='6) If any other event occurs, then the state remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain p(0,i)→(0,i) = 1 − p(0,i)→(1,i) − p(0,i)→(0,i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='7) Using these transition probabilities, we obtain ˜x(0,i) = p(0,i)→(1,i)˜x(1,i) + p(0,i)→(0,i−1)˜x(0,i−1) + p(0,i)→(0,i)˜x(0,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8) Equations (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='4) and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='8) lead to ˜x(1,i) = αi˜x(1,i−1) + (1 − αi)˜x(0,i−1) (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9) and ˜x(0,i) = βi˜x(1,i) + (1 − βi)˜x(0,i−1), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10) respectively, where αi = 1 + (N − i − 1)(N − 1) r(i − 1)(N + 1) , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='11) βi = r(i − 1)(N − 1) r(i − 1)(N − 1) + (N − i − 1)(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='12) We rewrite Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='9) and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='10) as ϱi = Aiϱi−1, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13) where ϱi = (˜x(1,i), ˜x(0,i))⊤, and Ai is the 2 × 2 matrix given by Ai = � αi 1 − αi αiβi 1 − αiβi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='14) Equation (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13) yields ϱi = AiAi−1 · · · A2ϱ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='15) Therefore, � 1 1 � = ϱN−1 = AN−1AN−2 · · · A2ϱ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='16) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='54) also holds true for model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' Therefore, we obtain ˜x(1,1) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='16), ˜x(0,2) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='13), and finally x2 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' By analytically solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='16) for N = 4 and N = 5, we obtain Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='70) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content='71), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQf7w7r/content/2301.05343v1.pdf'} +page_content=' A.' metadata={'source': 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reconnection as a plasma diagnostic in the solar corona +Konstantinos Karampelas,1, 2 James A. McLaughlin,1 Gert J. J. Botha,1 and St´ephane R´egnier1 +1Department of Mathematics, Physics and Electrical Engineering, Northumbria University, +Newcastle upon Tyne, NE1 8ST, UK +2Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, +Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium +(Received November 1, 2022; Revised December 6, 2022; Accepted December 15, 2022) +Submitted to ApJ +ABSTRACT +Oscillatory reconnection is a relaxation process in magnetised plasma, with an inherent periodicity +that is exclusively dependent on the properties of the background plasma. +This study focuses on +the seismological prospects of oscillatory reconnection in the solar corona. +We perform three sets +of parameter studies (for characteristic coronal values of the background magnetic field, density and +temperature) using the PLUTO code to solve the fully compressive, resistive MHD equations for a 2D +magnetic X-point. From each parameter study, we derive the period of the oscillatory reconnection. +We find that this period is inversely proportional to the characteristic strength of the background +magnetic field and the square root of the initial plasma temperature, while following a square root +dependency upon the equilibrium plasma density. +These results reveal an inverse proportionality +between the magnitude of the Alfv´en speed and the period, as well as the background sound speed +and the period. Furthermore, we note that the addition of anisotropic thermal conduction only leads +to a small increase in the mean value for the period. Finally, we establish an empirical formula that +gives the value for the period in relation to the background magnetic field, density and temperature. +This gives us a quantified relation for oscillatory reconnection, to be used as a plasma diagnostic in +the solar corona, opening up the possibility of using oscillatory reconnection for coronal seismology. +Keywords: Magnetohydrodynamics (1964); Solar magnetic reconnection (1504); Solar coronal seis- +mology (1994); Solar coronal waves (1995); Magnetohydrodynamical simulations (1966); +1. INTRODUCTION +Oscillatory reconnection is a physical phenomenon +characterised by a series of reconnection events Parker +1957; Sweet 1958; Petschek 1964) that take place along- +side periodic changes in the magnetic connectivity of a +perturbed magnetic field. The process was identified for +the first time in Craig & McClymont (1991), during the +study of the relaxation of an 2D X-point. One impor- +tant characteristic of oscillatory reconnection is that the +Corresponding author: Konstantinos Karampelas +konstantinos.karampelas@northumbria.ac.uk +periodicity is not imposed by an external driver, rather +it is an inherent property of the relaxation process. +Over the recent years, a number of numerical stud- +ies have been conducted regarding oscillatory reconnec- +tion. McLaughlin et al. (2009) studied the mechanism +for a 2D magnetic X-point in a cold plasma, solving the +fully compressible resistive MHD equations. Using an +external fast magnetoaccoustic pulse, they initiated os- +cillatory reconnection by perturbing a magnetic X-point. +This study had identified many properties of this mech- +anism, like the periodic changes in the resulting current +sheet orientation with the respective changes in connec- +tivity, and the formation of both fast and slow oblique +magnetic shocks. Thurgood et al. (2017) later expanded +the results of the previous study for a 3D null point, +arXiv:2301.02452v1 [astro-ph.SR] 6 Jan 2023 + +2 +Karampelas et al. +also reporting the generation of MHD waves. Oscilla- +tory reconnection has also been studied for a realistic +solar atmosphere, as a result of flux rope emergence +(Murray et al. 2009; McLaughlin et al. 2012b), while +other studies revolved around the effects of resistivity, +initial perturbation amplitude, and the length of the ini- +tial current sheet on the period of the reconnection pro- +cess (McLaughlin et al. 2012a; Thurgood et al. 2018a,b, +2019). Stewart et al. (2022) reported the onset of oscil- +latory reconnection and the generation of waves through +the coalescence of two cylindrical flux ropes, while Sabri +et al. (2020) reported the development of the plasmoid +instability in a magnetic O-point and the resulting man- +ifestation of plasmoid-mediated quasi-oscillatory mag- +netic reconnection. +The results of McLaughlin et al. +(2009) have recently been expanded for a hot coronal +plasma in Karampelas et al. (2022a), studying the re- +lation between the oscillation period and the strength +of the background magnetic field, while also taking into +account the effects of anisotropic thermal conduction. A +following study (Karampelas et al. 2022b) reported for +the first time on the independence between the type and +strength of the perturbing wave pulse and the frequency +of the resulting oscillatory reconnection in a hot coronal +plasma. These two studies have produced encouraging +results regarding the possibility of using oscillatory re- +connection as a new tool for coronal seismology. +Magnetic reconnection can cause the dissipation of +magnetic field and electric current, leading to the ac- +celeration of particles, ejection of mass and heating +through the generation of shocks. As such, it is con- +sidered as the main mechanism behind solar flares (e.g. +Shibata & Magara 2011; Jel´ınek et al. 2015), while the +ubiquitous null points in the solar atmosphere (Gals- +gaard & Nordlund 1997; Brown & Priest 2001; Long- +cope 2005; R´egnier et al. 2008), where reconnection can +take place, are consequently considered preferential lo- +cations the manifestation of flares (e.g. Murawski et al. +2011). Over the years, oscillatory reconnection has been +proposed as a driving force behind observed phenom- +ena like quasi-periodic pulsations (QPPs) of solar flares +Kupriyanova et al. 2016; Van Doorsselaere et al. 2016; +Pugh et al. 2017; Yuan et al. 2019; Hayes et al. 2020; +Li et al. 2020a,b, 2021, 2022; Clarke et al. 2021; Li +& Chen 2022; Shi et al. 2022) and stellar flares (e.g. +Broomhall et al. 2019; Guarcello et al. 2019; Jackman +et al. 2019; Notsu et al. 2019; Vida et al. 2019; Man- +cuso et al. 2020; Ramsay et al. 2021). The mechanism +is included in reviews summarising our current knowl- +edge around QPPs and the proposed mechanisms behind +them, such as McLaughlin et al. (2018), Kupriyanova +et al. (2020), and Zimovets et al. (2021). In particu- +lar, there are many examples from QPP observations +(see histogram in McLaughlin et al. 2018, and its cor- +responding online catalog), with reported periods close +to those derived from the studies of Karampelas et al. +(2022a) and Karampelas et al. (2022b), for the plasma +conditions considered in those studies. +Connection has also been proposed between oscilla- +tory reconnection and quasi-periodic flows, like those +associated with spicules (e.g. De Pontieu & McIntosh +2010; De Pontieu et al. 2011; Samanta et al. 2019; +Yurchyshyn et al. 2020), as well as with observed pe- +riodicities in breakout current sheets at the base of jets +(Hong et al. 2019). Mandal et al. (2022), have reported +a highly dynamic small-scale jet in a polar coronal hole, +and proposed oscillatory reconnection as a possible driv- +ing mechanism behind the observed repetitive outflows. +McLaughlin et al. (2012b) were able to reproduce such +observed periodic outflows through oscillatory reconnec- +tion in a 2D flux emergence model. The resulting peri- +ods from that model had a very good match with those +reported from wavelet analysis in Mandal et al. (2022), +although the latter showed no significant power at the +99% confidence level, preventing them to characterise +the outflows as periodic, but merely repetitive. Obser- +vational signatures of chromospheric jets by periodic re- +connection events were also reported in simulations by +Heggland et al. (2009), although, the periodicity was +attributed to the continuous driving rather than being +inherent to the system. +Oscillatory reconnection has +also been considered as a possible mechanism behind +with the creation of an observed quasi-periodic fast- +propagating (QFP) magnetosonic wave from the erup- +tion of a magnetic flux rope (Shen et al. 2018), as well +as behind the formation and disappearance of a small +scale magnetic flux rope consisting of new loops formed +by the reconnection events (Xue et al. 2019). +Zhang +et al. (2014) have reported oscillatory (or reciprocatory) +magnetic reconnection in observations of Coronal Bright +Points (CBPs), while reversals of an elongated current +sheet in a recent numerical 2D CBP model has been at- +tributed to oscillatory reconnection (N´obrega-Siverio & +Moreno-Insertis 2022). Finally, recent observations by +the Parker Solar Probe could also be attributed to oscil- +latory reconnection (e.g. Bale et al. 2016, 2019; Kasper +et al. 2019), like Alfv´enic spikes (He et al. 2021) and pe- +riodicities correlated with Type III radio bursts (Cattell +et al. 2021). +In this paper, we will further investigate oscillatory +reconnection in a hot coronal plasma and to explore +its potential for utilising oscillatory reconnection as a +tool for coronal seismology. +We will expand the re- +sults of Karampelas et al. (2022a) through a series of + +Oscillatory Reconnection as Plasma Diagnostic +3 +Figure 1. Magnetic field lines of the unperturbed X-point, +were the black solid and dashed lines depict the regions of +opposing polarity. The separatrices (red solid lines) and the +equipartition layer for a 1 MK coronal plasma (blue circular +line) are also included. +Figure 2. 2D profiles of the vx and vy velocity components +of the initial circular pulse. The magenta circular line is the +equipartition layer for a 1 MK coronal plasma. All values are +depicted in code units. +parameter studies for different characteristic strengths +of the magnetic field, equilibrium plasma density and +initial plasma temperature, for a 2D magnetic X-point. +Like in Karampelas et al. (2022a) and Karampelas et al. +(2022b), we will be exploring these cases both in the ab- +sence and presence of anisotropic thermal conduction. +In section 2 we present our physical domain, code used +to solve the fully compressible mhd equations and nu- +merical schemes utilised, while we present the results +of the parameter studies in the respective subsection in +§3. Finally, our conclusions and general discussion take +place in §4. +2. NUMERICAL SETUP +2.1. Numerical Scheme +For the numerical studies below, we solve the 2D com- +pressible resistive MHD equations in cartesian coordi- +nates, in the absence of gravity (see §2.1 in Karampelas +et al. 2022a), using the PLUTO code (Mignone et al. +2007, 2012). Like in our past studies (Karampelas et al. +2022a,b), we employ the fifth-order monotonicity pre- +serving scheme (MP5) for the spatial integration and +the third-order Runge-Kutta method for the time inte- +gration. To satisfy the solenoidal constraint of the mag- +netic field (∇·B = 0), we use the Constrained Transport +method implemented in the code. +In these simulations, we also consider setups where +we introduce anisotropic thermal conduction. The val- +ues for the parallel and perpendicular thermal conduc- +tion coefficients (in J s−1 K−1 m−1), as calculated from +the Spitzer conductivity (Orlando et al. 2008), are given +below: +κ∥ = 5.6 × 10−12 T +5 +2 , +(1) +κ⊥ = 3.3 × 10−21 +n2 +H +√ +TB2 , +(2) +where κ∥, κ⊥ and the hydrogen number density nH, +temperature T and magnetic field B are all given in SI1. +The effects of saturation are also taken into account for +very large temperature gradients. +The corresponding +source term (∇ · Fc) in the energy equation varying be- +tween the classical (Fclass) and saturated thermal con- +duction (Fsat): +∇ · Fc = ∇ · +� +Fsat +Fsat + |Fclass|Fclass +� +(3) +Fclass = κ∥ˆb +� +ˆb · ∇T +� ++ κ⊥ +� +∇T − ˆb +� +ˆb · ∇T +�� +(4) +Fsat = 5 φ ρ V 3 +S,iso, +(5) +where VS,iso = +� +p/ρ is the isothermal sound speed, ˆb = +B/|B| is the unit vector in the direction of magnetic field +and φ is a free code parameter (with a default value of +0.3). For zero magnetic field, Fc reduces to Fc = κ∥ ∇T. +During this analysis, we will be working in code units +U = Uph U −1 +0 , with Uph being the physical quantities +and U0 the normalization units U0. The constants U0 are +characteristic values, chosen for solar coronal plasma. +We consider the unit length L0 = 1 Mm, unit density +ρ0 = 10−12 kg m−3, and unit velocity v0 = 1.29 × 105 m +s−1, equal to VS/√γ for coronal plasma at 1 MK. We +also take the unit temperature T0 = 1 MK. The char- +acteristic magnetic field and unit time are respectively +B0 = +� +µρ0v2 +0 = 1.44 G and t0 = L0/v0 = 7.78 s. +Since we want to solve the resistive MHD equations, +we take the magnetic diffusivity in code units as η = +R−1 +m = 10−5, where Rm = (v0 L0)/η = 105 is the mag- +netic Reynolds number, assuming that the typical length +1 In cgs, the thermal conduction coefficients (in erg s−1 K−1 cm−1) +are given as κ∥ = 5.6 × 10−7 T +5 +2 and κ⊥ = 3.3 × 10−16 +n2 +H +√ +TB2 . + +10.0 +7.5 +5.0 +- +2.5 +- +y (c.u.) +1 +- +1 +1 +1 +1 +0.0 +- +- +- +- +- +1 +- +- +- +- +1 +-2.5 +1 +-5.0 +-7.5 +-10.0 +-10.07.5-5.0-2.5 0.0 2.5 5.0 +7.5 10.0 +x (c.u.)6 +0.3 +4 +0.2 +2 +0.1 +y (c.u.) +0 +0.0 +-0.1 +-2 +-0.2 +-4 +-0.3 +-6 +-4 +-2 +0 +2 +4 +6 +x (c.u.)6 +0.3 +4 +0.2 +2 +0.1 +y (c.u.) +0 +0.0 +-0.1 +-2 +-0.2 +-4 +-0.3 +-6 +-4 +-2 +0 +2 +4 +6 +x (c.u.)4 +Karampelas et al. +Figure 3. The evolution of the absolute value of the radial velocity for Model 2 (see Table 1), and the respective vector plot +(normalized). Starting from the top, from left to right, the snapshots correspond to time t = 0, 0.2, 0.8, 1.4, 1.6 and 1.8 t0. All +values are depicted in code units. +and velocity scales of our system are respectively L0 and +v0. Due to the finite size of our grid, our code also faces +the effects of the ‘effective’ numerical diffusivity, which +prevents us from using Rm values closer to those ex- +pected in the solar corona. Through a parameter study, +this numerical diffusivity is estimated to be in the order +of 10−6 to 10−5. +2.2. Initial Setup +This numerical study focuses on the perturbations of +a 2D magnetic X-point. Similarly to Karampelas et al. +(2022a), the equilibrium magnetic field is defined in +physical units as: +B = B0 +L0 +(y, x, 0) . +(6) +In Equation (6), B0 is the characteristic magnetic field +strength, and L0 is the characteristic length scale of the +magnetic field variations. A visual depiction of the mag- +netic field is shown in Figure 1, where the black solid +and dashed lines depict the magnetic field lines in the +regions with opposite polarities; the separatrices are in +red. +From Equation (6) we also see that the magni- +tude of the magnetic field is proportional to the radius +r = +� +x2 + y2. +We consider uniform equilibrium values for the density +and temperature across the physical domain, obtaining +a uniform initial sound speed +VS = √γ VS,iso = +� +γp/ρ = +� +γ R T +(7) +where γ = 5/3 is the ratio of the specific heats, and +R is the specific gas constant. +This also results in a +increasing Alfv´en speed +VA = +B +√µ0 ρ = B0 +L0 +r +√µ0 ρ, +(8) +(µ0 is the magnetic permeability of vacuum) as we move +away from the X-point. +Additionally, the choice of a +uniform initial density distribution prevents the devel- +opment of phase mixing in our setups (e.g. Heyvaerts +& Priest 1983). Figure 1 also depicts the equipartition +layer, i.e. the layer where the ratio of VA over VS equals +one. Given that the initial VS is constant in our setups, +and VA is proportional to the magnitude of the mag- +netic field, and thus the radius, the equipartition layer +will initially be a circle of radius, req, where: +req = L0 +B0 +� +γ ρ R T +µ0 +. +(9) +From Equation (9) we see that the initial radius of the +equipartition layer in our setups will be defined by the +values initial uniform plasma temperature and density, +and by the characteristic strength of the magnetic field. + +0.5 +6 +0.4 +4 +2 +y (c.u.) +0.3 +0 +0.2 +-2 +皖A·1 +: +-4 +0.1 +-6 +0.0 +-6 +-4 +-2 +0 +2 +4 +6 +x (c.u.)6 +0.2 +4 +2 +y (c.u.) +0.2 +0 +0.1 +-2 +-4 +0.1 +-6 +0.0 +-6 +-4 +-2 +0 +2 +4 +6 +x (c.u.)6 +0.5 +4 +0.4 +2 +y (c.u.) +0.3 +0 +-2 +0.2 +-4 +0.1 +-6 +0.0 +9- +-4 +-2 +0 +2 +4 +6 +X (c.u.)6 +0.7 +4 +0.6 +2 +0.5 +y (c.u.) +0.4 +0 +0.3 +-2 +0.2 +-4 +0.1 +-6 +0.0 +9- +-4 +-2 +0 +2 +4 +6 +X (c.u.)0.6 +6 +0.5 +4 +0.4 +2 +y (c.u.) +0 +0.3 +-2 +0.2 +-4 +0.1 +-6 +0.0 +9- +-4 +-2 +0 +2 +4 +6 +X (c.u.)0.6 +6 +0.5 +4 +2 +0.4 +y (c.u.) +0 +0.3 +-2 +0.2 +-4 +0.1 +-6 +0.0 +9- +-4 +-2 +0 +2 +4 +6 +X (c.u.)Oscillatory Reconnection as Plasma Diagnostic +5 +Figure 4. Time series of the Jz current density at the null point for setups with different characteristic magnetic field strength +(0.5 B0, 1 B0, 2 B0, and 3 B0). The equilibrium density and temperature are 1 ρ0 and 1 MK respectively. Cases without (left +column) and with anisotropic thermal conduction (right column) are considered. All values are depicted in code units. + +Without Thermal Conduction, O.5 Bo +5 +Jz (c. u.) +一5 +-10 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)With Thermal Conduction, O.5 Bo +5 +('n) +0 +-5 +N +-10 +-15 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 1 Be +5 +('n') +一5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)With Thermal Conduction, 1 Bo +5 +Jz (c.u.) +-5 +-10 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 2 Bo +5 +('n') +0 +-5 +5 +10 +15 +20 +25 +0 +30 +35 +40 +period (c.u.) +2468 +0 +5 +10 +15 +20 +25 +30 +35 +40 +time (c.u.)With Thermal Conduction, 2 Bo +5 +('n) { +-5 +-10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +period (c.u.) +2468 +0 +5 +10 +15 +20 +25 +30 +35 +40 +time (c.u.)Without Thermal Conduction, 3 Bo +5 +Jz (c. u.) +-5 +-10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +period (c.u.) +2 +4 +10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +time (c.u.)With Thermal Conduction, 3 Bo +5 +('n) { +0 +-5 +-10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +period (c.u.) +24 +¥68 +10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +time (c.u.)6 +Karampelas et al. +Figure 5. Graph depicting the distribution of the Jz oscil- +lation period with respect to the magnetic field magnitude at +radius r = 1. Overplotted are the fits for both distributions +of the function F(B0) = a (B0)−1 + b. The blue dashed and +orange dotted lines correspond to the cases without and with +anisotropic thermal conduction. All values are depicted in +code units, unless stated otherwise. +In order to initiate oscillatory reconnection at the +X-point, we use a circular fast magnetoacoustic pulse +(mentioned as Ring driver in Karampelas et al. 2022b) +to perturb the magnetic field from its equilibrium state. +The horizontal components of the velocity pulse, as +shown in Figure 2, are calculated as follows: +vx = (v∥Bx + v⊥By)/(B2 +x + B2 +y), +(10) +vy = (v∥Bx − v⊥By)/(B2 +x + B2 +y), +(11) +where v⊥ = (v×B)·ˆz is a quantity related to the veloc- +ity component perpendicular to the magnetic field lines +and v∥ = v · B is a quantity related to the velocity com- +ponent parallel to the magnetic field lines. +Following +Karampelas et al. (2022a), we consider a fast magnetoa- +coustic wave pulse (in code units) of the form: +v⊥(t = 0) = +1 +0.2 +√ +2π exp +� +−0.5(r − 5)2 +0.2 +� +, +(12) +v∥(t = 0) = 0. +(13) +2.3. Domain and Boundary Conditions +Our setup consists of a square domain with a struc- +tured uniform grid with a range (x, y) ∈ [−10, 10] in +code units, and resolution of 1801 × 1801 grid points. +We use reflective boundaries for the velocity components +(vx, vy), so that no flows can cross the boundary and dis- +rupt the initial equilibrium. To prevent the accumula- +tion of heat at the boundaries, once thermal conduction +is switched on, we fix pressure and density at the bound- +aries to their initial values. In order to keep the current +density at the edges or our domain from getting artifi- +cial values due to boundary effects, we take zero-gradient +boundary conditions for the magnetic field components +(of the form Bi − Bi−1 = Bi−1 − Bi−2). +Following Karampelas et al. (2022b), we take steps to +minimize the amount of reflected waves from the bound- +aries returning to the null point. +Our first step is to +deal with the outward propagating velocity pulse that +emerges from the splitting of the initial velocity annulus, +as we can seen in Figure 3 at t=0.2 t0. We do so, after +the start of the each simulation, by turning the value of +the velocity components to zero for a region with radius +of r ⩾ 7. +The second step is to create a numerical dissipation +scheme away from the null point, with the purpose of +reducing the kinetic energy of the waves in that region. +To that end, we divide each velocity component by a +dissipation coefficient nd > 1, for each iteration. The +relation for the coefficient is given in code units: +nd = 1.0005 + 0.0005 tanh(r − rd), +t > tC, +(14) +where tC is the time that we switch off the outward prop- +agating pulse in the previous step and rd being the ef- +fective distance from the null point at which the scheme +starts acting. The value for rd changes for each setup, to +better accommodate the effects of the different Alfv´en +and sound speed profiles for each setup and to make the +dissipation of the reflective waves more effective. +Finally, for some of our setups we introduce explicit +physical viscosity in the MHD equations, in addition +to the previous numerical dissipation scheme (see also +Karampelas et al. 2022b), with coefficient in code units: +nvisc = 0.1 + 0.1 tanh(r − rd), +t > tC. +(15) +3. RESULTS +The purpose of this study is to gain a better under- +standing into the nature of oscillatory reconnection in a +hot coronal plasma and to explore its behaviour under +different coronal conditions. +To that end, we expand +the results of Karampelas et al. (2022a) through a se- +ries of parameter studies. For each parameter study, we +change either the characteristic strength of the magnetic +field, the equilibrium density or the initial temperature. +The studies for the different magnetic field and density +have been performed both in the absence and presence +of anisotropic thermal conduction, whereas the parame- +ter study for the temperature has been performed only +for setups without thermal conduction. An overview of +the different cases can be found in Table 1. +The initial velocity perturbation described by Equa- +tions (12) and (13) splits into two counter-propagating +pulses of equal amplitude, with each travelling to op- + +12 +Without T.C., fit: a=3.159. b=0.642 +With T.C., fit: a=3.398, b=0.671 +10 +Period (c.u.) +8 +6 +4 +2 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Magnetic field (Bo)Oscillatory Reconnection as Plasma Diagnostic +7 +Figure 6. Time series of the Jz current density at the null point for setups with different equilibrium density (ρ = 1, 2, 3 and +4 ρ0). The equilibrium magnetic field magnitude at a radius r = 1 and temperature are 1 B0 and 1 MK respectively. Again, +cases without (left column) and with anisotropic thermal conduction (right column) are considered. All values are depicted in +code units. + +Without Thermal Conduction, 1 po +5 +('n') +-5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)With Thermal Conduction, 1 po +5 +Jz (c.u.) +-5 +-10 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 2 po +5 +('n') +0 +-5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)With Thermal Conduction, 2 po +5 +Jz (c.u.) +0 +-5 +-10 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 3 po +5 +('n') +0 +-5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)With Thermal Conduction, 3 po +5 +Jz (c.u.) +-5 +10- +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 4 po +5 +('n') +0 +-5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)With Thermal Conduction, 4 po +5 +Jz (c.u.) +0 +-5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)8 +Karampelas et al. +Figure 7. Same as Figure 5, but here we depict the oscil- +lation period with respect to the background density. Over- +plotted are the fits for both distributions of the function +G(ρ0) = a (ρ0)1/2 + b. +Table 1. An overview of the physical parameters (in code +units) for the different models in our simulations. +Model +B (B0) +ρ (ρ0) +T (T0) +κ⊥, κ∥ +nvisc +rd (L0) +1 +0.5 +1.0 +1.0 +0, ̸= 0 +̸= 0 +6 +2 +1.0 +1.0 +1.0 +0, ̸= 0 +̸= 0 +5 +3 +2.0 +1.0 +1.0 +0, ̸= 0 +̸= 0 +6 +4 +3.0 +1.0 +1.0 +0, ̸= 0 +̸= 0 +6 +5 +1.0 +2.0 +1.0 +0, ̸= 0 +̸= 0 +6 +6 +1.0 +3.0 +1.0 +0, ̸= 0 +̸= 0 +6 +7 +1.0 +4.0 +1.0 +0, ̸= 0 +̸= 0 +6 +8 +1.0 +1.0 +1.0 +0 +0 +5 +9 +1.0 +1.0 +3.0 +0 +0 +5 +10 +1.0 +1.0 +5.0 +0 +0 +6 +11 +1.0 +1.0 +7.0 +0 +0 +6 +12 +1.0 +1.0 +10.0 +0 +0 +6 +Note—Models 1 to 7 have been studied for both without +(κ⊥, κ∥ = 0) and with (κ⊥, κ∥ ̸= 0) anisotropic thermal +conduction. +posing directions. While we deal with outward propa- +gating pulse in the way that was described in the pre- +vious section, we focus on the evolution and effects of +the pulse approaching the null point. The inward prop- +agating pulse focuses at the X-point due to refraction, +as shown in Figure 3 for the default setup without ther- +mal conduction (B0 = 1, ρ0 = 1 and T = 1 MK, see +Model 2 from Table 1). Mode conversion takes place as +the fast magnetoacoustic wave pulse crosses the equipar- +tition layer, from the region of low-β to the region of +high-β plasma (e.g. McLaughlin & Hood 2006; Karam- +pelas et al. 2022a), deforming the layer in the process +due to the formation of strong compression and rarefac- +tion shocks in the y-direction and x-direction respec- +tively (see also Gruszecki et al. 2011). +Once the pulse reaches the null point, it perturbs it +from its equilibrium, forcing it to perform a series of re- +connection events, that are characterized by a periodic +manifestation of horizontal and vertical current sheets +(i.e. oscillatory reconnection). Like in the past stud- +ies, our main tool of studying oscillatory reconnection +will be the tracking of the oscillating Jz current den- +sity at the perturbed null point, as was first performed +by McLaughlin et al. (2009), and the calculation of its +period for each different case. +3.1. Magnetic Field Dependence +Our first goal is to revisit the effects of the charac- +teristic strength of the magnetic field (B0) on oscilla- +tory reconnection of an X-point in a hot coronal plasma. +A first study has been performed in Karampelas et al. +(2022a), for an X-point in the presence of anisotropic +thermal conduction. Here we will repeat this analysis +for the updated numerical dissipation scheme that was +first introduced in Karampelas et al. (2022b). The latter +is more efficient in dealing with the reflections return- +ing to the perturbed null point and thus leads to less +contamination of the Jz current density signal and a +cleaner resulting spectrum. Unlike the previous parame- +ter study on the magnetic field strength (see Karampelas +et al. 2022a), here we will expand the analysis for setups +both in the presence and absence of anisotropic thermal +conduction. In total we will consider four different val- +ues for the characteristic strength of the magnetic field +(0.5B0, 1 B0, 2 B0 and 3 B0, where B0 = 1.44 Gauss). +We note here that the magnitude of the magnetic field +is proportional to the radius for the X-point, and that +the characteristic value of the field is not the maximum +value in our setups. As we can see in Table 1 for models +1 to 4, in these four cases the initial density and tem- +perature are 1 ρ0 = 10−12 kg m−3 and 1 T0 = 1 MK, and +we will consider both the numerical dissipation scheme +and a non-zero viscosity coefficient (nvisc) away from +the null, in order to deal with the reflective waves. +The produced time series for the Jz current density of +the different cases are shown gathered in Figure 4, were +the results both and with and without thermal conduc- +tion are shown. Upon a visual inspection, we see that +in all cases oscillatory reconnection has developed, as is +hinted by the oscillatory Jz signal at the null. The time +series reveal for a stronger, and therefore stiffer mag- +netic field, the phenomenon of oscillatory reconnection +lasts for progressively shorter times, before the oscilla- + +9 +With0ut T.C., fit: a=3.309, b=0.484 +With T.C., fit: a=3.352. b=0.602 +8 +Period (c.u.) +7 +6 +5 +4 +3 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Density (po)Oscillatory Reconnection as Plasma Diagnostic +9 +Figure 8. Time series of the Jz current density at the null point for setups with different equilibrium temperature (T = 1, 3, 5, 7 +and 10 MK). The equilibrium magnetic field magnitude at a radius r = 1 and density are 1 B0 and 1 ρ0 respectively. Only setups +without anisotropic thermal conduction are considered. All values are depicted in code units. +tion is damped. This is in agreement with Karampelas +et al. (2022a), where it was shown that the decay rates +for of these oscillations increase for stronger magnetic +fields. On that note and to reduce the computational +costs, the simulations for 2 B0 and 3 B0 are left running +only up to t = 40 t0, since the oscillation decays faster +than the other cases. +In the same study it was also +shown that the period of the oscillation also decreases +for stronger, stiffer magnetic fields. +This can also be +derived from Figure 4, once we focus on the calculated +wavelet spectra for each case, shown here below their +respective time series. As we see, there is a clear trend +regarding the period of the oscillation, with the domi- +nant period band being shifted towards smaller values +for stronger fields. Finally, we see that for most of the +cases studied here, the dissipation scheme used to deal +with the reflected waves is working efficiently, allowing +us to produce clear spectra, where there is one clearly +defined band of periods. The only exception is for the +case of 0.5 B0 without anisotropic thermal conduction, +where a strong secondary band of periods is observed. +From Karampelas et al. (2022a) and Karampelas et al. +(2022b) it was shown that these secondary period bands +are associated with the reflected waves returning to the +null point. This means that for this particular case, with +0.5 B0 characteristic magnetic field strength, our dissi- +pation scheme was less effective than in the other cases. +However, the main period band is still clearly defined +and more prominent that the other one. +In order to quantify this trend, we use the wavelet +spectra to calculate the oscillation period for each case. +We do so by first locating the coordinates (time t0 and +period P0) of maximum power for each spectrum. We +then consider a time interval ∆t = [t0 − P0, t0 + 3 P0] +containing the periods that exhibit higher values of +power, for which we calculate the average value for the +period, and the standard deviation that will act as the +error in the calculation. The calculated average values +for the period of each oscillating signal are then placed +in the graph of period versus the magnetic field strength, +shown in Figure 5. The calculated standard deviation +for each value is added as error bars for each point, al- +though for most cases theses error bars are barely visi- +ble. The data points on Figure 5 clearly hint towards an +inverse proportionality relation between the oscillation +period and the magnetic field strength. Because of this, +we have fitted both sets of data points (with thermal +conduction, in orange and without thermal conduction, +in blue) with the function F(B0) = a (B0)−1 + b. Fig- +ure 5 also contains the values of the coefficients for both +cases, which are a = (3.159 ± 0.096, 3.398 ± 0.046) and +b = (0.642 ± 0.111, 0.671 ± 0.053) for the cases without +and with thermal conduction respectively. We see that +the addition of thermal conduction does not alter the + +Without Thermal Conduction, 1 Mk +5 +('n') +一5 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 3 Mk +4 +2 +('n) +0 +-2 +-4 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 5 Mk +2 +('n) +N +-2 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 7 Mk +2 +('n) +N +-2 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)Without Thermal Conduction, 1o Mk +2 +1 +('n') +-1 +-2 +0 +10 +20 +30 +40 +50 +60 +period (c.u.) +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +time (c.u.)10 +Karampelas et al. +Figure 9. Graph depicting the distribution of the Jz oscil- +lation period with respect to the equilibrium temperature. +In the left panel, we fit the function F(T0) = a (T0)−1 + b +to our original data points (red dashed line) and to the ad- +justed data points (black dotted line). In the right panel, we +do the same but for the function H(T0) = a (T0)−1/2 +b. All +setups are considered in the absence of thermal conduction. +All values are depicted in code units, unless stated otherwise. +trend in any significant way. We also see that the setups +with thermal conduction generally give higher values of +the period than those cases without thermal conduction, +in agreement with our past studies (Karampelas et al. +2022a,b). +3.2. Density Dependence +Our next goal is to study the response of oscillatory +reconnection for different equilibrium density. We have +considered again four different cases, where we take den- +sity values ρ = 1, 2, 3 and 4 ρ0 where ρ0 = 10−12 kg +m−3. In all cases we have taken a characteristic strength +of the magnetic field equal to 1 B0 = 1.44 Gauss, and +temperature 1 T0 = 1 MK. All cases are studied both in +the presence and absence of anisotropic thermal conduc- +tion. Just like before, we will again consider both the +numerical dissipation scheme and a non-zero viscosity +coefficient (nvisc) away from the null, to treat the reflec- +tions. The details of the different models (2, 5, 6 and 7) +are shown on Table 1. +The derived time series for the Jz current density are +shown in Figure 6 alongside their respective wavelet +spectra. +Again, the spectra of the time series reveal +a prominent period band for each case, associated with +the oscillatory reconnection process, the secondary pe- +riod bands from the reflected waves being of lower power. +Again, upon a visual inspection we see that by increas- +ing the value of the equilibrium density, the resulting +period of the oscillation increases as well, again with +thermal conduction leading to higher periods. +Following the same process as in the previous case, +we derive the average values for the period, and the +errors from the standard deviation for each case and +we place them in the same oscillation period-density +graph, in Figure 7. +Again, we see a clear trend for +each set of data points (with and without thermal con- +duction, shown in orange and blue respectively). +For +each set of data points, we fit the function G(ρ0) = +a (ρ0)1/2 + b, that we believe is showing the best agree- +ment with the observed trend of the values of the pe- +riod. +The values of the coefficients, as derived from +the fit, are a = (3.309 ± 0.303, 3.352 ± 0.190) and +b = (0.484±0.479, 0.602±0.300) without and with ther- +mal conduction, respectively and are also shown Figure +7. +3.3. Temperature Dependence +The final parameter study that we want to perform re- +volves around the response of oscillatory reconnection to +the initial background temperature. In the previous sub- +section, we took setups of different background densities, +but we kept temperature the same for all cases, mean- +ing that the sound speed was always the same for those +cases. In this section, we will consider setups with differ- +ent temperatures, and therefore different sound speeds +as well. +We have considered 5 different cases, corre- +sponding to models 8 to 12 on Table 1. In all models +we have taken a characteristic strength of the magnetic +field equal to 1 B0 = 1.44 Gauss, and an initial density +of 1 ρ0 = 10−12 kg m−3, while the temperature takes +values of 1, 3, 5, 7 and 10 MK. Unlike the two previous +parameter studies, no anisotropic thermal conduction +is considered in this one. +This is due to the ever in- +creasing computational costs once thermal conduction +is considered, caused by a combination of the increasing +temperatures and the Alfv´en speed profile for our given +magnetic field geometry, making these simulations very +costly to perform for the proper resolution. Addition- +ally, for these five cases viscosity has been dropped from +the artificial dissipation scheme dealing with the reflec- + +4.4 +Original, fit: a=0.743, b=3.532 +Adjusted, fit: a=0.743, b=3.219 +4.2 +Period (c.u.) +4.0 +3.8 +3.6 +3.4 +3.2 +2 +4 +6 +8 +10 +Temperature (1 MK)4.4 +Original, fit: a=1.020, b=3.241 +Adjusted, fit: a=1.020, b=2.929 +4.2 +Period (c.u.) +4.0 +3.8 +3.6 +3.4 +3.2 +2 +4 +6 +8 +10 +Temperature (1 MK)Oscillatory Reconnection as Plasma Diagnostic +11 +Figure 10. Left: Oscillation period versus the background Alfv´en speed at radius r = 1, calculated for all the setups with the +different background density and equilibrium magnetic field. The function F(VA) = a (VA)−1 + b is fitted for both data sets. +The color choice of Figure 5 is also followed here. Right: Same graph as in Figure 9, only now the background sound speed is +depicted instead of the background temperature. The function F(VS) = a (VS)−1 + b is fitted for both data sets. All values are +depicted in code units, unless stated otherwise. +tions. The viscous scheme was not working efficiently for +the cases with higher temperatures and so we decided +to drop it from the setups with the lower temperatures, +for consistency. +Figure 8 shows the produced times series of the Jz +current density profiles at the perturbed null point, and +the corresponding wavelet spectra for each profile. Un- +like before, the changes in period here seem to be more +subtle from one setup to the next. We see the gradual +appearance of a secondary period band, which becomes +increasingly stronger for higher temperatures, but with- +out ever reaching the same power as the main period +band. Given the more uneven Jz signal for higher tem- +peratures, it is safe to associate this secondary period +band with the reflected waves from the boundaries, pol- +luting the null point region and giving rise to more noisy +signals. +Following the same methodology as before for the +magnetic field and the density, we again calculate the +average values for the period of each case, and their re- +spective errors through the standard deviation, and we +plot then together in a graph, showing the relation be- +tween the period of oscillation and the background tem- +perature. The results are shown in red in both panels of +Figure 9. In the left panel, just like before, we also fit the +the function F(T0) = a (T0)−1 + b, with the coefficients +taking the values a = 0.743±0.060 and b = 3.532±0.029. +Although, the fitted function passes through all the data +points if we consider their error bars, we have also de- +cided to fit the function H(T0) = a (T0)−1/2 + b in our +data (right panel), with coefficients a = 1.02 ± 0.013 +and b = 3.241 ± 0.007. +As we can see by compar- +ing both panels of Figure 9, the function H(T0) pro- +vides a better fit on the given data, with the coeffi- +cients presenting smaller errors by comparison to those +for F(T0). +Therefore, from now on we will be using +the H(T0) = a (T0)−1/2 + b function to describe the de- +pendency of the period to the background temperature. +Also, it becomes obvious that for our range of chosen +temperature that matches the coronal conditions, the +variations of the period are considerably smaller than +the other case that we have examined. +Finally, we need to address the effects of the differ- +ent dissipation scheme used in this parameter study. As +mentioned earlier, for the cases considered in this sub- +section we took the numerical dissipation scheme de- +scribed by the coefficient of Equation (14), without the +supplementary viscous scheme described by the coeffi- +cient of Equation (15). +In other words, for models 8 +to 12 of Table 1, we took nvisc = 0. When comparing +the resulting periods for model 2 (P = 3.947 ± 0.022), +used in the previous two subsections and from its equiv- +alent model 8 used here (P = 4.259 ± 0.155), we see +that the two produce slightly different results. +It is +not certain how removing the artificial viscous scheme +leads to this small difference in period, of the order of +∆P ≈ 0.312 t0 = 2.43 s. It is very likely that we are +dealing with some code-specific numerical effects at this +point, which would be hard to properly treat within the +context of this study. However, the very small value of +this difference makes us confident to compare our cur- +rent results with those of the previous sections. To that +end, we have subtracted the difference 0.312 from the +periods for all of our data points shown in both pan- +els of Figure 9. We do this, because none of the cases +studied in this subsection had the viscous dissipation +scheme switched on and thus we have been consistent +among these five different setups. +The resulting ad- + +Without T.C., fit: a=0.911,b=0.574 +10 +With T.C..fit: a=0.941,b=0.672 +L +: +I: +Period (c.u.) +8 +6 +4 +2 +0.2 +0.4 +0.6 +0.8 +VAlfvén (c.u.)4.4 +Original, fit: a=1.317, b=3.241 +Adjusted, fit: a=1.317, b=2.929 +4.2 +Period (c.u.) +4.0 +3.8 +3.6 +3.4 +3.2 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Sound speed (c.u.)12 +Karampelas et al. +justed points (in black) follow the same trend as be- +fore for each panel, with the fitted function F(T0) hav- +ing coefficients a = 0.743 ± 0.060 (same as before) and +b = (3.532 − 0.312) ± 0.029 = 3.219 ± 0.029, and the fit- +ted function H(T0) having coefficients a = 1.02 ± 0.013 +(same as before) and b = 2.929 ± 0.007. We repeat that +from now on we will be using the H(T0) = a (T0)−1/2+b +function to describe the dependency of the period to the +background temperature. +4. DISCUSSION AND CONCLUSIONS +4.1. Parameter Studies +In this paper we once again revisit the mechanism of +oscillatory reconnection of a 2D X-point in hot coronal +plasma, further exploring its response to different coro- +nal conditions. This comes as a need, due to the large +number of observations that can be attributed to the +process of oscillatory reconnection. The first step was +taken in Karampelas et al. (2022a) where the periodic- +ity and the decay rate of the mechanism was studied in +the presence of anisotropic thermal conduction in coro- +nal conditions, expanding past studies that focused in +cold plasma. +In that same study, a clear connection +was revealed between the magnetic field strength and +the period of the oscillation. The second step was taken +in Karampelas et al. (2022b), where it was found that +the period of oscillatory reconnection of a magnetic X- +point perturbed by an external pulse is independent of +the amplitude and type of the perturbing pulse. These +two studies had already hinted towards the possibility +of using oscillatory reconnection as a tool for coronal +seismology. To that end, in the current study we have +expanded upon the results of Karampelas et al. (2022a), +by considering cases with different magnetic field, den- +sity and background temperature. +Our focus on the three quantities mentioned previ- +ously is based on the assumption that the properties of +oscillatory reconnection, like its period, in the absence +of dedicated external driving, should be related to the +conditions of the background plasma in the vicinity of +the null point. This is due to oscillatory reconnection +being a fundamental process, related to the relaxation of +a perturbed magnetic null point (here, X-point). There- +fore, we do not expect the large scale magnetic field +topology to affect our results, as the field geometry used +here is characteristic of the field geometry in the imme- +diate neighbourhood of an X-point. This is analogous to +null-points acting as resonant cavities (see Santamaria +& Van Doorsselaere 2018) where the wave-null point in- +teraction properties are determined by the background +plasma properties near the null point. Here we need to +note that having external driving would lead to a de- +pendancy of the oscillation period to the period of the +driving (Heggland et al. 2009). However, our focus in +this study is on the non-driven, relaxation based oscil- +latory reconnection. +Using the PLUTO code, we have solved the compress- +ible and resistive 2D MHD equations, for a series of +parameter studies. +The first one included four differ- +ent setups, each for a different value of the character- +istic strength of the magnetic field (0.5B0, 1 B0, 2 B0 +and 3 B0, where B0 = 1.44 Gauss), studied both with +and without anisotropic thermal conduction. We note +here that the characteristic value of the field is not the +same as its maximum value in each setups, rather the +field magnitude is proportional to the distance from the +X-point. The results revealed an inverse proportional- +ity between the period and the strength of the magnetic +field, as shown in Figure 5. In the second one we consid- +ered again four different cases, where we took the density +to be equal to ρ = 1, 2, 3 and 4 ρ0 where ρ0 = 10−12 kg +m−3, again studied both with and without anisotropic +thermal conduction. +The results, as shown in Figure +7 reveal a square root relation between the period and +the equilibrium density. The third and final parameter +study involved five setups with different values of back- +ground temperature (1, 3, 5, 7 and 10 MK), all studied +in the absence of anisotropic thermal conduction. This +last parameter study, shown in Figure 9, revealed an in- +verse proportional relation, this time between the period +and the square root of the background temperature. +As expected from our previous studies, the cases with +anisotropic thermal conduction practically follow the +same trend as their respective ones without thermal con- +duction, their only difference being that the former ex- +hibit slightly higher values of period. The only exception +are for those setups in the temperature parameter study, +where thermal conduction has not been considered, due +to increased computational costs for our given resolu- +tion. This is caused by a combination of the increasing +temperatures and the given magnetic field geometry. +Also, as was explained in the previous section, the +derived values for the period from the temperature pa- +rameter study are shifted with respect to the rest, due to +the slightly different artificial dissipation schemed, with- +out any supplementary viscosity-based scheme that was +used throughout it. Comparing the resulting periods of +two equivalent setups, each with a version of the dissi- +pation scheme, we get a difference of ∆P ≈ 0.312 t0 = +2.43 s, which is of the order of ∼ 8% from the value +of P ∼ 30 s that we get from the other two parameter +studies. Since we have used the same dissipation scheme +when studying the response of oscillatory reconnection +to temperature, we subtract ∆P from all of these re- + +Oscillatory Reconnection as Plasma Diagnostic +13 +Table 2. +Summary of fitted functions and the values of +their coefficients for the parameter studies, as well as for +the Alfv´en and sound speeds. +Study +Fit +a, b +a, b +(without T.C.) +(with T.C.) +Magnetic field +F(B0) +3.159, 0.642 +3.398, 0.671 +Density +G(ρ0) +3.309, 0.484 +3.352, 0.602 +Temperature +H(T0) +1.020, 3.241 +1.020, 2.929 +Alfv´en speed +F(VA) +0.911, 0.574 +0.941, 0.672 +Sound speed +F(VS) +1.317, 3.241 +1.317, 2.929 +Note—The three types of fitted functions are F(x) = +a (x)−1 + b, G(x) = a (x)1/2 + b and H(x) = a (x)−1/2 + b. +For the temperature and sound speed, both pairs of coef- +ficients are without thermal conduction, with the second +pair being for the adjusted data sets. This is indicated in +italics. +sults, as shown by the black line of the adjusted fit in +both panels of Figure 9. This allows for a better com- +parison with the other two parameter studies presented +here. A synopsis of the fitted functions and the values +of their coefficients can be found in Table 2. +4.2. Period vs Alfv´en and Sound Speed +Continuing on the trend set by our analysis so far, +we want to study the evolution of the period of oscil- +latory reconnection in terms of the Alfv´en and sound +speed profiles. +The initial Alfv´en speed profile is de- +pendent both on the initial equilibrium density and the +characteristic magnetic field strength. We then take the +results from the first 7 models of Table 1, for the dif- +ferent values of density and characteristic magnetic field +strength, calculate the characteristic Alfv´en speed and +plot them with respect to the period. +This graph is +featured on the left panel of Figure 10. +We again fit +the function F(VA) = a (VA)−1 + b, the coefficients of +which take values a = (0.911±0.027, 0.941±0.020) and +b = (0.574 ± 0.136, 0.672 ± 0.103) for the datasets with- +out and with thermal conduction, shown in dashed blue +and dotted orange lines. This fit for the Alfv´en speed is +in agreement with the previous fits for the magnetic field +and the density, since the Alfv´en speed is proportional +to the magnetic field and the inversely proportional to +the square root of density. The right panel of the same +figure, shows the results for the sound speed, which are +derived from those of the temperature parameter study +without thermal conduction. +On that panel we show +both the original values of the period (points in red) +Table 3. Examples of calculating the period of +oscillatory reconnection through Equation (17). +Bph (G) +ρph (kg m−3) +Tph (MK) +Pph (s) +10 +2.0 × 10−12 +5.0 +15.9 +20 +2.0 × 10−12 +5.0 +14.2 +30 +2.0 × 10−12 +5.0 +13.6 +20 +2.0 × 10−12 +3.0 +15.2 +20 +2.0 × 10−12 +10.0 +13.1 +20 +3.0 × 10−12 +5.0 +22.4 +20 +5.0 × 10−12 +5.0 +35.3 +20 +30.0 × 10−12 +5.0 +118.8 +Note—The subscript ‘ph’ refers to the physical +quantities Uph = U U0, with U the quantities +in code units and U0 the normalization unit +(see also §2.1). +and the adjusted ones (points in black) for which we +subtracted the difference ∆P ≈ 0.312 t0 = 2.43 s as was +mentioned in the section for the temperature parame- +ter study. Finally, we have fitted the function F(VS) = +a (VS)−1 + b, for the original (red dashed line) and the +adjusted data (black dotted line), the coefficients of +which take values a = (1.317 ± 0.016, 1.317 ± 0.016) +b = (3.241 ± 0.007, 2.929 ± 0.007) for the original and +adjusted data respectively. This fit agrees with the one +of the H(T0) = a (T0)−1/2 + b function for the back- +ground temperature, presented in Section 3.3, since the +sound speed is proportional to the square root of the +temperature. This further justifies the use of function +H(T0) to describe the relation between the period of +oscillatory reconnection and the plasma temperature. +4.3. Empirical Formula +As a last step, we want to merge the derived relations +from each one of the three parameter studies into one. +We do this because one of the main goals of this present +study was to start developing its capabilities as a plasma +diagnostic tool. To that end, we are aided by the results +of Karampelas et al. (2022b), that allow us to ignore +the strength of the perturbing pulse from this relation. +Taking the cases without thermal conduction, we can +merge the derived relations of each previous fit into the +following formula for our four key parameters: +Pph +t0 += 3.159 B0 +Bph ++3.309 +�ρph +ρ0 ++1.02 +� +T0 +Tph +−3.541±0.434 +(16) + +14 +Karampelas et al. +where the penultimate right-hand term comes from solv- +ing the above equation for a known value of the period +Pph (in s). For this, we considered the resulting period +for model 2 (Pph = (3.947 ± 0.022) t0). We also include +the maximum error in the last right-hand term that is +derived from the different combinations of errors in the +values of Pph and the coefficients of the fits. Here, the +subscript ‘ph’ refers to the physical quantities Uph, as de- +fined in §2.1. For the quantities in physical units we have +Uph = U U0, with U the quantities in code units and U0 +the normalization units. We used the adjusted results +for the temperature parameter study, as explained ear- +lier, while all the coefficients are given in code units. +Using the normalization units defined in §2.1, we can +rewrite the above formula in SI units, except for the +magnetic field which is given in Gauss: +Pph = 35.39 +Bph ++ 25.74 × 106√ρph + 7.94 +� +Tph +− 27.55 ± 3.38 +(17) +where we have the period Pph (in s) for a known combi- +nation of Bph (in G), ρph (in kg m−3) and Tph (in MK). +A similar analysis on the cases with added thermal con- +duction can be found in Appendix A. +Finally we show in Table 3, some examples of using the +above formula to calculate the periods of oscillatory re- +connection for different combinations of parameters for +a flaring coronal plasma. One thing that needs to be +stressed is that Equation (17) has been derived from +a set of values that reflects the average conditions in +the solar corona. As a result, we need to be cautious +when extrapolating the above relation for values out- +side of that parameter space, as we may end up with +non-physical results. However, the derived relation can +be a useful plasma diagnostic tool in coronal conditions, +and needs to be tested further against observational pe- +riodic signals, that could be attributed to oscillatory +reconnection. Such periodic signals in the solar atmo- +sphere include, but are not limited to quasi-periodic pul- +sations (QPPs) of solar (e.g. Kupriyanova et al. 2016) +and stellar flares (e.g. Broomhall et al. 2019), quasi- +periodic chromospheric (e.g. De Pontieu et al. 2011) +and coronal jets (e.g. Hong et al. 2019; Mandal et al. +2022), quasi-periodic fast-propagating (QFP) magne- +tosonic wave from the eruption of a magnetic flux rope +(e.g. Shen et al. 2018) and periodicities correlated with +Type III radio bursts (Cattell et al. 2021). A detailed +discussion of the different phenomena attributed to os- +cillatory reconnection has already been presented in the +Section 1. The fundamental nature of oscillatory recon- +nection in perturbed magnetic X-points, indicates that +our derived plasma diagnostic tool can be employed to +study the periodicities in the different cases of periodic +signals attributed to oscillatory reconnection. +To summarize, our series of parameter studies have ex- +plored the effects of temperature, density and magnetic +field strength on the periodicity of oscillatory reconnec- +tion in a hot coronal plasma, expanding the earlier re- +sults of Karampelas et al. (2022a). Our findings show +that the period of the oscillation depends on the under- +lying characteristics of the plasma near the null point. +Taking into additional account the independence of the +periodicity of oscillatory reconnection from the strength +and type of the initial, perturbing pulse (Karampelas +et al. 2022b), we have now developed a first quantita- +tive formula to be used as a plasma diagnostic, opening +the possibility of using this mechanism within the con- +text of coronal seismology. +All authors acknowledge UK Science and Technol- +ogy Facilities Council (STFC) support from grant +ST/T000384/1. +K.K. also acknowledges support by +an FWO (Fonds voor Wetenschappelijk Onderzoek – +Vlaanderen) postdoctoral fellowship (1273221N). This +work used the Oswald High Performance Computing fa- +cility operated by Northumbria University (UK). +APPENDIX +A. EMPIRICAL FORMULA FOR ADDED THERMAL CONDUCTION +In §4.3 we derived an empirical formula (see Equations 16 and 17) that connects the period of oscillatory reconnection +with the characteristic strength of the magnetic field, the background density and the equilibrium plasma temperature. +We did this by merging the derived relations of each fit in the data sets without anisotropic conduction, featured in +§3. A similar formula can be derived for the period, magnetic field strength and density when we include anisotropic +thermal conduction for a plasma temperature of Tph = 1 MK: +Pph +t0 += 3.398 B0 +Bph ++ 3.352 +�ρph +ρ0 +− 2.701 ± 0.547 +(A1) + +Oscillatory Reconnection as Plasma Diagnostic +15 +where we used the period of model 2 (see Table 1) with thermal conduction switched on (Pph = (4.049 ± 0.311) t0) in +order to calculate the penultimate right-hand term. Similarly to Equation (16), we also include the maximum error +derived from the different combinations of errors in the values of Pph and the coefficients of the fits. When written in +SI units, except from the magnetic field that is in Gauss, the previous relation takes the form: +Pph = 38.07 +Bph ++ 26.08 × 106√ρph − 21.01 ± 4.26 +(A2) +where we again take Bph in G and ρph in kg m−3, for Tph = 1 MK. +One drawback of the current study is the fact that, due to numerical reasons, implementing thermal conduction +for the setups with high coronal temperatures (> 1 MK) lead to very costly and slow to perform simulations for our +resolution of choice. That means that Equations (A1) and (A2) lack the temperature term of Equations (16) and +(17) and can only be valid for plasma with temperatures near 1 MK. However, that might not necessarily hinder +our analysis. 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K., et al. +2021, SSRv, 217, 66, doi: 10.1007/s11214-021-00840-9 + diff --git a/L9E0T4oBgHgl3EQfjAEs/content/tmp_files/load_file.txt b/L9E0T4oBgHgl3EQfjAEs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ea6d20199d0500fafbc3cb902b9c716e696ad91 --- /dev/null +++ b/L9E0T4oBgHgl3EQfjAEs/content/tmp_files/load_file.txt @@ -0,0 +1,1402 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf,len=1401 +page_content='Draft version January 9, 2023 Typeset using LATEX twocolumn style in AASTeX631 Oscillatory reconnection as a plasma diagnostic in the solar corona Konstantinos Karampelas,1, 2 James A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' McLaughlin,1 Gert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Botha,1 and St´ephane R´egnier1 1Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK 2Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium (Received November 1, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Revised December 6, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Accepted December 15, 2022) Submitted to ApJ ABSTRACT Oscillatory reconnection is a relaxation process in magnetised plasma, with an inherent periodicity that is exclusively dependent on the properties of the background plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This study focuses on the seismological prospects of oscillatory reconnection in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We perform three sets of parameter studies (for characteristic coronal values of the background magnetic field, density and temperature) using the PLUTO code to solve the fully compressive, resistive MHD equations for a 2D magnetic X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' From each parameter study, we derive the period of the oscillatory reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We find that this period is inversely proportional to the characteristic strength of the background magnetic field and the square root of the initial plasma temperature, while following a square root dependency upon the equilibrium plasma density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' These results reveal an inverse proportionality between the magnitude of the Alfv´en speed and the period, as well as the background sound speed and the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Furthermore, we note that the addition of anisotropic thermal conduction only leads to a small increase in the mean value for the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, we establish an empirical formula that gives the value for the period in relation to the background magnetic field, density and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This gives us a quantified relation for oscillatory reconnection, to be used as a plasma diagnostic in the solar corona, opening up the possibility of using oscillatory reconnection for coronal seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Keywords: Magnetohydrodynamics (1964);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Solar magnetic reconnection (1504);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Solar coronal seis- mology (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Solar coronal waves (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Magnetohydrodynamical simulations (1966);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' INTRODUCTION Oscillatory reconnection is a physical phenomenon characterised by a series of reconnection events Parker 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Sweet 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Petschek 1964) that take place along- side periodic changes in the magnetic connectivity of a perturbed magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The process was identified for the first time in Craig & McClymont (1991), during the study of the relaxation of an 2D X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' One impor- tant characteristic of oscillatory reconnection is that the Corresponding author: Konstantinos Karampelas konstantinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='karampelas@northumbria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='uk periodicity is not imposed by an external driver, rather it is an inherent property of the relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Over the recent years, a number of numerical stud- ies have been conducted regarding oscillatory reconnec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2009) studied the mechanism for a 2D magnetic X-point in a cold plasma, solving the fully compressible resistive MHD equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Using an external fast magnetoaccoustic pulse, they initiated os- cillatory reconnection by perturbing a magnetic X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This study had identified many properties of this mech- anism, like the periodic changes in the resulting current sheet orientation with the respective changes in connec- tivity, and the formation of both fast and slow oblique magnetic shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Thurgood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2017) later expanded the results of the previous study for a 3D null point, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='02452v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='SR] 6 Jan 2023 2 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' also reporting the generation of MHD waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Oscilla- tory reconnection has also been studied for a realistic solar atmosphere, as a result of flux rope emergence (Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2012b), while other studies revolved around the effects of resistivity, initial perturbation amplitude, and the length of the ini- tial current sheet on the period of the reconnection pro- cess (McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Thurgood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2018a,b, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Stewart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022) reported the onset of oscil- latory reconnection and the generation of waves through the coalescence of two cylindrical flux ropes, while Sabri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2020) reported the development of the plasmoid instability in a magnetic O-point and the resulting man- ifestation of plasmoid-mediated quasi-oscillatory mag- netic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The results of McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2009) have recently been expanded for a hot coronal plasma in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a), studying the re- lation between the oscillation period and the strength of the background magnetic field, while also taking into account the effects of anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A following study (Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022b) reported for the first time on the independence between the type and strength of the perturbing wave pulse and the frequency of the resulting oscillatory reconnection in a hot coronal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' These two studies have produced encouraging results regarding the possibility of using oscillatory re- connection as a new tool for coronal seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Magnetic reconnection can cause the dissipation of magnetic field and electric current, leading to the ac- celeration of particles, ejection of mass and heating through the generation of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As such, it is con- sidered as the main mechanism behind solar flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Shibata & Magara 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Jel´ınek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2015), while the ubiquitous null points in the solar atmosphere (Gals- gaard & Nordlund 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Brown & Priest 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Long- cope 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' R´egnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2008), where reconnection can take place, are consequently considered preferential lo- cations the manifestation of flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Murawski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Over the years, oscillatory reconnection has been proposed as a driving force behind observed phenom- ena like quasi-periodic pulsations (QPPs) of solar flares Kupriyanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Van Doorsselaere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Pugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2020a,b, 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Li & Chen 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022) and stellar flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Broomhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Guarcello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Jackman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Notsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Vida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Man- cuso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Ramsay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The mechanism is included in reviews summarising our current knowl- edge around QPPs and the proposed mechanisms behind them, such as McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2018), Kupriyanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2020), and Zimovets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In particu- lar, there are many examples from QPP observations (see histogram in McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2018, and its cor- responding online catalog), with reported periods close to those derived from the studies of Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a) and Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b), for the plasma conditions considered in those studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Connection has also been proposed between oscilla- tory reconnection and quasi-periodic flows, like those associated with spicules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' De Pontieu & McIntosh 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Samanta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Yurchyshyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2020), as well as with observed pe- riodicities in breakout current sheets at the base of jets (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022), have reported a highly dynamic small-scale jet in a polar coronal hole, and proposed oscillatory reconnection as a possible driv- ing mechanism behind the observed repetitive outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2012b) were able to reproduce such observed periodic outflows through oscillatory reconnec- tion in a 2D flux emergence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The resulting peri- ods from that model had a very good match with those reported from wavelet analysis in Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022), although the latter showed no significant power at the 99% confidence level, preventing them to characterise the outflows as periodic, but merely repetitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Obser- vational signatures of chromospheric jets by periodic re- connection events were also reported in simulations by Heggland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2009), although, the periodicity was attributed to the continuous driving rather than being inherent to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Oscillatory reconnection has also been considered as a possible mechanism behind with the creation of an observed quasi-periodic fast- propagating (QFP) magnetosonic wave from the erup- tion of a magnetic flux rope (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2018), as well as behind the formation and disappearance of a small scale magnetic flux rope consisting of new loops formed by the reconnection events (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2014) have reported oscillatory (or reciprocatory) magnetic reconnection in observations of Coronal Bright Points (CBPs), while reversals of an elongated current sheet in a recent numerical 2D CBP model has been at- tributed to oscillatory reconnection (N´obrega-Siverio & Moreno-Insertis 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, recent observations by the Parker Solar Probe could also be attributed to oscil- latory reconnection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2016, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019), like Alfv´enic spikes (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2021) and pe- riodicities correlated with Type III radio bursts (Cattell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In this paper, we will further investigate oscillatory reconnection in a hot coronal plasma and to explore its potential for utilising oscillatory reconnection as a tool for coronal seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We will expand the re- sults of Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a) through a series of Oscillatory Reconnection as Plasma Diagnostic 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Magnetic field lines of the unperturbed X-point, were the black solid and dashed lines depict the regions of opposing polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The separatrices (red solid lines) and the equipartition layer for a 1 MK coronal plasma (blue circular line) are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2D profiles of the vx and vy velocity components of the initial circular pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The magenta circular line is the equipartition layer for a 1 MK coronal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' parameter studies for different characteristic strengths of the magnetic field, equilibrium plasma density and initial plasma temperature, for a 2D magnetic X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Like in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a) and Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b), we will be exploring these cases both in the ab- sence and presence of anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In section 2 we present our physical domain, code used to solve the fully compressible mhd equations and nu- merical schemes utilised, while we present the results of the parameter studies in the respective subsection in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, our conclusions and general discussion take place in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' NUMERICAL SETUP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Numerical Scheme For the numerical studies below, we solve the 2D com- pressible resistive MHD equations in cartesian coordi- nates, in the absence of gravity (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022a), using the PLUTO code (Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2007, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Like in our past studies (Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022a,b), we employ the fifth-order monotonicity pre- serving scheme (MP5) for the spatial integration and the third-order Runge-Kutta method for the time inte- gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To satisfy the solenoidal constraint of the mag- netic field (∇·B = 0), we use the Constrained Transport method implemented in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In these simulations, we also consider setups where we introduce anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The val- ues for the parallel and perpendicular thermal conduc- tion coefficients (in J s−1 K−1 m−1), as calculated from the Spitzer conductivity (Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2008), are given below: κ∥ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 × 10−12 T 5 2 , (1) κ⊥ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 × 10−21 n2 H √ TB2 , (2) where κ∥, κ⊥ and the hydrogen number density nH, temperature T and magnetic field B are all given in SI1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The effects of saturation are also taken into account for very large temperature gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The corresponding source term (∇ · Fc) in the energy equation varying be- tween the classical (Fclass) and saturated thermal con- duction (Fsat): ∇ · Fc = ∇ · � Fsat Fsat + |Fclass|Fclass � (3) Fclass = κ∥ˆb � ˆb · ∇T � + κ⊥ � ∇T − ˆb � ˆb · ∇T �� (4) Fsat = 5 φ ρ V 3 S,iso, (5) where VS,iso = � p/ρ is the isothermal sound speed, ˆb = B/|B| is the unit vector in the direction of magnetic field and φ is a free code parameter (with a default value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' For zero magnetic field, Fc reduces to Fc = κ∥ ∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' During this analysis, we will be working in code units U = Uph U −1 0 , with Uph being the physical quantities and U0 the normalization units U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The constants U0 are characteristic values, chosen for solar coronal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We consider the unit length L0 = 1 Mm, unit density ρ0 = 10−12 kg m−3, and unit velocity v0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='29 × 105 m s−1, equal to VS/√γ for coronal plasma at 1 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We also take the unit temperature T0 = 1 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The char- acteristic magnetic field and unit time are respectively B0 = � µρ0v2 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='44 G and t0 = L0/v0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='78 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Since we want to solve the resistive MHD equations, we take the magnetic diffusivity in code units as η = R−1 m = 10−5, where Rm = (v0 L0)/η = 105 is the mag- netic Reynolds number, assuming that the typical length 1 In cgs, the thermal conduction coefficients (in erg s−1 K−1 cm−1) are given as κ∥ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 × 10−7 T 5 2 and κ⊥ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 × 10−16 n2 H √ TB2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 y (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 1 1 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 x (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 y (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 6 4 2 0 2 4 6 x (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 y (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 6 4 2 0 2 4 6 x (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )4 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The evolution of the absolute value of the radial velocity for Model 2 (see Table 1), and the respective vector plot (normalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Starting from the top, from left to right, the snapshots correspond to time t = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' and velocity scales of our system are respectively L0 and v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Due to the finite size of our grid, our code also faces the effects of the ‘effective’ numerical diffusivity, which prevents us from using Rm values closer to those ex- pected in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Through a parameter study, this numerical diffusivity is estimated to be in the order of 10−6 to 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Initial Setup This numerical study focuses on the perturbations of a 2D magnetic X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Similarly to Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a), the equilibrium magnetic field is defined in physical units as: B = B0 L0 (y, x, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (6) In Equation (6), B0 is the characteristic magnetic field strength, and L0 is the characteristic length scale of the magnetic field variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A visual depiction of the mag- netic field is shown in Figure 1, where the black solid and dashed lines depict the magnetic field lines in the regions with opposite polarities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' the separatrices are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' From Equation (6) we also see that the magni- tude of the magnetic field is proportional to the radius r = � x2 + y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We consider uniform equilibrium values for the density and temperature across the physical domain, obtaining a uniform initial sound speed VS = √γ VS,iso = � γp/ρ = � γ R T (7) where γ = 5/3 is the ratio of the specific heats, and R is the specific gas constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This also results in a increasing Alfv´en speed VA = B √µ0 ρ = B0 L0 r √µ0 ρ, (8) (µ0 is the magnetic permeability of vacuum) as we move away from the X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Additionally, the choice of a uniform initial density distribution prevents the devel- opment of phase mixing in our setups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Heyvaerts & Priest 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 1 also depicts the equipartition layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' the layer where the ratio of VA over VS equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Given that the initial VS is constant in our setups, and VA is proportional to the magnitude of the mag- netic field, and thus the radius, the equipartition layer will initially be a circle of radius, req, where: req = L0 B0 � γ ρ R T µ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (9) From Equation (9) we see that the initial radius of the equipartition layer in our setups will be defined by the values initial uniform plasma temperature and density, and by the characteristic strength of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 4 2 y (c.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 9- 4 2 0 2 4 6 X (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )Oscillatory Reconnection as Plasma Diagnostic 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Time series of the Jz current density at the null point for setups with different characteristic magnetic field strength (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 B0, 1 B0, 2 B0, and 3 B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The equilibrium density and temperature are 1 ρ0 and 1 MK respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Cases without (left column) and with anisotropic thermal conduction (right column) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Without Thermal Conduction, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 Bo 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 一5 10 0 10 20 30 40 50 60 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )With Thermal Conduction, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content="5 Bo 5 ('n) 0 5 N 10 15 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 1 Be 5 ('n') 一5 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )With Thermal Conduction, 1 Bo 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 0 10 20 30 40 50 60 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 2 Bo 5 ('n') 0 5 5 10 15 20 25 0 30 35 40 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 2468 0 5 10 15 20 25 30 35 40 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )With Thermal Conduction, 2 Bo 5 ('n) { 5 10 0 5 10 15 20 25 30 35 40 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 2468 0 5 10 15 20 25 30 35 40 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )Without Thermal Conduction, 3 Bo 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 0 5 10 15 20 25 30 35 40 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 2 4 10 0 5 10 15 20 25 30 35 40 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )With Thermal Conduction, 3 Bo 5 ('n) { 0 5 10 0 5 10 15 20 25 30 35 40 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 24 ¥68 10 0 5 10 15 20 25 30 35 40 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )6 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Graph depicting the distribution of the Jz oscil- lation period with respect to the magnetic field magnitude at radius r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Overplotted are the fits for both distributions of the function F(B0) = a (B0)−1 + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The blue dashed and orange dotted lines correspond to the cases without and with anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In order to initiate oscillatory reconnection at the X-point, we use a circular fast magnetoacoustic pulse (mentioned as Ring driver in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022b) to perturb the magnetic field from its equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The horizontal components of the velocity pulse, as shown in Figure 2, are calculated as follows: vx = (v∥Bx + v⊥By)/(B2 x + B2 y), (10) vy = (v∥Bx − v⊥By)/(B2 x + B2 y), (11) where v⊥ = (v×B)·ˆz is a quantity related to the veloc- ity component perpendicular to the magnetic field lines and v∥ = v · B is a quantity related to the velocity com- ponent parallel to the magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Following Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a), we consider a fast magnetoa- coustic wave pulse (in code units) of the form: v⊥(t = 0) = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 √ 2π exp � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5(r − 5)2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 � , (12) v∥(t = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (13) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Domain and Boundary Conditions Our setup consists of a square domain with a struc- tured uniform grid with a range (x, y) ∈ [−10, 10] in code units, and resolution of 1801 × 1801 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We use reflective boundaries for the velocity components (vx, vy), so that no flows can cross the boundary and dis- rupt the initial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To prevent the accumula- tion of heat at the boundaries, once thermal conduction is switched on, we fix pressure and density at the bound- aries to their initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In order to keep the current density at the edges or our domain from getting artifi- cial values due to boundary effects, we take zero-gradient boundary conditions for the magnetic field components (of the form Bi − Bi−1 = Bi−1 − Bi−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Following Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b), we take steps to minimize the amount of reflected waves from the bound- aries returning to the null point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Our first step is to deal with the outward propagating velocity pulse that emerges from the splitting of the initial velocity annulus, as we can seen in Figure 3 at t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We do so, after the start of the each simulation, by turning the value of the velocity components to zero for a region with radius of r ⩾ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The second step is to create a numerical dissipation scheme away from the null point, with the purpose of reducing the kinetic energy of the waves in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To that end, we divide each velocity component by a dissipation coefficient nd > 1, for each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The relation for the coefficient is given in code units: nd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0005 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0005 tanh(r − rd), t > tC, (14) where tC is the time that we switch off the outward prop- agating pulse in the previous step and rd being the ef- fective distance from the null point at which the scheme starts acting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The value for rd changes for each setup, to better accommodate the effects of the different Alfv´en and sound speed profiles for each setup and to make the dissipation of the reflective waves more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, for some of our setups we introduce explicit physical viscosity in the MHD equations, in addition to the previous numerical dissipation scheme (see also Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022b), with coefficient in code units: nvisc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 tanh(r − rd), t > tC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (15) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' RESULTS The purpose of this study is to gain a better under- standing into the nature of oscillatory reconnection in a hot coronal plasma and to explore its behaviour under different coronal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To that end, we expand the results of Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a) through a se- ries of parameter studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' For each parameter study, we change either the characteristic strength of the magnetic field, the equilibrium density or the initial temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The studies for the different magnetic field and density have been performed both in the absence and presence of anisotropic thermal conduction, whereas the parame- ter study for the temperature has been performed only for setups without thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' An overview of the different cases can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The initial velocity perturbation described by Equa- tions (12) and (13) splits into two counter-propagating pulses of equal amplitude, with each travelling to op- 12 Without T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=', fit: a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='642 With T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=', fit: a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='398, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='671 10 Period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 8 6 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 Magnetic field (Bo)Oscillatory Reconnection as Plasma Diagnostic 7 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Time series of the Jz current density at the null point for setups with different equilibrium density (ρ = 1, 2, 3 and 4 ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The equilibrium magnetic field magnitude at a radius r = 1 and temperature are 1 B0 and 1 MK respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Again, cases without (left column) and with anisotropic thermal conduction (right column) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" Without Thermal Conduction, 1 po 5 ('n') 5 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )With Thermal Conduction, 1 po 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 0 10 20 30 40 50 60 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 2 po 5 ('n') 0 5 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )With Thermal Conduction, 2 po 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 0 5 10 0 10 20 30 40 50 60 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 3 po 5 ('n') 0 5 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )With Thermal Conduction, 3 po 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10- 0 10 20 30 40 50 60 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 4 po 5 ('n') 0 5 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )With Thermal Conduction, 4 po 5 Jz (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 0 5 0 10 20 30 40 50 60 period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )8 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Same as Figure 5, but here we depict the oscil- lation period with respect to the background density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Over- plotted are the fits for both distributions of the function G(ρ0) = a (ρ0)1/2 + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' An overview of the physical parameters (in code units) for the different models in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Model B (B0) ρ (ρ0) T (T0) κ⊥, κ∥ nvisc rd (L0) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 6 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 6 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 6 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 6 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 6 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0, ̸= 0 ̸= 0 6 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0 0 5 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0 0 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0 0 6 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0 0 6 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 0 0 6 Note—Models 1 to 7 have been studied for both without (κ⊥, κ∥ = 0) and with (κ⊥, κ∥ ̸= 0) anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' posing directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' While we deal with outward propa- gating pulse in the way that was described in the pre- vious section, we focus on the evolution and effects of the pulse approaching the null point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The inward prop- agating pulse focuses at the X-point due to refraction, as shown in Figure 3 for the default setup without ther- mal conduction (B0 = 1, ρ0 = 1 and T = 1 MK, see Model 2 from Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Mode conversion takes place as the fast magnetoacoustic wave pulse crosses the equipar- tition layer, from the region of low-β to the region of high-β plasma (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' McLaughlin & Hood 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Karam- pelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022a), deforming the layer in the process due to the formation of strong compression and rarefac- tion shocks in the y-direction and x-direction respec- tively (see also Gruszecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Once the pulse reaches the null point, it perturbs it from its equilibrium, forcing it to perform a series of re- connection events, that are characterized by a periodic manifestation of horizontal and vertical current sheets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' oscillatory reconnection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Like in the past stud- ies, our main tool of studying oscillatory reconnection will be the tracking of the oscillating Jz current den- sity at the perturbed null point, as was first performed by McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2009), and the calculation of its period for each different case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Magnetic Field Dependence Our first goal is to revisit the effects of the charac- teristic strength of the magnetic field (B0) on oscilla- tory reconnection of an X-point in a hot coronal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A first study has been performed in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a), for an X-point in the presence of anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Here we will repeat this analysis for the updated numerical dissipation scheme that was first introduced in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The latter is more efficient in dealing with the reflections return- ing to the perturbed null point and thus leads to less contamination of the Jz current density signal and a cleaner resulting spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Unlike the previous parame- ter study on the magnetic field strength (see Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022a), here we will expand the analysis for setups both in the presence and absence of anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In total we will consider four different val- ues for the characteristic strength of the magnetic field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5B0, 1 B0, 2 B0 and 3 B0, where B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='44 Gauss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We note here that the magnitude of the magnetic field is proportional to the radius for the X-point, and that the characteristic value of the field is not the maximum value in our setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As we can see in Table 1 for models 1 to 4, in these four cases the initial density and tem- perature are 1 ρ0 = 10−12 kg m−3 and 1 T0 = 1 MK, and we will consider both the numerical dissipation scheme and a non-zero viscosity coefficient (nvisc) away from the null, in order to deal with the reflective waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The produced time series for the Jz current density of the different cases are shown gathered in Figure 4, were the results both and with and without thermal conduc- tion are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Upon a visual inspection, we see that in all cases oscillatory reconnection has developed, as is hinted by the oscillatory Jz signal at the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The time series reveal for a stronger, and therefore stiffer mag- netic field, the phenomenon of oscillatory reconnection lasts for progressively shorter times, before the oscilla- 9 With0ut T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=', fit: a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='309, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='484 With T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=', fit: a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='602 8 Period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 7 6 5 4 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 Density (po)Oscillatory Reconnection as Plasma Diagnostic 9 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Time series of the Jz current density at the null point for setups with different equilibrium temperature (T = 1, 3, 5, 7 and 10 MK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The equilibrium magnetic field magnitude at a radius r = 1 and density are 1 B0 and 1 ρ0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Only setups without anisotropic thermal conduction are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' tion is damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This is in agreement with Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a), where it was shown that the decay rates for of these oscillations increase for stronger magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' On that note and to reduce the computational costs, the simulations for 2 B0 and 3 B0 are left running only up to t = 40 t0, since the oscillation decays faster than the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In the same study it was also shown that the period of the oscillation also decreases for stronger, stiffer magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This can also be derived from Figure 4, once we focus on the calculated wavelet spectra for each case, shown here below their respective time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As we see, there is a clear trend regarding the period of the oscillation, with the domi- nant period band being shifted towards smaller values for stronger fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, we see that for most of the cases studied here, the dissipation scheme used to deal with the reflected waves is working efficiently, allowing us to produce clear spectra, where there is one clearly defined band of periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The only exception is for the case of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 B0 without anisotropic thermal conduction, where a strong secondary band of periods is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' From Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a) and Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b) it was shown that these secondary period bands are associated with the reflected waves returning to the null point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This means that for this particular case, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 B0 characteristic magnetic field strength, our dissi- pation scheme was less effective than in the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' However, the main period band is still clearly defined and more prominent that the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In order to quantify this trend, we use the wavelet spectra to calculate the oscillation period for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We do so by first locating the coordinates (time t0 and period P0) of maximum power for each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We then consider a time interval ∆t = [t0 − P0, t0 + 3 P0] containing the periods that exhibit higher values of power, for which we calculate the average value for the period, and the standard deviation that will act as the error in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The calculated average values for the period of each oscillating signal are then placed in the graph of period versus the magnetic field strength, shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The calculated standard deviation for each value is added as error bars for each point, al- though for most cases theses error bars are barely visi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The data points on Figure 5 clearly hint towards an inverse proportionality relation between the oscillation period and the magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Because of this, we have fitted both sets of data points (with thermal conduction, in orange and without thermal conduction, in blue) with the function F(B0) = a (B0)−1 + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Fig- ure 5 also contains the values of the coefficients for both cases, which are a = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='159 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='096, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='398 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='046) and b = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='642 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='111, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='671 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='053) for the cases without and with thermal conduction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" We see that the addition of thermal conduction does not alter the Without Thermal Conduction, 1 Mk 5 ('n') 一5 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 3 Mk 4 2 ('n) 0 2 4 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 5 Mk 2 ('n) N 2 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 7 Mk 2 ('n) N 2 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=" )Without Thermal Conduction, 1o Mk 2 1 ('n') 1 2 0 10 20 30 40 50 60 period (c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 5 10 15 0 10 20 30 40 50 60 time (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )10 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Graph depicting the distribution of the Jz oscil- lation period with respect to the equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In the left panel, we fit the function F(T0) = a (T0)−1 + b to our original data points (red dashed line) and to the ad- justed data points (black dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In the right panel, we do the same but for the function H(T0) = a (T0)−1/2 +b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All setups are considered in the absence of thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' trend in any significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We also see that the setups with thermal conduction generally give higher values of the period than those cases without thermal conduction, in agreement with our past studies (Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Density Dependence Our next goal is to study the response of oscillatory reconnection for different equilibrium density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We have considered again four different cases, where we take den- sity values ρ = 1, 2, 3 and 4 ρ0 where ρ0 = 10−12 kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In all cases we have taken a characteristic strength of the magnetic field equal to 1 B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='44 Gauss, and temperature 1 T0 = 1 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All cases are studied both in the presence and absence of anisotropic thermal conduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Just like before, we will again consider both the numerical dissipation scheme and a non-zero viscosity coefficient (nvisc) away from the null, to treat the reflec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The details of the different models (2, 5, 6 and 7) are shown on Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The derived time series for the Jz current density are shown in Figure 6 alongside their respective wavelet spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Again, the spectra of the time series reveal a prominent period band for each case, associated with the oscillatory reconnection process, the secondary pe- riod bands from the reflected waves being of lower power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Again, upon a visual inspection we see that by increas- ing the value of the equilibrium density, the resulting period of the oscillation increases as well, again with thermal conduction leading to higher periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Following the same process as in the previous case, we derive the average values for the period, and the errors from the standard deviation for each case and we place them in the same oscillation period-density graph, in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Again, we see a clear trend for each set of data points (with and without thermal con- duction, shown in orange and blue respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' For each set of data points, we fit the function G(ρ0) = a (ρ0)1/2 + b, that we believe is showing the best agree- ment with the observed trend of the values of the pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The values of the coefficients, as derived from the fit, are a = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='309 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='303, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='352 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='190) and b = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='484±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='479, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='602±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='300) without and with ther- mal conduction, respectively and are also shown Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Temperature Dependence The final parameter study that we want to perform re- volves around the response of oscillatory reconnection to the initial background temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In the previous sub- section, we took setups of different background densities, but we kept temperature the same for all cases, mean- ing that the sound speed was always the same for those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In this section, we will consider setups with differ- ent temperatures, and therefore different sound speeds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We have considered 5 different cases, corre- sponding to models 8 to 12 on Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In all models we have taken a characteristic strength of the magnetic field equal to 1 B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='44 Gauss, and an initial density of 1 ρ0 = 10−12 kg m−3, while the temperature takes values of 1, 3, 5, 7 and 10 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Unlike the two previous parameter studies, no anisotropic thermal conduction is considered in this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This is due to the ever in- creasing computational costs once thermal conduction is considered, caused by a combination of the increasing temperatures and the Alfv´en speed profile for our given magnetic field geometry, making these simulations very costly to perform for the proper resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Addition- ally, for these five cases viscosity has been dropped from the artificial dissipation scheme dealing with the reflec- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 Original, fit: a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='743, b=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='532 Adjusted, fit: a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='743, b=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='219 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 Period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 2 4 6 8 10 Temperature (1 MK)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 Original, fit: a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='020, b=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='241 Adjusted, fit: a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='020, b=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='929 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 Period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 2 4 6 8 10 Temperature (1 MK)Oscillatory Reconnection as Plasma Diagnostic 11 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Left: Oscillation period versus the background Alfv´en speed at radius r = 1, calculated for all the setups with the different background density and equilibrium magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The function F(VA) = a (VA)−1 + b is fitted for both data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The color choice of Figure 5 is also followed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Right: Same graph as in Figure 9, only now the background sound speed is depicted instead of the background temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The function F(VS) = a (VS)−1 + b is fitted for both data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All values are depicted in code units, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The viscous scheme was not working efficiently for the cases with higher temperatures and so we decided to drop it from the setups with the lower temperatures, for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Figure 8 shows the produced times series of the Jz current density profiles at the perturbed null point, and the corresponding wavelet spectra for each profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Un- like before, the changes in period here seem to be more subtle from one setup to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We see the gradual appearance of a secondary period band, which becomes increasingly stronger for higher temperatures, but with- out ever reaching the same power as the main period band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Given the more uneven Jz signal for higher tem- peratures, it is safe to associate this secondary period band with the reflected waves from the boundaries, pol- luting the null point region and giving rise to more noisy signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Following the same methodology as before for the magnetic field and the density, we again calculate the average values for the period of each case, and their re- spective errors through the standard deviation, and we plot then together in a graph, showing the relation be- tween the period of oscillation and the background tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The results are shown in red in both panels of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In the left panel, just like before, we also fit the the function F(T0) = a (T0)−1 + b, with the coefficients taking the values a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='743±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='060 and b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='532±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Although, the fitted function passes through all the data points if we consider their error bars, we have also de- cided to fit the function H(T0) = a (T0)−1/2 + b in our data (right panel), with coefficients a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='013 and b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='241 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As we can see by compar- ing both panels of Figure 9, the function H(T0) pro- vides a better fit on the given data, with the coeffi- cients presenting smaller errors by comparison to those for F(T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Therefore, from now on we will be using the H(T0) = a (T0)−1/2 + b function to describe the de- pendency of the period to the background temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Also, it becomes obvious that for our range of chosen temperature that matches the coronal conditions, the variations of the period are considerably smaller than the other case that we have examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, we need to address the effects of the differ- ent dissipation scheme used in this parameter study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As mentioned earlier, for the cases considered in this sub- section we took the numerical dissipation scheme de- scribed by the coefficient of Equation (14), without the supplementary viscous scheme described by the coeffi- cient of Equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In other words, for models 8 to 12 of Table 1, we took nvisc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' When comparing the resulting periods for model 2 (P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='947 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='022), used in the previous two subsections and from its equiv- alent model 8 used here (P = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='259 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='155), we see that the two produce slightly different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' It is not certain how removing the artificial viscous scheme leads to this small difference in period, of the order of ∆P ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='312 t0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='43 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' It is very likely that we are dealing with some code-specific numerical effects at this point, which would be hard to properly treat within the context of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' However, the very small value of this difference makes us confident to compare our cur- rent results with those of the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To that end, we have subtracted the difference 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='312 from the periods for all of our data points shown in both pan- els of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We do this, because none of the cases studied in this subsection had the viscous dissipation scheme switched on and thus we have been consistent among these five different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The resulting ad- Without T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=', fit: a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='911,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='574 10 With T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='.fit: a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='941,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='672 L : I: Period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 8 6 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8 VAlfvén (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 Original, fit: a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='317, b=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='241 Adjusted, fit: a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='317, b=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='929 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 Period (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 Sound speed (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' )12 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' justed points (in black) follow the same trend as be- fore for each panel, with the fitted function F(T0) hav- ing coefficients a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='743 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='060 (same as before) and b = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='532 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='312) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='029 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='219 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='029, and the fit- ted function H(T0) having coefficients a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='013 (same as before) and b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='929 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We repeat that from now on we will be using the H(T0) = a (T0)−1/2+b function to describe the dependency of the period to the background temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Parameter Studies In this paper we once again revisit the mechanism of oscillatory reconnection of a 2D X-point in hot coronal plasma, further exploring its response to different coro- nal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This comes as a need, due to the large number of observations that can be attributed to the process of oscillatory reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The first step was taken in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a) where the periodic- ity and the decay rate of the mechanism was studied in the presence of anisotropic thermal conduction in coro- nal conditions, expanding past studies that focused in cold plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In that same study, a clear connection was revealed between the magnetic field strength and the period of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The second step was taken in Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b), where it was found that the period of oscillatory reconnection of a magnetic X- point perturbed by an external pulse is independent of the amplitude and type of the perturbing pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' These two studies had already hinted towards the possibility of using oscillatory reconnection as a tool for coronal seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To that end, in the current study we have expanded upon the results of Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a), by considering cases with different magnetic field, den- sity and background temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Our focus on the three quantities mentioned previ- ously is based on the assumption that the properties of oscillatory reconnection, like its period, in the absence of dedicated external driving, should be related to the conditions of the background plasma in the vicinity of the null point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This is due to oscillatory reconnection being a fundamental process, related to the relaxation of a perturbed magnetic null point (here, X-point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' There- fore, we do not expect the large scale magnetic field topology to affect our results, as the field geometry used here is characteristic of the field geometry in the imme- diate neighbourhood of an X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This is analogous to null-points acting as resonant cavities (see Santamaria & Van Doorsselaere 2018) where the wave-null point in- teraction properties are determined by the background plasma properties near the null point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Here we need to note that having external driving would lead to a de- pendancy of the oscillation period to the period of the driving (Heggland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' However, our focus in this study is on the non-driven, relaxation based oscil- latory reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Using the PLUTO code, we have solved the compress- ible and resistive 2D MHD equations, for a series of parameter studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The first one included four differ- ent setups, each for a different value of the character- istic strength of the magnetic field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='5B0, 1 B0, 2 B0 and 3 B0, where B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='44 Gauss), studied both with and without anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We note here that the characteristic value of the field is not the same as its maximum value in each setups, rather the field magnitude is proportional to the distance from the X-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The results revealed an inverse proportional- ity between the period and the strength of the magnetic field, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' In the second one we consid- ered again four different cases, where we took the density to be equal to ρ = 1, 2, 3 and 4 ρ0 where ρ0 = 10−12 kg m−3, again studied both with and without anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The results, as shown in Figure 7 reveal a square root relation between the period and the equilibrium density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The third and final parameter study involved five setups with different values of back- ground temperature (1, 3, 5, 7 and 10 MK), all studied in the absence of anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This last parameter study, shown in Figure 9, revealed an in- verse proportional relation, this time between the period and the square root of the background temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As expected from our previous studies, the cases with anisotropic thermal conduction practically follow the same trend as their respective ones without thermal con- duction, their only difference being that the former ex- hibit slightly higher values of period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The only exception are for those setups in the temperature parameter study, where thermal conduction has not been considered, due to increased computational costs for our given resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This is caused by a combination of the increasing temperatures and the given magnetic field geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Also, as was explained in the previous section, the derived values for the period from the temperature pa- rameter study are shifted with respect to the rest, due to the slightly different artificial dissipation schemed, with- out any supplementary viscosity-based scheme that was used throughout it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Comparing the resulting periods of two equivalent setups, each with a version of the dissi- pation scheme, we get a difference of ∆P ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='312 t0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='43 s, which is of the order of ∼ 8% from the value of P ∼ 30 s that we get from the other two parameter studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Since we have used the same dissipation scheme when studying the response of oscillatory reconnection to temperature, we subtract ∆P from all of these re- Oscillatory Reconnection as Plasma Diagnostic 13 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Summary of fitted functions and the values of their coefficients for the parameter studies, as well as for the Alfv´en and sound speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Study Fit a, b a, b (without T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') (with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=') Magnetic field F(B0) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='159, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='642 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='398, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='671 Density G(ρ0) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='309, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='484 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='352, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='602 Temperature H(T0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='020, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='241 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='020, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='929 Alfv´en speed F(VA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='911, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='941, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='672 Sound speed F(VS) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='317, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='241 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='317, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='929 Note—The three types of fitted functions are F(x) = a (x)−1 + b, G(x) = a (x)1/2 + b and H(x) = a (x)−1/2 + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' For the temperature and sound speed, both pairs of coef- ficients are without thermal conduction, with the second pair being for the adjusted data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This is indicated in italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' sults, as shown by the black line of the adjusted fit in both panels of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This allows for a better com- parison with the other two parameter studies presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A synopsis of the fitted functions and the values of their coefficients can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Period vs Alfv´en and Sound Speed Continuing on the trend set by our analysis so far, we want to study the evolution of the period of oscil- latory reconnection in terms of the Alfv´en and sound speed profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The initial Alfv´en speed profile is de- pendent both on the initial equilibrium density and the characteristic magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We then take the results from the first 7 models of Table 1, for the dif- ferent values of density and characteristic magnetic field strength, calculate the characteristic Alfv´en speed and plot them with respect to the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This graph is featured on the left panel of Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We again fit the function F(VA) = a (VA)−1 + b, the coefficients of which take values a = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='911±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='027, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='941±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='020) and b = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='574 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='136, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='672 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='103) for the datasets with- out and with thermal conduction, shown in dashed blue and dotted orange lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This fit for the Alfv´en speed is in agreement with the previous fits for the magnetic field and the density, since the Alfv´en speed is proportional to the magnetic field and the inversely proportional to the square root of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The right panel of the same figure, shows the results for the sound speed, which are derived from those of the temperature parameter study without thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' On that panel we show both the original values of the period (points in red) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Examples of calculating the period of oscillatory reconnection through Equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Bph (G) ρph (kg m−3) Tph (MK) Pph (s) 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='9 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='6 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='2 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='4 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 20 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='0 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='8 Note—The subscript ‘ph’ refers to the physical quantities Uph = U U0, with U the quantities in code units and U0 the normalization unit (see also §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' and the adjusted ones (points in black) for which we subtracted the difference ∆P ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='312 t0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='43 s as was mentioned in the section for the temperature parame- ter study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally, we have fitted the function F(VS) = a (VS)−1 + b, for the original (red dashed line) and the adjusted data (black dotted line), the coefficients of which take values a = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='016, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='016) b = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='241 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='007, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='929 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='007) for the original and adjusted data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This fit agrees with the one of the H(T0) = a (T0)−1/2 + b function for the back- ground temperature, presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3, since the sound speed is proportional to the square root of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This further justifies the use of function H(T0) to describe the relation between the period of oscillatory reconnection and the plasma temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Empirical Formula As a last step, we want to merge the derived relations from each one of the three parameter studies into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We do this because one of the main goals of this present study was to start developing its capabilities as a plasma diagnostic tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To that end, we are aided by the results of Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022b), that allow us to ignore the strength of the perturbing pulse from this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Taking the cases without thermal conduction, we can merge the derived relations of each previous fit into the following formula for our four key parameters: Pph t0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='159 B0 Bph +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='309 �ρph ρ0 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='02 � T0 Tph −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='541±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='434 (16) 14 Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' where the penultimate right-hand term comes from solv- ing the above equation for a known value of the period Pph (in s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' For this, we considered the resulting period for model 2 (Pph = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='947 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='022) t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We also include the maximum error in the last right-hand term that is derived from the different combinations of errors in the values of Pph and the coefficients of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Here, the subscript ‘ph’ refers to the physical quantities Uph, as de- fined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' For the quantities in physical units we have Uph = U U0, with U the quantities in code units and U0 the normalization units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We used the adjusted results for the temperature parameter study, as explained ear- lier, while all the coefficients are given in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Using the normalization units defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='1, we can rewrite the above formula in SI units, except for the magnetic field which is given in Gauss: Pph = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='39 Bph + 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='74 × 106√ρph + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='94 � Tph − 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='55 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='38 (17) where we have the period Pph (in s) for a known combi- nation of Bph (in G), ρph (in kg m−3) and Tph (in MK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A similar analysis on the cases with added thermal con- duction can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Finally we show in Table 3, some examples of using the above formula to calculate the periods of oscillatory re- connection for different combinations of parameters for a flaring coronal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' One thing that needs to be stressed is that Equation (17) has been derived from a set of values that reflects the average conditions in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' As a result, we need to be cautious when extrapolating the above relation for values out- side of that parameter space, as we may end up with non-physical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' However, the derived relation can be a useful plasma diagnostic tool in coronal conditions, and needs to be tested further against observational pe- riodic signals, that could be attributed to oscillatory reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Such periodic signals in the solar atmo- sphere include, but are not limited to quasi-periodic pul- sations (QPPs) of solar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Kupriyanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2016) and stellar flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Broomhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019), quasi- periodic chromospheric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2011) and coronal jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022), quasi-periodic fast-propagating (QFP) magne- tosonic wave from the eruption of a magnetic flux rope (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2018) and periodicities correlated with Type III radio bursts (Cattell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A detailed discussion of the different phenomena attributed to os- cillatory reconnection has already been presented in the Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' The fundamental nature of oscillatory recon- nection in perturbed magnetic X-points, indicates that our derived plasma diagnostic tool can be employed to study the periodicities in the different cases of periodic signals attributed to oscillatory reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' To summarize, our series of parameter studies have ex- plored the effects of temperature, density and magnetic field strength on the periodicity of oscillatory reconnec- tion in a hot coronal plasma, expanding the earlier re- sults of Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Our findings show that the period of the oscillation depends on the under- lying characteristics of the plasma near the null point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Taking into additional account the independence of the periodicity of oscillatory reconnection from the strength and type of the initial, perturbing pulse (Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022b), we have now developed a first quantita- tive formula to be used as a plasma diagnostic, opening the possibility of using this mechanism within the con- text of coronal seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' All authors acknowledge UK Science and Technol- ogy Facilities Council (STFC) support from grant ST/T000384/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' also acknowledges support by an FWO (Fonds voor Wetenschappelijk Onderzoek – Vlaanderen) postdoctoral fellowship (1273221N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' This work used the Oswald High Performance Computing fa- cility operated by Northumbria University (UK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' EMPIRICAL FORMULA FOR ADDED THERMAL CONDUCTION In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='3 we derived an empirical formula (see Equations 16 and 17) that connects the period of oscillatory reconnection with the characteristic strength of the magnetic field, the background density and the equilibrium plasma temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We did this by merging the derived relations of each fit in the data sets without anisotropic conduction, featured in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' A similar formula can be derived for the period, magnetic field strength and density when we include anisotropic thermal conduction for a plasma temperature of Tph = 1 MK: Pph t0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='398 B0 Bph + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='352 �ρph ρ0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='701 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='547 (A1) Oscillatory Reconnection as Plasma Diagnostic 15 where we used the period of model 2 (see Table 1) with thermal conduction switched on (Pph = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='311) t0) in order to calculate the penultimate right-hand term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Similarly to Equation (16), we also include the maximum error derived from the different combinations of errors in the values of Pph and the coefficients of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' When written in SI units, except from the magnetic field that is in Gauss, the previous relation takes the form: Pph = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='07 Bph + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='08 × 106√ρph − 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='01 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content='26 (A2) where we again take Bph in G and ρph in kg m−3, for Tph = 1 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' One drawback of the current study is the fact that, due to numerical reasons, implementing thermal conduction for the setups with high coronal temperatures (> 1 MK) lead to very costly and slow to perform simulations for our resolution of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' That means that Equations (A1) and (A2) lack the temperature term of Equations (16) and (17) and can only be valid for plasma with temperatures near 1 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' However, that might not necessarily hinder our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' By comparing the two sets of equations, we see that the respective coefficients for each independent variable are very close in value to each other, when considering either the dimensionless or dimensionalized expressions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Also, from our past studies (Karampelas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' 2022a,b) and the results of the parameter studies for the magnetic field and density, we know that the addition of thermal conduction only increases the values of the oscillation period by a small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' Given that for T = 1 MK, the parallel to the magnetic field thermal conduction coefficient κ∥ is already many orders of magnitude larger than the perpendicular one κ⊥, it is unlikely that an increased temperature will significantly change the response of our setups to anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E0T4oBgHgl3EQfjAEs/content/2301.02452v1.pdf'} +page_content=' We thus conclude that our empirical formula without 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sha256:3a7bcbb06594a37d7b548b4ef7a43864e1e38c2fac04bb73262d32b485297cda +size 210117 diff --git a/MdE0T4oBgHgl3EQfSwDa/vector_store/index.pkl b/MdE0T4oBgHgl3EQfSwDa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..dc76c0efc1cffac02e6efdc4f92c0c6fff6a2de7 --- /dev/null +++ b/MdE0T4oBgHgl3EQfSwDa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4dc5caae44efb097af98486ae5d64cb3f1db36d9767bcc7c9438ac0cecff05dc +size 208762 diff --git a/MtFRT4oBgHgl3EQfGDcg/content/tmp_files/2301.13482v1.pdf.txt b/MtFRT4oBgHgl3EQfGDcg/content/tmp_files/2301.13482v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a3e8616c9a078be6957d2677bd3448b3d7b1b01 --- /dev/null +++ b/MtFRT4oBgHgl3EQfGDcg/content/tmp_files/2301.13482v1.pdf.txt @@ -0,0 +1,1627 @@ +arXiv:2301.13482v1 [math.FA] 31 Jan 2023 +The general theory of superoscillations and +supershifts in several variables +F. Colombo ∗, S. Pinton∗, I. Sabadini∗, D.C. Struppa† +Abstract +In this paper we describe a general method to generate superoscillatory functions of sev- +eral variables starting from a superoscillating sequence of one variable. Our results are based +on the study of suitable infinite order differential operators on holomorphic functions with +growth conditions of exponential type, where additional constraints are required when deal- +ing with infinite order differential operators whose symbol is a function that is holomorphic +in some open set, but not necessarily entire. The results proved for the superoscillating +sequence in several variables are extended to sequences of supershifts in several variables. +AMS Classification: 26A09, 41A60. +Key words: General superoscillatory functions, supershifts in several variables. +1 +Introduction +Superoscillating functions are band-limited functions that can oscillate faster than their fastest +Fourier component. Physical phenomena associated with superoscillatory functions have been +known for a long time for example in antennas theory see [31], and in the context of weak +values in quantum mechanics, see [1]. In more recent years there has been a wide interest in +the theory of superoscillating functions and of supershifts, a notion that generalizes the one of +superoscillations, and that was introduced in the literature in order to study the evolution of +superoscillations as initial data of the Schr¨odinger equation of other field equations, like Dirac +or Klein-Gordon equations. +An introduction to superoscillatory functions in one variable and some applications to +Schr¨odinger evolution of superoscillatory initial data can be found in [7]. Superoscillatory func- +tions in several variables have been rigorously defined and studied in [6] and in [9] where we +have initiated also the theory of supershifts in more then one variable. The aim of this paper is +to remove the restrictions in [6, 9] and to obtain a very general theory of superoscillations and +supershifts. +Our results are directed to a general audience of physicists, mathematicians, and engineers, +and our main tool is the theory of infinite order differential operators acting on spaces of holo- +morphic functions. +The literature on superoscillations is quite large, and without claiming +∗Politecnico +di +Milano, +Dipartimento +di +Matematica, +Via +E. +Bonardi, +9 +20133 +Milano, +Italy, +fabrizio.colombo@polimi.it, +irene.sabadini@polimi.it, stefano.pinton@polimi.it +†The +Donald +Bren +Presidential +Chair +in +Mathematics, +Chapman +University, +Orange, +USA, +struppa@chapman.edu +1 + +completeness we have tried to mention some of the most relevant (and recent) results. Papers +[2]-[7], [12], [15], [25], [28] and [29] deal with the issue of permanence of superoscillatory behav- +ior when evolved under a suitable Schr¨odinger equation; papers [18]-[20], [26]-[27] and [30] are +mostly concerned with the physical nature of superoscillations, while papers [10], [11], [13]-[14], +[21]-[24] develop in depth the mathematical theory of superoscillations. Finally we have cited [7] +as a good reference for the state of the art on the mathematics of superoscillations until 2017, +and the Roadmap on Superoscillations [17], where the most recent advances in superoscillations +and their applications to technology are well explained by the leading experts in this field. +In this paper we extend the results in [9] considering analytic functions in one variable +G1, . . . , Gd, d ≥ 2, whose Taylor series at zero have radius of convergence grater than or equal +to 1. Thus we define general superoscillating functions of several variables as expressions of the +form +Fn(x1, x2, . . . , xd) := +n +� +j=0 +Zj(n, a)eix1G1(hj(n))eix2G2(hj(n)) . . . eixdGd(hj(n)) +where Zj(n, a), j = 0, ..., n, for n ∈ N0 are suitable coefficients of a superoscillating function in +one variable as we will see in the sequel. We will give conditions on the functions G1, . . . , Gd in +order that +lim +n→∞ Fn(x1, x2, . . . , xd) = eix1G1(a)eix2G2(a) . . . eixdGd(a), +so that, when |a| > 1, Fn(x1, x2, . . . , xd) is superoscillating. Moreover, we shall also treat the +case of sequences that admit a supershift in d ≥ 2 variables. +The paper is organized in four sections including the introduction. Section 2 contains the +preliminary material on superoscillations, the relevant function spaces and their topology, and +the study of the continuity of some infinite order differential operators acting on such spaces. +Section 3 is the main part of the paper and contains the definition of superoscillating functions +in d ≥ 2 variables as well as some results. Section 4 discusses the notion of supershift in this +framework. +2 +Preliminary results on infinite order differential operators +We begin this section with some preliminary material on superoscillations and supershifts in one +variable. Then we introduce and study some infinite order differential operators that will be of +crucial importance to define and study superoscillations and supershifts in several variables. +Definition 2.1. We call generalized Fourier sequence a sequence of the form +fn(x) := +n +� +j=0 +Zj(n, a)eihj(n)x, +n ∈ N, +x ∈ R, +(1) +where a ∈ R, Zj(n, a) and hj(n) are complex and real valued functions of the variables n, a and +n, respectively. The sequence (1) is said to be a superoscillating sequence if supj,n |hj(n)| ≤ 1 +and there exists a compact subset of R, which will be called a superoscillation set, on which fn(x) +converges uniformly to eig(a)x, where g is a continuous real valued function such that |g(a)| > 1. +The classical Fourier expansion is obviously not a superoscillating sequence since its frequen- +cies are not, in general, bounded. +In the recent paper [8] we enlarged the class of superoscillating functions, with respect to +the existing literature, and we solved the following problem. +2 + +Problem 2.2. Let hj(n) be a given set of points in [−1, 1], j = 0, 1, ..., n, for n ∈ N and let +a ∈ R be such that |a| > 1. Determine the coefficients Xj(n) of the sequence +fn(x) = +n +� +j=0 +Xj(n)eihj(n)x, +x ∈ R +in such a way that +f (p) +n (0) = (ia)p, +for +p = 0, 1, ..., n. +Remark 2.3. The conditions f (p) +n (0) = (ia)p mean that the functions x �→ eiax and x �→ +fn(x) have the same derivatives at the origin, for p = 0, 1, ..., n, and therefore the same Taylor +polynomial of order n. +Theorem 2.4 (Solution of Problem 2.2). Let hj(n) be a given set of points in [−1, 1], j = +0, 1, ..., n for n ∈ N and let a ∈ R be such that |a| > 1. If hj(n) ̸= hi(n), for every i ̸= j, then +the coefficients Xj(n, a) are uniquely determined and given by +Xj(n, a) = +n +� +k=0, k̸=j +� +hk(n) − a +hk(n) − hj(n) +� +. +(2) +As a consequence, the sequence +fn(x) = +n +� +j=0 +n +� +k=0, k̸=j +� +hk(n) − a +hk(n) − hj(n) +� +eixhj(n), +x ∈ R +solves Problem 2.2. Moreover, when the holomorphic extensions of the functions fn converge in +A1, we have +lim +n→∞ fn(x) = eiax, +for all x ∈ R. +Our approach to the study of superoscillatory functions in one or several variables makes use +of infinite order differential operators. Such operators naturally act on spaces of holomorphic +functions. This is the reason for which we consider the holomorphic extension to entire functions +of the sequence fn(x) defined in (2.1) by replacing the real variable x with the complex variable +ξ. For the sequences of entire functions we shall consider, a natural notion of convergence is +the convergence in the space A1 or in the space A1,B for some real positive constant B (see the +following definition and considerations). +Definition 2.5. The space A1 is the complex algebra of entire functions such that there exists +B > 0 such that +sup +ξ∈C +� +|f(ξ)| exp(−B|ξ|) +� +< +∞. +(3) +The space A1 has a rather complicated topology, see e.g. +[16], since it is a linear space +obtained via an inductive limit. For our purposes, it is enough to consider, for any fixed B > 0, +the set A1,B of functions f satisfying (3), and to observe that +∥f∥B := sup +ξ∈C +� +|f(ξ)| exp(−B|ξ|) +� +defines a norm on A1,B, called the B-norm. One can prove that A1,B is a Banach space with +respect to this norm. +3 + +Moreover, let f and a sequence (fn)n belong to A1; fn converges to f in A1 if and only if +there exists B such that f, fn ∈ A1,B and +lim +n→∞ sup +ξ∈C +��fn(ξ) − f(ξ) +��e−B|ξ| = 0. +With these notations and definitions we can make the notion of continuity explicit (see [14]): +A linear operator U : A1 → A1 is continuous if and only if for any B > 0 there exists B′ > 0 +and C > 0 such that +U(A1,B) ⊂ A1,B′ and +∥U(f)∥B′ ≤ C∥f∥B, +for any f ∈ A1,B. +(4) +The following result, see Lemma 2.6 in [13], gives a characterization of the functions in A1 +in terms of the coefficients appearing in their Taylor series expansion. +Lemma 2.6. The entire function +f(ξ) = +∞ +� +j=0 +ajξj +belongs to A1 if and only if there exists Cf > 0 and b > 0 such that +|aj| ≤ Cf +bj +Γ(j + 1). +Remark 2.7. To say that f ∈ A1 means that f ∈ A1,B for some B > 0. The computations in +the proof of Lemma 2.6 in [13], show that b = 2eB, and that we can choose Cf = ∥f∥B. +We now define two infinite order differential operators that will be used to study superoscil- +latory functions and supershifts in several variables. We shall denote by x the vector (x1, . . . , xd) +in Rd. +Proposition 2.8. Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} for any ℓ = 1, . . . , d. Let +(g1,m), . . . , (gd,m) be d sequences of complex numbers such that +lim sup +m→∞ |gℓ,m|1/m = 1 +Rℓ +, +for ℓ = 1, . . . , d. +(5) +Let x1, . . . , xd ∈ R. Denote by Dξ := +∂ +∂ξ the derivative operator with respect to the auxiliary +complex variable ξ. We define the formal operator: +U(x1, x2, . . . , xd, Dξ) := +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + Dk1 +ξ +ik1 +(6) +where we have set +yp := ix1g1,p + . . . + ixdgd,p, +for p = 1, . . . r with r ∈ N. +Then, setting +R := +min +ℓ=1,...,d Rℓ, +for any real value 0 < B < R +4e, the operator U(x1, . . . , xd, Dξ) : A1,B → A1,4eB is continuous for +all x ∈ Rd. +4 + +Proof. Let us consider f ∈ A1,B; then we have +U(x1, . . . , xd, Dξ)f(ξ) = +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + Dk1 +ξ +ik1 f(ξ) += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + +∞ +� +j=k1 +aj +j! +(j − k1)!ξj−k1 += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + +∞ +� +j=0 +aj+k1 +(j + k1)! +j! +ξj. +Taking the modulus we get +|U(x1, . . . , xd, Dξ)f(ξ)| +≤ +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm||ykm−1−km| . . . |yk1−k2| + + +∞ +� +j=0 +|aj+k1|(j + k1)! +j! +ξj. +and Lemma 2.6 gives the estimate on the coefficients aj+k1 +|aj+k1| ≤ Cf +bj+k1 +Γ(j + k1 + 1). +where b = 2eB. Using the well known inequality (a + b)! ≤ 2a+ba!b! we also have +(j + k1)! ≤ 2j+k1j!k1! +so we get +|U(x1, . . . , xd, Dξ)f(ξ)| ≤ +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm||ykm−1−km| . . . |yk1−k2| + + × +×Cf +∞ +� +j=0 +bj+k1 +Γ(j + k1 + 1) +2j+k1k1!j! +j! +|ξ|j. +Now we use the Gamma function estimate +1 +Γ(a + b + 2) ≤ +1 +Γ(a + 1) +1 +Γ(b + 1) +(7) +to separate the series, and we have +1 +Γ(j − 1 +2 + k1 − 1 +2 + 2) ≤ +1 +Γ(j + 1 +2) +1 +Γ(k1 + 1 +2) +and so +|U(x1, . . . , xd, Dξ)f(ξ)| ≤ Cf +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm||ykm−1−km| . . . |yk1−k2| + + × +5 + +×(k1)!(2b)k1 +Γ(k1 + 1 +2) +∞ +� +j=0 +1 +Γ(j + 1 +2)(2b|ξ|)j. +Now observe that the latter series satisfies the estimate +∞ +� +j=0 +1 +Γ(k + 1 +2)(2b|ξ|)j ≤ Ce4b|ξ| +where C is a positive constant, because of the properties of the Mittag-Leffler function; moreover, +the series +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm||ykm−1−km| . . . |yk1−k2| + + (k1)!(2b)k1 +Γ(k1 + 1 +2) +(8) +is convergent and is bounded by a positive real constant Cx,G1,...,Gd. +In fact, using Stirling +formula for the Gamma function, we have +m! ∼ +√ +2πm e−mmm, +for +m → ∞ +and then we deduce +Γ(m + 1) +Γ(m + 1/2) ∼ +√ +2π m e−mmm +� +2π(m − 1/2) e−(m−1/2) (m − 1/2)(m−1/2) ∼ +� +m − 1/2, +for +m → ∞ +(9) +so that +k1! +Γ(k1 + 1 +2) ∼ +� +k1 − 1/2, +for +k1 → ∞. +Now observe that the series (8) has positive coefficients and so it converges if and only if the +series +∞ +� +m=1 +1 +m! +∞ +� +k1=1 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm||ykm−1−km| . . . |yk1−k2| + + (2b)k1� +k1 − 1/2 +converges. Given an absolutely convergent series �∞ +p=0 ap, then its m-th power can be computed +by means of the Cauchy product as follows: + + +∞ +� +p=0 +ap + + +m += +∞ +� +k1=0 +k1 +� +k2=0 +. . . +km−1 +� +km=0 +akmakm−1−km . . . ak1−k2. +(10) +Using the inequality: +� +k1 − 1 +2 ≤ k1 ≤ km +(km−1 −km)+. . .+(k1 −k2) ≤ (km +2)·(km−1 −km +2)·····(k1 −k2 +2), +where k1 ≥ k2 ≥ · · · ≥ km, we deduce that there exists a positive constant Cx,G1,...,Gd such that +6 + +the following chain of inequalities hold: +∞ +� +m=1 +1 +m! +∞ +� +k1=1 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm||ykm−1−km| . . . |yk1−k2| + + (2b)k1� +k1 − 1/2 +≤ +∞ +� +m=1 +1 +m! +∞ +� +k1=1 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +|ykm(km + 2)(2b)km||ykm−1−km(km−1 − km + 2)(2b)km−1−km|× +. . . × |yk1−k2(k2 − k1 + 2)(2b)k1−k2| +� += +∞ +� +m=1 +1 +(m)! + + +∞ +� +p=0 +|yp|(p + 2)(2b)p + + +m +≤ +∞ +� +m=1 +1 +(m)! + + +∞ +� +p=1 +|x1|(p + 2)(2b)p|g1,p| + +. . . + |xd|(p + 2)(2b)p|gd,p| +�m +≤ Cx,G1,...,Gd +where for the equality we used (10), while the last inequality follows by the assumption +B < R +4e +which implies 2b < R. From the previous estimate we have that the series (8) converges for all +x1, . . . , xd ∈ R. So we finally have +|U(x1, . . . , xd, Dξ)f(ξ)| ≤ Cf Cx,G1,...,Gd C e4b|ξ|, +x ∈ Rd, +ξ ∈ C. +(11) +Recalling that b = 2eB, the estimate (11) implies that U(x1, . . . , xd, )f ∈ A1,8eB, in fact +|U(x1, . . . , xd, Dξ)f(ξ)| e−8eB|ξ| ≤ Cf Cx,G1,...,Gd C +x ∈ Rd, +ξ ∈ C. +Moreover, we deduce that the 8eB-norm satisfies the estimate +∥U(x1, . . . , xd, Dξ)f∥8eB ≤ Cf Cx,G1,...,Gd C = Cx,G1,...,Gd C∥f∥B. +Thus U(x1, . . . , xd, Dξ) : A1,B → A1,8eB is continuous for all x ∈ Rd. +Remark 2.9. Whenever we fix a compact subset K ⊂ Rd, we have that, for any x ∈ K, the +constants Cx,G1,...,Gd appearing in the proof of the previous theorem are bounded by a constant +which depends only on K and G1, . . . , Gd. +Moreover, if Rℓ = ∞ for any ℓ = 1, . . . , d, the +continuity of the operator U(x1, . . . , xd, Dξ) holds for any B > 0 and the proof of the previous +theorem shows that U(x1, . . . , xd, Dξ) is a continuous operator in A1. +Proposition 2.10. Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} for any ℓ = 1, . . . , d. Let +(g1,m), . . . , (gd,m) be d sequences of complex numbers such that +lim sup +m→∞ |gℓ,m|1/m = 1 +Rℓ +, +for ℓ = 1, . . . , d. +(12) +We define the formal operator +V(x1, . . . , xd, Dξ) := +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,mdxm1 +1 +. . . xmd +d +1 +im1+···+md Dm1+···+md +ξ +, +(13) +where x1, . . . , xd ∈ R, ξ ∈ C. Then, for any real value B > 0, the operator V(x1, . . . , xd, Dξ) : +A1,B → A1,8eB is continuous whenever |xℓ| < +R +4eB for any ℓ = 1, . . . , d where R := minℓ=1,...,d Rℓ. +7 + +Proof. We apply the operator V(x1, . . . , xd, Dξ) to a function f in A1,B for |x| < +R +4eB . We have +V(x1, . . . , xd, Dξ)f(ξ) = +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,mdxm1 +1 +. . . xmd +d +1 +im1+···+md Dm1+···+m2 +ξ +f(ξ) += +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,mdxm1 +1 +. . . xmd +d +1 +im1+···+md Dm1+···+md +ξ +∞ +� +j=0 +ajξj += +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,mdxm1 +1 +. . . xmd +d +1 +im1+···+md × +× +∞ +� +j=m1+···+md +aj +j! +(j − (m1 + · · · + md))!ξj−(m1+···+md) += +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,mdxm1 +1 +. . . xmd +d +1 +im1+···+md +∞ +� +k=0 +am1+···+md+k +(m1 + · · · + md + k)! +k! +ξk. +We then have +|V(x1, . . . , xd, Dξ)f(ξ)| ≤ +∞ +� +m1=0 +|g1,m1| . . . +∞ +� +md=0 +|gd,md||x1|m1 . . . |xd|md× +× +∞ +� +k=0 +|am1+···+md+k|(m1 + · · · + md + k)! +k! +|ξ|k +and using the estimate in Lemma 2.6 +|am1+...md+k| ≤ Cf +bm1+...md+k +Γ(m1 + · · · + md + k + 1), +where b = 2eB, we get +|V(x1, . . . , xd, Dξ)f(ξ)| ≤ +∞ +� +m1=0 +|g1,m1| · · · +∞ +� +md=0 +|gd,md||x1|m1 . . . |xd|md× +×Cf +∞ +� +k=0 +bm1+···+md+k +Γ(m1 + · · · + md + k + 1) +(m1 + · · · + md + k)! +k! +|ξ|k. +With the estimates +(m1 + · · · + md + k)! ≤ 2m1+···+md+k(m1 + · · · + md)!k! +and +1 +Γ(m1 + · · · + md − 1 +2 + k − 1 +2 + 2) ≤ +1 +Γ(m1 + · · · + md + 1 +2) +1 +Γ(k + 1 +2) +we separate the series +|V(x1, . . . , xd, Dξ)f(ξ)| ≤ +∞ +� +m1=0 +|g1,m1| · · · +∞ +� +md=0 +|gd,md||x1|m1 . . . |xd|md× +× +∞ +� +k=0 +Cfbm1+···+md+k +1 +Γ(m1 + · · · + md + 1 +2) +1 +Γ(k + 1 +2) +2m1+···+md+k(m1 + · · · + md)!k! +k! +|ξ|k. +8 + +Finally we get +|V(x1, . . . , xd, Dξ)f(ξ)| ≤ Cf +∞ +� +m1=0 +|g1,m1| · · · +∞ +� +md=0 +|gd,md|(2b|x1|)m1 · · · (2b|xd|)md× +× +(m1 + · · · + md)! +Γ(m1 + · · · + md + 1 +2) +∞ +� +k=0 +1 +Γ(k + 1 +2)(2b|ξ|)k. +Using (9) we have +(m1 + · · · + md)! +Γ(m1 + · · · + md + 1 +2) ∼ +� +m1 + · · · + md − 1/2, +for +m1 + · · · + md → ∞, +and, moreover, +� +m1 + · · · + md − 1/2 ≤ m1 · · · md if mℓ ≥ 2 for any ℓ = 1, . . . , d. +Since +|xℓ| < +R +4eB for any ℓ = 1, . . . , d and b = 2eB, the series +∞ +� +mℓ=1 +mℓ|gℓ,mℓ|(2b|xℓ|)mℓ +converges to a constant which depends on xℓ ∈ R. Thus there exist constants Cxℓ such that +|V(x1, . . . , xd, Dξ)f(ξ)| ≤ CfCx1 . . . Cxd(2b|ξ|)e2b|ξ| ≤ CfCx1,...,xde4b|ξ| +from which, recalling that Cf = ∥f∥B, we deduce +∥V(x1, . . . , xd, Dξ)f∥8eB ≤ Cx1,...,xd∥f∥B. +We conclude that the operator V(x1, . . . , xd, Dξ) : A1,B → A1,8eB is continuous. +Remark 2.11. Whenever we fix a compact subset +K ⊂ {x ∈ Rd : |xℓ| < +R +4eB for any ℓ = 1, . . . , d}, +we have that, for any x ∈ K, the constants Cxℓ’s, appearing in the proof of the previous theorem +are bounded by a constant which depends only on K. Moreover, if Rℓ = ∞ for any ℓ = 1, . . . , d, +the continuity of the operator V(x1, . . . , xd, Dξ) holds to be true for any x ∈ Rd and the proof +of the previous theorem shows that V(x1, . . . , xd, Dξ) satisfies the conditions in (4). Thus we +conclude that V(x1, . . . , xd, Dξ) is a continuous operator in A1. +3 +Superoscillating functions in several variables +We recall some preliminary definitions related to superoscillating functions in several variables. +Definition 3.1 (Generalized Fourier sequence in several variables). For d ∈ N such that d ≥ 2, +we assume that (x1, ..., xd) ∈ Rd. Let (hj,ℓ(n)), j = 0, ..., n for n ∈ N0, be real-valued sequences +for ℓ = 1, ..., d. We call generalized Fourier sequence in several variables a sequence of the form +Fn(x1, . . . , xd) = +n +� +j=0 +cj(n)eix1hj,1(n)eix2hj,2(n) . . . eixdhj,d(n), +(14) +where (cj(n))j,n, j = 0, . . . , n, for n ∈ N0 is a complex-valued sequence. +9 + +Definition 3.2 (Superoscillating sequence). A generalized Fourier sequence in several variables +Fn(x1, . . . , xd), with d ∈ N such that d ≥ 2, is said to be a superoscillating sequence if +sup +j=0,...,n, n∈N0 +|hj,ℓ(n)| ≤ 1, +for ℓ = 1, ..., d, +and there exists a compact subset of Rd, which will be called a superoscillation set, on which +Fn(x1, . . . , xd) converges uniformly to eix1g1eix2g2 . . . eixdgd, where |gℓ| > 1 for ℓ = 1, . . . , d. +In the paper [6] we studied the function theory of superoscillating functions in several vari- +ables under the additional hypothesis that there exist rℓ ∈ N, such that +p = r1q1 + . . . + rdqd. +(15) +In that case, we proved that for p, qℓ ∈ N, ℓ = 1, . . . , d the function +Fn(x, y1, . . . , yd) = +n +� +j=0 +Cj(n, a)eix(1−2j/n)peiy1(1−2j/n)q1 . . . eiyd(1−2j/n)qd +is superoscillating when |a| > 1, where Cj(n, a) are suitable coefficients. In the paper [9], we +were able to remove the condition (15), while here we will show that it is possible to replace +the functions (1 − 2j/n)p in the terms eix(1−2j/n)p with more general holomorphic functions. As +we shall see, different function spaces are involved in the proofs according to the fact that the +holomorphic functions are entire or not. +Theorem 3.3 (The general case of d ≥ 2 variables). Let d be a positive integer and let Rℓ ∈ +R+ ∪ {∞} be such that Rℓ ≥ 1 for any ℓ = 1, . . . , d. Let G1, . . . , Gd be holomorphic functions +whose series expansion at zero is given by +Gℓ(λ) = +∞ +� +mℓ=0 +gℓ,mλmℓ, +∀ℓ = 1, . . . , d +(16) +and, moreover, the sequences (gℓ,m) satisfy the condition +lim sup +m→∞ |gℓ,m|1/m = 1 +Rℓ +, +∀ℓ = 1, . . . , d. +Let +fn(x) := +n +� +j=0 +Zj(n, a)eihj(n)x, +n ∈ N, +x ∈ R, +(17) +be superoscillating functions as in Definition 2.1 and assume that their entire extensions to +the functions fn(ξ) converge to eiaξ in A1,B for some positive real value 0 < B < +R +4e where +R := minℓ=1,...,d Rℓ. We define +Fn(x1, . . . , xd) := +n +� +j=0 +Zj(n, a)eix1G1(hj(n))eix2G2(hj(n)) . . . eixdGd(hj(n)). +Then, whenever |a| < R we have +lim +n→∞ Fn(x1, x2, . . . , xd) = eix1G1(a)eix2G2(a) . . . eixdGd(a), +uniformly on compact subsets of Rd. In particular, Fn(x1, x2, . . . , xd) is superoscillating when +|a| > 1. +10 + +Proof. Since Rℓ ≥ 1 for any ℓ = 1, . . . , d and |hj(n)| < 1, using (10) we have the chain of +equalities +Fn(x1, x1, . . . , xd) = +n +� +j=0 +Zj(n, a)eix1G1(hj(n))+ix2G2(hj(n))+...+ixdGd(hj(n)) += +n +� +j=0 +Zj(n, a) +∞ +� +m=0 +1 +m! +� +ix1G1(hj(n)) + ix2G2(hj(n)) + . . . + ixdGd(hj(n)) +�m += +n +� +j=0 +Zj(n, a) +∞ +� +m=0 +1 +m! +� +ix1 +∞ +� +p=1 +g1,p(hj(n))p + . . . + ixd +∞ +� +p=1 +gd,p(hj(n))p�m += +n +� +j=0 +Zj(n, a) +∞ +� +m=0 +1 +m! +� ∞ +� +p=1 +(ix1g1,p + . . . + ixdgd,p)(hj(n))p�m += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + (hj(n))k1, +where we have set +yp := ix1g1,p + . . . + ixdgd,p, +for p = 1, . . . r with r ∈ N. +We define the infinite order differential operator +U(x1, x2, . . . , xd, Dξ) := +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + Dk1 +ξ +ik1 . +(18) +Since 0 < B < R +4e, Proposition 2.8 implies that the operator +U(x1, x2, . . . , xd, Dξ) : A1,B �→ A1,8eB +is continuous. We observe that +Fn(x1, x2, . . . , xd) = U(x1, x2, . . . , xd, Dξ) +n +� +j=0 +Zj(n, a)eiξhj(n)��� +ξ=0 +The explicit computation of the term U(x1, . . . , xd, Dξ)eiξa gives +U(x1, . . . , xd, Dξ)eiξa = += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + Dk1 +ξ +ik1 eiξa += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + ak1eiξa, +11 + +so we finally get +lim +n→∞ Fn(x1, . . . , xd) = += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +ykmykm−1−km . . . yk1−k2 + + ak1eiξa��� +ξ=0 += +∞ +� +m=0 +1 +m! +∞ +� +k1=0 + + +k1 +� +k2=0 +. . . +km−1 +� +km=0 +(ykmakm)(ykm−1−kmakm−1−km) . . . (yk1−k2ak1−k2) + + += +∞ +� +m=0 +1 +m! + + +∞ +� +p=1 +ypap + + +m += +∞ +� +m=0 +1 +m! (ix1G1(a) + . . . + ixdGd(a))m = eix1G1(a)+...+ixdGd(a) +where the third equality is due to the formula (10) and the fourth equality holds because we +are assuming |a| < R. The previous limit is uniform over the compact subset of Rd because of +Remark 2.9. +Remark 3.4. From the inspection of the proof we observe that: +(I) The space of the entire functions on which the infinite order differential operator U(x1, . . . , xd, Dξ) +acts is the space A1,B in one complex variable, for some positive real value 0 < B < R +4e. +(III) The variables (x1, x1, . . . , xd) become the coefficients of the infinite order differential oper- +ator U(x1, x2, . . . , xd, Dξ), defined in (18), that still acts on the space A1,B. +4 +Supershifts in several variables +The procedure to define superoscillating functions can be extended to the case of supershift. +Recall that the supershift property of a function extends the notion of superoscillation and that +this concept, that we recall below, turned out to be a crucial ingredient for the study of the +evolution of superoscillatory functions as initial conditions of the Schr¨odinger equation. +Definition 4.1 (Supershift). Let I ⊆ R be an interval with [−1, 1] ⊂ I and let ϕ : I × R → R, +be a continuous function on I. We set +ϕh(x) := ϕ(h, x), +h ∈ I, +x ∈ R +and we consider a sequence of points (hj(n)) such that +hj(n) ∈ [−1, 1] for j = 0, ..., n and n ∈ N0. +We define the functions +ψn(x) = +n +� +j=0 +cj(n)ϕhj(n)(x), +(19) +where (cj(n)) is a sequence of complex numbers for j = 0, ..., n and n ∈ N0. If +lim +n→∞ ψn(x) = ϕa(x) +for some a ∈ I with |a| > 1, we say that the function ψn(x), for x ∈ R, admits a supershift. +12 + +Remark 4.2. The term supershift comes from the fact that the interval I can be arbitrarily +large (it can also be R) and that the constant a can be arbitrarily far away from the interval +[−1, 1] where the functions ϕhj,n(·) are indexed, see (19). +Problem 2.2, for the supershift case, is formulated as follows. +Problem 4.3. Let hj(n) be a given set of points in [−1, 1], j = 0, 1, ..., n, for n ∈ N and let +a ∈ R be such that |a| > 1. Suppose that for every x ∈ R the function h �→ G(hx) extends to a +holomorphic and entire function in h. Consider the functions +fn(x) = +n +� +j=0 +Yj(n, a)G(hj(n)x), +x ∈ R +where h �→ G(hx) depends on the parameter x ∈ R. Determine the coefficients Yj(n) in such a +way that +f (p) +n (0) = (a)pG(p)(0) +for +p = 0, 1, ..., n. +(20) +The solution of Problem 4.3, obtained in [8], is summarized in the following theorem. +Theorem 4.4. Let hj(n) be a given set of points in [−1, 1], j = 0, 1, ..., n for n ∈ N and let +a ∈ R be such that |a| > 1. If hj(n) ̸= hi(n) for every i ̸= j and G(p)(0) ̸= 0 for all p = 0, 1, ..., n, +then there exists a unique solution Yj(n, a) of the linear system (20) and it is given by +Yj(n, a) = +n +� +k=0, k̸=j +� +hk(n) − a +hk(n) − hj(n) +� +, +so that +fn(x) = +n +� +j=0 +n +� +k=0, k̸=j +� +hk(n) − a +hk(n) − hj(n) +� +G(hj(n)x), +x ∈ R. +Remark 4.5. In the sequel, we shall move from the real to the complex setting and we will +consider those functions G and sequences hj(n) for which the holomorphic extension fn(z) of +fn(x) converges in A1 to G(az). +We can now extend the notion of supershift of a function in several variables. +Definition 4.6 (Supershifts in several variables). Let |a| > 1. +For d ∈ N with d ≥ 2, we +assume that (x1, ..., xd) ∈ Rd. Let (hj,ℓ(n)), j = 0, ..., n for n ∈ N0, be real-valued sequences for +ℓ = 1, ..., d such that for +sup +j=0,...,n, n∈N0 +|hj,ℓ(n)| ≤ 1, +for ℓ = 1, ..., d +and let Gℓ(λ), for ℓ = 1, ..., d, be entire holomorphic functions. We say that the sequence +Fn(x1, . . . , xd) = +n +� +j=0 +cj(n)G1(x1hj,1(n))G2(x2hj,2(n)) . . . Gd(xdhj,d(n)), +(21) +where (cj(n))j,n, j = 0, . . . , n, for n ∈ N0 is a complex-valued sequence, admits the supershift +property if +lim +n→∞ Fn(x1, . . . , xd) = G1(x1a)G2(x2a) . . . Gd(xda). +13 + +Theorem 4.7 (The case of d ≥ 1 variables). Let |a| > 1 and let +fn(x) := +n +� +j=0 +Zj(n, a)eihj(n)x, +n ∈ N, +x ∈ R, +(22) +be a superoscillating function as in Definition 2.1 and assume that its holomorphic extension +to the entire functions fn(z) converges to eiaz in the space A1,B for some positive real value B. +Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} for any ℓ = 1, . . . , d. Let G1, . . . , Gd be +holomorphic functions whose series expansion at zero is given by +Gℓ(λ) = +∞ +� +mℓ=0 +gℓ,mλmℓ, +∀ℓ = 1, . . . , d. +(23) +Moreover, we suppose the sequences (gl,m)’s satisfy the condition +lim sup +m→∞ +|gℓ,m|1/m = 1 +Rℓ +, +∀ℓ = 1, . . . , d. +We define +Fn(x1, . . . , xd) = +n +� +j=0 +Zj(n, a)G1(x1hj(n)) · · · Gd(xdhj(n)), +where Zj(n, a) are given as in (22). Then, Fn(x1, . . . , xd) admits the supershift property that is +lim +n→∞ Fn(x1, . . . , xd) = G1(x1a) · · · Gd(xda) +uniformly on compact subsets of {x ∈ Rd : |xℓ| < R′ for any ℓ = 1, . . . , d} where +R′ := min +� R +|a|, +R +4eB , R +� +where +R := +min +ℓ=1,...,d Rℓ. +Proof. Since |xℓ| < R for any ℓ = 1, . . . , d, we have +Fn(x1, . . . , xd) = +n +� +j=0 +Zj(n, a)G1(x1hj(n)) . . . Gd(xdhj(n)) += +n +� +j=0 +Zj(n, a) +∞ +� +m1=0 +gm1 · · · +∞ +� +md=0 +gmdxm1 +1 +· · · xmd +d (hj(n))m1+···+md. +We now consider the auxiliary complex variable ξ and we note that +λℓ = 1 +iℓ Dℓ +ξeiξλ��� +ξ=0 +for +λ ∈ C, +ℓ ∈ N, +(24) +where Dξ is the derivative with respect to ξ and |ξ=0 denotes the restriction to ξ = 0. We have +Fn(x1, . . . , xd) = +n +� +j=0 +Zj(n, a) +∞ +� +m1=0 +gm1 · · · +∞ +� +md=0 +gmdxm1 +1 +· · · xmd +d [hj(n)]m1+···+md += +n +� +j=0 +Zj(n, a) +∞ +� +m1=0 +gm1 · · · +∞ +� +md=0 +gmdxm1 +1 +· · · xmd +d +1 +im1+···+md Dm1+···+md +ξ +eiξhj(n)��� +ξ=0 += +∞ +� +m1=0 +gm1 · · · +∞ +� +md=0 +gmdxm1 +1 +· · · xmd +d +1 +im1+···+md Dm1+···+md +ξ +n +� +j=0 +Zj(n, a)eiξhj(n)��� +ξ=0. +14 + +We define the operator +V(x1, . . . , xd, Dξ) := +∞ +� +m1=0 +gm1 · · · +∞ +� +md=0 +gmdxm1 +1 +· · · xmd +d +1 +im1+···+md Dm1+···+md +ξ +so that we can write +Fn(x1, . . . , xd) = V(x1, . . . , xd, Dξ) +n +� +j=0 +Zj(n, a)eiξhj(n)��� +ξ=0. +Since |xℓ| < +R +4eB for any ℓ = 1, . . . , d, we can use Proposition 2.10 in order to compute the +following limit +lim +n→∞ Fn(x1, . . . , xd) = V(x1, . . . , xd, Dξ) lim +n→∞ +n +� +j=0 +Zj(n, a)eiξhj(n)��� +ξ=0 += V(x1, . . . , xd, Dξ)eiξa��� +ξ=0 += +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,mdxm1 +1 +. . . xm2 +d +1 +im1+···+md Dm1+···+m2 +ξ +eiξa��� +ξ=0 += +∞ +� +m1=0 +g1,m1 · · · +∞ +� +md=0 +gd,md(ax1)m1 . . . (axd)m2 = G1(ax1) · · · Gd(axd) +where the last equality holds because we are assuming |xℓ| < +R +|a| for any ℓ = 1, . . . , d. +The +previous limit is uniform over the compact subset of {x ∈ Rd : |xℓ| < R′ for any ℓ = 1, . . . , d} +because of Remark 2.11. +Remark 4.8. A special case of the previous theorem occurs when the holomorphic functions Gℓ’s +are entire functions. Moreover, differently from Theorem 3.3, in Theorem 4.7 the parameters xℓ +appear in the arguments of the functions Gℓ’s. This implies that the hypothesis of Theorem 4.7 +imposes more constraints on the parameters xℓ’s, namely |xℓ| < R′ for any ℓ = 1, . . . , d. +References +[1] Y. Aharonov, D. Albert, L. Vaidman, How the result of a measurement of a component +of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett., 60 (1988), +1351-1354. +[2] Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser, Schr¨odinger evolution of superoscilla- +tions with δ- and δ′-potentials, Quantum Stud. Math. 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Toraldo di Francia, Super-Gain Antennas and Optical Resolving Power, Nuovo Cimento +Suppl., 9 (1952), 426–438. +17 + diff --git a/MtFRT4oBgHgl3EQfGDcg/content/tmp_files/load_file.txt b/MtFRT4oBgHgl3EQfGDcg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3dd108263347a2b9bc6e4a9622234701535ba240 --- /dev/null +++ b/MtFRT4oBgHgl3EQfGDcg/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf,len=1133 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='13482v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='FA] 31 Jan 2023 The general theory of superoscillations and supershifts in several variables F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Colombo ∗, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Pinton∗, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Sabadini∗, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Struppa† Abstract In this paper we describe a general method to generate superoscillatory functions of sev- eral variables starting from a superoscillating sequence of one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Our results are based on the study of suitable infinite order differential operators on holomorphic functions with growth conditions of exponential type, where additional constraints are required when deal- ing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The results proved for the superoscillating sequence in several variables are extended to sequences of supershifts in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' AMS Classification: 26A09, 41A60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Key words: General superoscillatory functions, supershifts in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 1 Introduction Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Physical phenomena associated with superoscillatory functions have been known for a long time for example in antennas theory see [31], and in the context of weak values in quantum mechanics, see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In more recent years there has been a wide interest in the theory of superoscillating functions and of supershifts, a notion that generalizes the one of superoscillations, and that was introduced in the literature in order to study the evolution of superoscillations as initial data of the Schr¨odinger equation of other field equations, like Dirac or Klein-Gordon equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' An introduction to superoscillatory functions in one variable and some applications to Schr¨odinger evolution of superoscillatory initial data can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Superoscillatory func- tions in several variables have been rigorously defined and studied in [6] and in [9] where we have initiated also the theory of supershifts in more then one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The aim of this paper is to remove the restrictions in [6, 9] and to obtain a very general theory of superoscillations and supershifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Our results are directed to a general audience of physicists, mathematicians, and engineers, and our main tool is the theory of infinite order differential operators acting on spaces of holo- morphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The literature on superoscillations is quite large, and without claiming ∗Politecnico di Milano, Dipartimento di Matematica, Via E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Bonardi, 9 20133 Milano, Italy, fabrizio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='colombo@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='it, irene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='sabadini@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='it, stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='pinton@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='it †The Donald Bren Presidential Chair in Mathematics, Chapman University, Orange, USA, struppa@chapman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='edu 1 completeness we have tried to mention some of the most relevant (and recent) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Papers [2]-[7], [12], [15], [25], [28] and [29] deal with the issue of permanence of superoscillatory behav- ior when evolved under a suitable Schr¨odinger equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' papers [18]-[20], [26]-[27] and [30] are mostly concerned with the physical nature of superoscillations, while papers [10], [11], [13]-[14], [21]-[24] develop in depth the mathematical theory of superoscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Finally we have cited [7] as a good reference for the state of the art on the mathematics of superoscillations until 2017, and the Roadmap on Superoscillations [17], where the most recent advances in superoscillations and their applications to technology are well explained by the leading experts in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In this paper we extend the results in [9] considering analytic functions in one variable G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , Gd, d ≥ 2, whose Taylor series at zero have radius of convergence grater than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Thus we define general superoscillating functions of several variables as expressions of the form Fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) := n � j=0 Zj(n, a)eix1G1(hj(n))eix2G2(hj(n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eixdGd(hj(n)) where Zj(n, a), j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n, for n ∈ N0 are suitable coefficients of a superoscillating function in one variable as we will see in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We will give conditions on the functions G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , Gd in order that lim n→∞ Fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = eix1G1(a)eix2G2(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eixdGd(a), so that, when |a| > 1, Fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) is superoscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Moreover, we shall also treat the case of sequences that admit a supershift in d ≥ 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The paper is organized in four sections including the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Section 2 contains the preliminary material on superoscillations, the relevant function spaces and their topology, and the study of the continuity of some infinite order differential operators acting on such spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Section 3 is the main part of the paper and contains the definition of superoscillating functions in d ≥ 2 variables as well as some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Section 4 discusses the notion of supershift in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 2 Preliminary results on infinite order differential operators We begin this section with some preliminary material on superoscillations and supershifts in one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Then we introduce and study some infinite order differential operators that will be of crucial importance to define and study superoscillations and supershifts in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We call generalized Fourier sequence a sequence of the form fn(x) := n � j=0 Zj(n, a)eihj(n)x, n ∈ N, x ∈ R, (1) where a ∈ R, Zj(n, a) and hj(n) are complex and real valued functions of the variables n, a and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The sequence (1) is said to be a superoscillating sequence if supj,n |hj(n)| ≤ 1 and there exists a compact subset of R, which will be called a superoscillation set, on which fn(x) converges uniformly to eig(a)x, where g is a continuous real valued function such that |g(a)| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The classical Fourier expansion is obviously not a superoscillating sequence since its frequen- cies are not, in general, bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In the recent paper [8] we enlarged the class of superoscillating functions, with respect to the existing literature, and we solved the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 2 Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let hj(n) be a given set of points in [−1, 1], j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n, for n ∈ N and let a ∈ R be such that |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Determine the coefficients Xj(n) of the sequence fn(x) = n � j=0 Xj(n)eihj(n)x, x ∈ R in such a way that f (p) n (0) = (ia)p, for p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The conditions f (p) n (0) = (ia)p mean that the functions x �→ eiax and x �→ fn(x) have the same derivatives at the origin, for p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n, and therefore the same Taylor polynomial of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='4 (Solution of Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let hj(n) be a given set of points in [−1, 1], j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n for n ∈ N and let a ∈ R be such that |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' If hj(n) ̸= hi(n), for every i ̸= j, then the coefficients Xj(n, a) are uniquely determined and given by Xj(n, a) = n � k=0, k̸=j � hk(n) − a hk(n) − hj(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (2) As a consequence, the sequence fn(x) = n � j=0 n � k=0, k̸=j � hk(n) − a hk(n) − hj(n) � eixhj(n), x ∈ R solves Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Moreover, when the holomorphic extensions of the functions fn converge in A1, we have lim n→∞ fn(x) = eiax, for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Our approach to the study of superoscillatory functions in one or several variables makes use of infinite order differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Such operators naturally act on spaces of holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' This is the reason for which we consider the holomorphic extension to entire functions of the sequence fn(x) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='1) by replacing the real variable x with the complex variable ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' For the sequences of entire functions we shall consider, a natural notion of convergence is the convergence in the space A1 or in the space A1,B for some real positive constant B (see the following definition and considerations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The space A1 is the complex algebra of entire functions such that there exists B > 0 such that sup ξ∈C � |f(ξ)| exp(−B|ξ|) � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (3) The space A1 has a rather complicated topology, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' [16], since it is a linear space obtained via an inductive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' For our purposes, it is enough to consider, for any fixed B > 0, the set A1,B of functions f satisfying (3), and to observe that ∥f∥B := sup ξ∈C � |f(ξ)| exp(−B|ξ|) � defines a norm on A1,B, called the B-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' One can prove that A1,B is a Banach space with respect to this norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 3 Moreover, let f and a sequence (fn)n belong to A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' fn converges to f in A1 if and only if there exists B such that f, fn ∈ A1,B and lim n→∞ sup ξ∈C ��fn(ξ) − f(ξ) ��e−B|ξ| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' With these notations and definitions we can make the notion of continuity explicit (see [14]): A linear operator U : A1 → A1 is continuous if and only if for any B > 0 there exists B′ > 0 and C > 0 such that U(A1,B) ⊂ A1,B′ and ∥U(f)∥B′ ≤ C∥f∥B, for any f ∈ A1,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (4) The following result, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='6 in [13], gives a characterization of the functions in A1 in terms of the coefficients appearing in their Taylor series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The entire function f(ξ) = ∞ � j=0 ajξj belongs to A1 if and only if there exists Cf > 0 and b > 0 such that |aj| ≤ Cf bj Γ(j + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' To say that f ∈ A1 means that f ∈ A1,B for some B > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The computations in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='6 in [13], show that b = 2eB, and that we can choose Cf = ∥f∥B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We now define two infinite order differential operators that will be used to study superoscil- latory functions and supershifts in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We shall denote by x the vector (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let (g1,m), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , (gd,m) be d sequences of complex numbers such that lim sup m→∞ |gℓ,m|1/m = 1 Rℓ , for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (5) Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Denote by Dξ := ∂ ∂ξ the derivative operator with respect to the auxiliary complex variable ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We define the formal operator: U(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) := ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 Dk1 ξ ik1 (6) where we have set yp := ix1g1,p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + ixdgd,p, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' r with r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Then, setting R := min ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',d Rℓ, for any real value 0 < B < R 4e, the operator U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) : A1,B → A1,4eB is continuous for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let us consider f ∈ A1,B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' then we have U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ) = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 Dk1 ξ ik1 f(ξ) = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 ∞ � j=k1 aj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (j − k1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='ξj−k1 = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 ∞ � j=0 aj+k1 (j + k1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Taking the modulus we get |U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm||ykm−1−km| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |yk1−k2| \uf8f6 \uf8f8 ∞ � j=0 |aj+k1|(j + k1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='6 gives the estimate on the coefficients aj+k1 |aj+k1| ≤ Cf bj+k1 Γ(j + k1 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' where b = 2eB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Using the well known inequality (a + b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ≤ 2a+ba!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' we also have (j + k1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ≤ 2j+k1j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='k1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' so we get |U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm||ykm−1−km| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |yk1−k2| \uf8f6 \uf8f8 × ×Cf ∞ � j=0 bj+k1 Γ(j + k1 + 1) 2j+k1k1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |ξ|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Now we use the Gamma function estimate 1 Γ(a + b + 2) ≤ 1 Γ(a + 1) 1 Γ(b + 1) (7) to separate the series, and we have 1 Γ(j − 1 2 + k1 − 1 2 + 2) ≤ 1 Γ(j + 1 2) 1 Γ(k1 + 1 2) and so |U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ Cf ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm||ykm−1−km| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |yk1−k2| \uf8f6 \uf8f8 × 5 ×(k1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (2b)k1 Γ(k1 + 1 2) ∞ � j=0 1 Γ(j + 1 2)(2b|ξ|)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Now observe that the latter series satisfies the estimate ∞ � j=0 1 Γ(k + 1 2)(2b|ξ|)j ≤ Ce4b|ξ| where C is a positive constant, because of the properties of the Mittag-Leffler function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' moreover, the series ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm||ykm−1−km| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |yk1−k2| \uf8f6 \uf8f8 (k1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (2b)k1 Γ(k1 + 1 2) (8) is convergent and is bounded by a positive real constant Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In fact, using Stirling formula for the Gamma function, we have m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∼ √ 2πm e−mmm, for m → ∞ and then we deduce Γ(m + 1) Γ(m + 1/2) ∼ √ 2π m e−mmm � 2π(m − 1/2) e−(m−1/2) (m − 1/2)(m−1/2) ∼ � m − 1/2, for m → ∞ (9) so that k1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Γ(k1 + 1 2) ∼ � k1 − 1/2, for k1 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Now observe that the series (8) has positive coefficients and so it converges if and only if the series ∞ � m=1 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=1 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm||ykm−1−km| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |yk1−k2| \uf8f6 \uf8f8 (2b)k1� k1 − 1/2 converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Given an absolutely convergent series �∞ p=0 ap, then its m-th power can be computed by means of the Cauchy product as follows: \uf8eb \uf8ed ∞ � p=0 ap \uf8f6 \uf8f8 m = ∞ � k1=0 k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 akmakm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ak1−k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (10) Using the inequality: � k1 − 1 2 ≤ k1 ≤ km +(km−1 −km)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='+(k1 −k2) ≤ (km +2)·(km−1 −km +2)·····(k1 −k2 +2), where k1 ≥ k2 ≥ · · · ≥ km, we deduce that there exists a positive constant Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd such that 6 the following chain of inequalities hold: ∞ � m=1 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=1 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm||ykm−1−km| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |yk1−k2| \uf8f6 \uf8f8 (2b)k1� k1 − 1/2 ≤ ∞ � m=1 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=1 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 |ykm(km + 2)(2b)km||ykm−1−km(km−1 − km + 2)(2b)km−1−km|× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' × |yk1−k2(k2 − k1 + 2)(2b)k1−k2| � = ∞ � m=1 1 (m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' \uf8ee \uf8f0 ∞ � p=0 |yp|(p + 2)(2b)p \uf8f9 \uf8fb m ≤ ∞ � m=1 1 (m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' \uf8ee \uf8f0 ∞ � p=1 |x1|(p + 2)(2b)p|g1,p| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + |xd|(p + 2)(2b)p|gd,p| �m ≤ Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd where for the equality we used (10), while the last inequality follows by the assumption B < R 4e which implies 2b < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' From the previous estimate we have that the series (8) converges for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' So we finally have |U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ Cf Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd C e4b|ξ|, x ∈ Rd, ξ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (11) Recalling that b = 2eB, the estimate (11) implies that U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, )f ∈ A1,8eB, in fact |U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| e−8eB|ξ| ≤ Cf Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd C x ∈ Rd, ξ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Moreover, we deduce that the 8eB-norm satisfies the estimate ∥U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f∥8eB ≤ Cf Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd C = Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd C∥f∥B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Thus U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) : A1,B → A1,8eB is continuous for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Whenever we fix a compact subset K ⊂ Rd, we have that, for any x ∈ K, the constants Cx,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',Gd appearing in the proof of the previous theorem are bounded by a constant which depends only on K and G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , Gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Moreover, if Rℓ = ∞ for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d, the continuity of the operator U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) holds for any B > 0 and the proof of the previous theorem shows that U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) is a continuous operator in A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let (g1,m), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , (gd,m) be d sequences of complex numbers such that lim sup m→∞ |gℓ,m|1/m = 1 Rℓ , for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (12) We define the formal operator V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) := ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,mdxm1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' xmd d 1 im1+···+md Dm1+···+md ξ , (13) where x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd ∈ R, ξ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Then, for any real value B > 0, the operator V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) : A1,B → A1,8eB is continuous whenever |xℓ| < R 4eB for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d where R := minℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',d Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We apply the operator V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) to a function f in A1,B for |x| < R 4eB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We have V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ) = ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,mdxm1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' xmd d 1 im1+···+md Dm1+···+m2 ξ f(ξ) = ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,mdxm1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' xmd d 1 im1+···+md Dm1+···+md ξ ∞ � j=0 ajξj = ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,mdxm1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' xmd d 1 im1+···+md × × ∞ � j=m1+···+md aj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (j − (m1 + · · · + md))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='ξj−(m1+···+md) = ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,mdxm1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' xmd d 1 im1+···+md ∞ � k=0 am1+···+md+k (m1 + · · · + md + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We then have |V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ ∞ � m1=0 |g1,m1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � md=0 |gd,md||x1|m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |xd|md× × ∞ � k=0 |am1+···+md+k|(m1 + · · · + md + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |ξ|k and using the estimate in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='6 |am1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='md+k| ≤ Cf bm1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='md+k Γ(m1 + · · · + md + k + 1), where b = 2eB, we get |V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ ∞ � m1=0 |g1,m1| · · · ∞ � md=0 |gd,md||x1|m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |xd|md× ×Cf ∞ � k=0 bm1+···+md+k Γ(m1 + · · · + md + k + 1) (m1 + · · · + md + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |ξ|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' With the estimates (m1 + · · · + md + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ≤ 2m1+···+md+k(m1 + · · · + md)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' and 1 Γ(m1 + · · · + md − 1 2 + k − 1 2 + 2) ≤ 1 Γ(m1 + · · · + md + 1 2) 1 Γ(k + 1 2) we separate the series |V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ ∞ � m1=0 |g1,m1| · · · ∞ � md=0 |gd,md||x1|m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |xd|md× × ∞ � k=0 Cfbm1+···+md+k 1 Γ(m1 + · · · + md + 1 2) 1 Γ(k + 1 2) 2m1+···+md+k(m1 + · · · + md)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' |ξ|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 8 Finally we get |V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ Cf ∞ � m1=0 |g1,m1| · · · ∞ � md=0 |gd,md|(2b|x1|)m1 · · · (2b|xd|)md× × (m1 + · · · + md)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Γ(m1 + · · · + md + 1 2) ∞ � k=0 1 Γ(k + 1 2)(2b|ξ|)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Using (9) we have (m1 + · · · + md)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Γ(m1 + · · · + md + 1 2) ∼ � m1 + · · · + md − 1/2, for m1 + · · · + md → ∞, and, moreover, � m1 + · · · + md − 1/2 ≤ m1 · · · md if mℓ ≥ 2 for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Since |xℓ| < R 4eB for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d and b = 2eB, the series ∞ � mℓ=1 mℓ|gℓ,mℓ|(2b|xℓ|)mℓ converges to a constant which depends on xℓ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Thus there exist constants Cxℓ such that |V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f(ξ)| ≤ CfCx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Cxd(2b|ξ|)e2b|ξ| ≤ CfCx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',xde4b|ξ| from which, recalling that Cf = ∥f∥B, we deduce ∥V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)f∥8eB ≤ Cx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',xd∥f∥B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We conclude that the operator V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) : A1,B → A1,8eB is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Whenever we fix a compact subset K ⊂ {x ∈ Rd : |xℓ| < R 4eB for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d}, we have that, for any x ∈ K, the constants Cxℓ’s, appearing in the proof of the previous theorem are bounded by a constant which depends only on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Moreover, if Rℓ = ∞ for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d, the continuity of the operator V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) holds to be true for any x ∈ Rd and the proof of the previous theorem shows that V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) satisfies the conditions in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Thus we conclude that V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) is a continuous operator in A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 3 Superoscillating functions in several variables We recall some preliminary definitions related to superoscillating functions in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='1 (Generalized Fourier sequence in several variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' For d ∈ N such that d ≥ 2, we assume that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', xd) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let (hj,ℓ(n)), j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n for n ∈ N0, be real-valued sequences for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We call generalized Fourier sequence in several variables a sequence of the form Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = n � j=0 cj(n)eix1hj,1(n)eix2hj,2(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eixdhj,d(n), (14) where (cj(n))j,n, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , n, for n ∈ N0 is a complex-valued sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 9 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='2 (Superoscillating sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' A generalized Fourier sequence in several variables Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd), with d ∈ N such that d ≥ 2, is said to be a superoscillating sequence if sup j=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',n, n∈N0 |hj,ℓ(n)| ≤ 1, for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', d, and there exists a compact subset of Rd, which will be called a superoscillation set, on which Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) converges uniformly to eix1g1eix2g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eixdgd, where |gℓ| > 1 for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In the paper [6] we studied the function theory of superoscillating functions in several vari- ables under the additional hypothesis that there exist rℓ ∈ N, such that p = r1q1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + rdqd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (15) In that case, we proved that for p, qℓ ∈ N, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d the function Fn(x, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , yd) = n � j=0 Cj(n, a)eix(1−2j/n)peiy1(1−2j/n)q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eiyd(1−2j/n)qd is superoscillating when |a| > 1, where Cj(n, a) are suitable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In the paper [9], we were able to remove the condition (15), while here we will show that it is possible to replace the functions (1 − 2j/n)p in the terms eix(1−2j/n)p with more general holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' As we shall see, different function spaces are involved in the proofs according to the fact that the holomorphic functions are entire or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='3 (The general case of d ≥ 2 variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} be such that Rℓ ≥ 1 for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , Gd be holomorphic functions whose series expansion at zero is given by Gℓ(λ) = ∞ � mℓ=0 gℓ,mλmℓ, ∀ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d (16) and, moreover, the sequences (gℓ,m) satisfy the condition lim sup m→∞ |gℓ,m|1/m = 1 Rℓ , ∀ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let fn(x) := n � j=0 Zj(n, a)eihj(n)x, n ∈ N, x ∈ R, (17) be superoscillating functions as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='1 and assume that their entire extensions to the functions fn(ξ) converge to eiaξ in A1,B for some positive real value 0 < B < R 4e where R := minℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',d Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We define Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) := n � j=0 Zj(n, a)eix1G1(hj(n))eix2G2(hj(n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eixdGd(hj(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Then, whenever |a| < R we have lim n→∞ Fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = eix1G1(a)eix2G2(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' eixdGd(a), uniformly on compact subsets of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In particular, Fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) is superoscillating when |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Since Rℓ ≥ 1 for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d and |hj(n)| < 1, using (10) we have the chain of equalities Fn(x1, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = n � j=0 Zj(n, a)eix1G1(hj(n))+ix2G2(hj(n))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='+ixdGd(hj(n)) = n � j=0 Zj(n, a) ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' � ix1G1(hj(n)) + ix2G2(hj(n)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + ixdGd(hj(n)) �m = n � j=0 Zj(n, a) ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' � ix1 ∞ � p=1 g1,p(hj(n))p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + ixd ∞ � p=1 gd,p(hj(n))p�m = n � j=0 Zj(n, a) ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' � ∞ � p=1 (ix1g1,p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + ixdgd,p)(hj(n))p�m = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 (hj(n))k1, where we have set yp := ix1g1,p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + ixdgd,p, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' r with r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We define the infinite order differential operator U(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) := ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 Dk1 ξ ik1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (18) Since 0 < B < R 4e, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='8 implies that the operator U(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) : A1,B �→ A1,8eB is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We observe that Fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = U(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) n � j=0 Zj(n, a)eiξhj(n)��� ξ=0 The explicit computation of the term U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)eiξa gives U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)eiξa = = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 Dk1 ξ ik1 eiξa = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 ak1eiξa, 11 so we finally get lim n→∞ Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 ykmykm−1−km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' yk1−k2 \uf8f6 \uf8f8 ak1eiξa��� ξ=0 = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' ∞ � k1=0 \uf8eb \uf8ed k1 � k2=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' km−1 � km=0 (ykmakm)(ykm−1−kmakm−1−km) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (yk1−k2ak1−k2) \uf8f6 \uf8f8 = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' \uf8eb \uf8ed ∞ � p=1 ypap \uf8f6 \uf8f8 m = ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (ix1G1(a) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' + ixdGd(a))m = eix1G1(a)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='+ixdGd(a) where the third equality is due to the formula (10) and the fourth equality holds because we are assuming |a| < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The previous limit is uniform over the compact subset of Rd because of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' From the inspection of the proof we observe that: (I) The space of the entire functions on which the infinite order differential operator U(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) acts is the space A1,B in one complex variable, for some positive real value 0 < B < R 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (III) The variables (x1, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) become the coefficients of the infinite order differential oper- ator U(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ), defined in (18), that still acts on the space A1,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 4 Supershifts in several variables The procedure to define superoscillating functions can be extended to the case of supershift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Recall that the supershift property of a function extends the notion of superoscillation and that this concept, that we recall below, turned out to be a crucial ingredient for the study of the evolution of superoscillatory functions as initial conditions of the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='1 (Supershift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let I ⊆ R be an interval with [−1, 1] ⊂ I and let ϕ : I × R → R, be a continuous function on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We set ϕh(x) := ϕ(h, x), h ∈ I, x ∈ R and we consider a sequence of points (hj(n)) such that hj(n) ∈ [−1, 1] for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n and n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We define the functions ψn(x) = n � j=0 cj(n)ϕhj(n)(x), (19) where (cj(n)) is a sequence of complex numbers for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n and n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' If lim n→∞ ψn(x) = ϕa(x) for some a ∈ I with |a| > 1, we say that the function ψn(x), for x ∈ R, admits a supershift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 12 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The term supershift comes from the fact that the interval I can be arbitrarily large (it can also be R) and that the constant a can be arbitrarily far away from the interval [−1, 1] where the functions ϕhj,n(·) are indexed, see (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='2, for the supershift case, is formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let hj(n) be a given set of points in [−1, 1], j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n, for n ∈ N and let a ∈ R be such that |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Suppose that for every x ∈ R the function h �→ G(hx) extends to a holomorphic and entire function in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Consider the functions fn(x) = n � j=0 Yj(n, a)G(hj(n)x), x ∈ R where h �→ G(hx) depends on the parameter x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Determine the coefficients Yj(n) in such a way that f (p) n (0) = (a)pG(p)(0) for p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (20) The solution of Problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='3, obtained in [8], is summarized in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let hj(n) be a given set of points in [−1, 1], j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n for n ∈ N and let a ∈ R be such that |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' If hj(n) ̸= hi(n) for every i ̸= j and G(p)(0) ̸= 0 for all p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n, then there exists a unique solution Yj(n, a) of the linear system (20) and it is given by Yj(n, a) = n � k=0, k̸=j � hk(n) − a hk(n) − hj(n) � , so that fn(x) = n � j=0 n � k=0, k̸=j � hk(n) − a hk(n) − hj(n) � G(hj(n)x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' In the sequel, we shall move from the real to the complex setting and we will consider those functions G and sequences hj(n) for which the holomorphic extension fn(z) of fn(x) converges in A1 to G(az).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We can now extend the notion of supershift of a function in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='6 (Supershifts in several variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' For d ∈ N with d ≥ 2, we assume that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', xd) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let (hj,ℓ(n)), j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', n for n ∈ N0, be real-valued sequences for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', d such that for sup j=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',n, n∈N0 |hj,ℓ(n)| ≤ 1, for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', d and let Gℓ(λ), for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=', d, be entire holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We say that the sequence Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = n � j=0 cj(n)G1(x1hj,1(n))G2(x2hj,2(n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Gd(xdhj,d(n)), (21) where (cj(n))j,n, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , n, for n ∈ N0 is a complex-valued sequence, admits the supershift property if lim n→∞ Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = G1(x1a)G2(x2a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Gd(xda).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 13 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='7 (The case of d ≥ 1 variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let |a| > 1 and let fn(x) := n � j=0 Zj(n, a)eihj(n)x, n ∈ N, x ∈ R, (22) be a superoscillating function as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='1 and assume that its holomorphic extension to the entire functions fn(z) converges to eiaz in the space A1,B for some positive real value B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let d be a positive integer and let Rℓ ∈ R+ ∪ {∞} for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , Gd be holomorphic functions whose series expansion at zero is given by Gℓ(λ) = ∞ � mℓ=0 gℓ,mλmℓ, ∀ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (23) Moreover, we suppose the sequences (gl,m)’s satisfy the condition lim sup m→∞ |gℓ,m|1/m = 1 Rℓ , ∀ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We define Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = n � j=0 Zj(n, a)G1(x1hj(n)) · · · Gd(xdhj(n)), where Zj(n, a) are given as in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Then, Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) admits the supershift property that is lim n→∞ Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = G1(x1a) · · · Gd(xda) uniformly on compact subsets of {x ∈ Rd : |xℓ| < R′ for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d} where R′ := min � R |a|, R 4eB , R � where R := min ℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=',d Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Since |xℓ| < R for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d, we have Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = n � j=0 Zj(n, a)G1(x1hj(n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Gd(xdhj(n)) = n � j=0 Zj(n, a) ∞ � m1=0 gm1 · · · ∞ � md=0 gmdxm1 1 · · xmd d (hj(n))m1+···+md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We now consider the auxiliary complex variable ξ and we note that λℓ = 1 iℓ Dℓ ξeiξλ��� ξ=0 for λ ∈ C, ℓ ∈ N, (24) where Dξ is the derivative with respect to ξ and |ξ=0 denotes the restriction to ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' We have Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = n � j=0 Zj(n, a) ∞ � m1=0 gm1 · · · ∞ � md=0 gmdxm1 1 · · xmd d [hj(n)]m1+···+md = n � j=0 Zj(n, a) ∞ � m1=0 gm1 · · · ∞ � md=0 gmdxm1 1 · · xmd d 1 im1+···+md Dm1+···+md ξ eiξhj(n)��� ξ=0 = ∞ � m1=0 gm1 · · · ∞ � md=0 gmdxm1 1 · · xmd d 1 im1+···+md Dm1+···+md ξ n � j=0 Zj(n, a)eiξhj(n)��� ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' 14 We define the operator V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) := ∞ � m1=0 gm1 · · · ∞ � md=0 gmdxm1 1 · · xmd d 1 im1+···+md Dm1+···+md ξ so that we can write Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) n � j=0 Zj(n, a)eiξhj(n)��� ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Since |xℓ| < R 4eB for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d, we can use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='10 in order to compute the following limit lim n→∞ Fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd) = V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ) lim n→∞ n � j=0 Zj(n, a)eiξhj(n)��� ξ=0 = V(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , xd, Dξ)eiξa��� ξ=0 = ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,mdxm1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' xm2 d 1 im1+···+md Dm1+···+m2 ξ eiξa��� ξ=0 = ∞ � m1=0 g1,m1 · · · ∞ � md=0 gd,md(ax1)m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' (axd)m2 = G1(ax1) · · · Gd(axd) where the last equality holds because we are assuming |xℓ| < R |a| for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' The previous limit is uniform over the compact subset of {x ∈ Rd : |xℓ| < R′ for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d} because of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' A special case of the previous theorem occurs when the holomorphic functions Gℓ’s are entire functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Moreover, differently from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='3, in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='7 the parameters xℓ appear in the arguments of the functions Gℓ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' This implies that the hypothesis of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content='7 imposes more constraints on the parameters xℓ’s, namely |xℓ| < R′ for any ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Aharonov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Albert, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQfGDcg/content/2301.13482v1.pdf'} +page_content=' Vaidman, How the result of a 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Hauser1 +1Institut für Wissenschaftliches Rechnen, +Technische Universität Dresden, +Zellescher Weg 25, 01217 Dresden, Germany +julia.hauser@tu-dresden.de +Abstract +The ability to deal with complex geometries and to go to higher orders is the main +advantage of space-time finite element methods. Therefore, we want to develop a solid +background from which we can construct appropriate space-time methods. In this pa- +per, we will treat time as another space direction, which is the main idea of space-time +methods. First, we will briefly discuss how exactly the vectorial wave equation is derived +from Maxwell’s equations in a space-time structure, taking into account Ohm’s law. Then +we will derive a space-time variational formulation for the vectorial wave equation using +different trial and test spaces. This paper has two main goals. First, we prove unique +solvability for the resulting Galerkin–Petrov variational formulation. Second, we analyze +the discrete equivalent of the equation in a tensor product and show conditional stability, +i.e. a CFL condition. +Understanding the vectorial wave equation and the corresponding space-time finite ele- +ment methods is crucial for improving the existing theory of Maxwell’s equations and +paves the way to computations of more complicated electromagnetic problems. +1 +Modeling and Introduction +Maxwell’s equations are used to model electromagnetic problems. They are the foundation of +classical electromagnetism. Maxwell’s equations are given by +curlx E = −∂tB, +(1a) +curlx H = ∂tD + j, +(1b) +divxB = 0, +(1c) +divxD = ρ. +(1d) +In equation (1), there are four unknowns: The electric field E, the magnetic field H, the +electric flux density D and the magnetic flux density B. The variable j is the given electric +current density and ρ is the given charge density. Additional to the Maxwell system (1) we use +constitutive relations, or material laws, relating the electric flux density D with the electric +1 + +field E and the magnetic field H with the magnetic flux density B. In this paper, we will +consider the constitutive relations +D = εE, +(2a) +H = µB, +(2b) +where ε is the permittivity and µ the permeability, which will be defined in Assumption 1. +How to solve the Maxwell system (1) has been an active field of study in the last century. +One possible solution is going into the frequency domain. For that purpose, we assume that the +solutions of Maxwell’s equations look like waves in time. There exists several introductions +to Maxwell’s equations in the literature. We will not discuss the modeling background in +the frequency domain, but simply refer to some of these works such as [33, 18] or, from a +mathematical and numerical point of view [6, 29]. +A good introduction to the computational theory of the frequency domain would be [6, 34]. +In the last century, the time-harmonic Maxwell’s equations were often applied to scattering +problems, see e.g. [25]. We find analytical solutions to scattering problems from 1970 in [17] +and theory on inverse scattering and optimal control in scattering by Colton and Kress in +1983, [13]. An introduction to inverse scattering problems can be found in [14]. +Another possibility to solve Maxwell’s equations (1). For an introduction we refer to [1, +Ch. 12.2] and references there. However, the time-stepping approach observes instabilities. A +possibility to deal with these complications is to stabilize the system which is created by the +time-stepping method. Examples can be found in, e.g., [2, 3, 30]. However, since the time- +stepping method is a type of finite difference method, it cannot deal with complex geometries. +The possibility to deal with complex geometries as well as going into higher order is the +main advantage of space-time finite element methods. In this paper, we will set the theoretical +background that is needed for such a method and explore the possibilities and restrictions of +the vectorial wave equation that is derived from Maxwell’s equations. We will also study +elements of second order in time when we go over to the study of the finite element method. +The goal of this paper is to derive a space-time variational formulation for the vectorial wave +equation, show its unique solvability and analyze its numerical properties. +First, we will start by deriving the vectorial wave equation from Maxwell’s equations. For +the modeling, we will assume that all functions in (1) are smooth and we consider a space-time +domain Q in R4 which is star-like with respect to a ball B and Lipschitz in space. The star-like +property implies the convex hull of any x ∈ Ω and B is contained in Ω. Let us not treat time +differently from space, but rewrite the equations such that the partial derivative in time is +simply a partial derivative of a space-time derivative. This can be naturally done in terms of +differential forms for which we use the exterior derivative. Hence, we need a form for which +the exterior derivative of that form gives us one or more of Maxwell’s equations. Following +the derivation of [9] we combine the 1-form ˜E corresponding to the electric field E with the +2-form ˜B corresponding to the magnetic flux density B to get the Faraday 2-form +F = ˜E ∧ dt + ˜B, +where dt ∈ Λ1(Q) is the basis element for 1-forms in the direction of time, for more details +see [16]. Second, we combine the 2-form ˜D corresponding to the electric flux density D and +the 1-form ˜H corresponding to the magnetic field H into the Maxwell 2-form +G = ˜H ∧ dt − ˜D. +2 + +Additionally, we need the source 3-form +J = ˜j ∧ dt − ˜ρ, +where ˜j is the 2-form corresponding to j and ˜ρ is the corresponding 3-form to ρ. By inserting +these forms into the equations we end up with a representation in four dimensions +d F = 0, +d G = J , +(3) +G = ⋆ε,−µ−1F. +For the derivation of the equations consider [9] or [8, p. 135]. +The operator ⋆ε,−µ−1 is a +weighted Hodge star operator. For R4 with the euclidean metric ǫ is the weight in the direction +⋆(dx01, dx02, dx03)T and (−µ−1) in the direction ⋆(dx23, dx31, dx12)T . +Let us take a closer look at the equations in (3). The third equation in (1) includes the +constitutive equations (2). The second equation, d G = J , includes the second and fourth +equation of (1). The first equation, d F = 0, includes the first and third equations of (1). +Moreover, the first equation in (3) implies that the form F is closed. Since the domain Q is +starlike, the Poincaré-Lemma, [7, Thm. 4.1], can be used. The Poincaré-Lemma tells us that +if the form F is exact then there is a potential A such that dA = F. +If we insert F = dA into the second equation combined with the third equation of (3), we +derive the following wave-type equation +d ⋆ε,−µ−1 dA = J . +(4) +Additionally, we have the following two relations +E = −∂tA + ∇xA0, +B = ∇ × A, +where A0 is the time component and A := (A1, A2, A3)T the spatial component of A, see +e.g. [24, p. 389]. Note that (4) will result in the scalar wave equation if we use a differential +form of order zero instead of one for A, while the equation (4) as such will end up in the +vectorial wave equation. +Hence, in terms of differential forms in 4D, the scalar and the +vectorial wave equation are closely related. +Moreover, we know that A is not unique. If we take a 0-form ˜ +A adding it to A then +F = dA = dA+dd ˜ +A = d(A+d ˜ +A). Therefore we can add the gradient of any H1 +0(Ω)-function +to A and get another viable potential. Hence we need a gauge to fix the potential A. There +are many gauges, see e.g. a paper from 2001 on the history of gauge invariance [26]. +Depending on the gauge we can derive different equations from (4). To derive the vectorial +wave equation, we use the Weyl gauge, which is the temporal gauge A0 = 0, to arrive in +euclidean space at +∂t(ε(∂tA)) + curlx +� +µ−1 curlx A +� += j, +(5a) +divx(ε(∂tA)) = −ρ. +(5b) +Note that the equation (5a) is the vectorial wave equation. In addition to these two equations, +we assume that the charge is preserved and therefore the continuity equation +∂tρ + divxj = 0 +3 + +holds true. The continuity equation is, however, already included in the combination of both +equations of (5). On the other hand, we can rewrite the second equation (5b) into initial +conditions for ∂tA. Indeed, if we assume enough regularity of A, ρ, and j, we can take the +spatial divergence of the first equation (5a) and use the continuity equation to derive +∂t(divx(ε∂tA(t)) + ρ) = 0. +Hence divx(ε∂tA(t))(t)+ρ(t) = divx(ε∂tA(t))(0)+ρ(0), for all t ∈ (0, T]. Therefore, if ∂tA(0) +satisfies +divx(ε∂tA(0)) = −ρ(0) +then we satisfy (5b) for all t > 0. +Now that we derived the vectorial wave equation, let us take a look at Ohm’s law. In +the case of a conducting material, the electromagnetic field itself induces currents and in the +easiest case this can be modeled by Ohm’s law. Hence, we want to include Ohm’s law into +our equation which is given by +j = σE + ja, +where σ is the conductivity and ja the applied current. The relationship between E and the +magnetic vector potential A was given by E = −∂tA. Therefore, in this paper, we consider +the following equation +ε∂ttA + σ∂tA + curlx(µ−1 curlx A) = ja +in Q = (0, T) × Ω, +(6) +∂tA(0, x) = ψ(x) +in Ω, +A(0, x) = φ(x) +in Ω, +γtA = 0 +on Σ = {0, T} × ∂Ω, +where γt is the tangential trace of A on Ω. +Before we state the general assumption of this paper on the functions in (6), we have to +define a few functional spaces that will be used throughout this paper. First, we define for +Q ⊂ Rd, d ∈ N, the space L2(Q; Rd) as the usual Lebesgue space for vector-valued functions +v: Q → Rd with the inner product (v, w)L2(Ω) := (v, w)L2(Ω;Rd) := +� +Q(v(x), w(x))Rddx for +v, w ∈ L2(Q; Rd) and the induced norm ∥·∥L2(Q) := ∥·∥L2(Q;Rd) := +� +(·, ·)L2(Q). Second, we +define L∞(Ω; R) := L∞(Ω) for Ω ⊂ Rd, d ∈ N, as the space of measurable functions bounded +almost everywhere and equipped with the usual norm ∥·∥L∞(Ω). Third, we define the space +L∞(Ω; Rd×d), d ∈ N, as the space of matrix-valued measurable functions bounded almost +everywhere with the norm +∥w∥L∞(Ω) := ∥w∥L∞(Ω;Rd×d) := ess sup +x∈Ω +sup +0̸=ξ∈Rd +ξ⊤ǫ(x)ξ +ξ⊤ξ +. +Finally, we define space-time spaces L2(0, T; X) with the inner product (v, w)L2(0,T;X) := +� T +0 (v(x), w(x))Xdx and L1(0, T; L2(Ω; Rd)) with the norm ∥v∥2,1,Q := ∥v∥L1(0,T;L2(Ω;Rd)) := +� T +0 +���∥v∥L2(Ω) +��� dt in the same way as above. Additionally, we can define the space H1(0, T; X) +as the Hilbert space H1(0, T) over the Hilbert space X, see [22] for more details. +4 + +With these definitions we define the tangential trace operator γt for d = 3 as the contin- +uous mapping γt : H(curl; Ω) → H−1/2(∂Ω; R3). The tangential trace operator is the unique +extension of the vector-valued function γtv = v|∂Ω × nx for v ∈ H1(Ω)3 that is defined by +the Green’s identity for the curl operator. For d = 2 we define the tangential trace operator +γt : H(curl; Ω) → H−1/2(∂Ω) as the unique extension of the scalar function γtv = v|∂Ω · τ x +for v ∈ H1(Ω)2, where τ x is the unit tangent vector, i.e., τ x · nx = 0. Note that γv := (v)|∂Ω +is the usual trace operator γ : H1(Ω, Rd) → H1/2(∂Ω; Rd) for d ∈ N and H−1/2(∂Ω) is the +dual space of the fractional-order Sobolev space H1/2(∂Ω). For more details on the tangential +trace operator γt consider [15, 4, 6]. +For detailed definitions of the spaces H0(curl; Ω) := {v ∈ H(curl; Ω) | γtv = 0} and +H(div; Ω), we reference [6]. +However, we quickly note how the curl operator behaves dif- +ferently in two and three dimensions. For d = 2, we set the curl of a sufficiently smooth +vector-valued function v: Ω → R2 with v = (v1, v2)⊤ as the scalar-valued curl operator +curlx v = ∂x1v2 − ∂x2v1. +Sometimes this curl operator is called rot. +Additionally, for a +sufficiently smooth scalar function w: Ω → R, we define the vector-valued curl operator +curlx w = (∂x2w, −∂x1w)⊤. Note that the vector-valued curl operator is the adjoint operator +of the scalar-valued curl operator for d = 2. +In the case of d = 3, the curl of a sufficiently smooth vector-valued function v: Ω → R3 with +v = (v1, v2, v3)⊤ is given by the vector-valued function +curlx v = (∂x2v3 − ∂x3v2, ∂x3v1 − ∂x1v3, ∂x1v2 − ∂x2v1)⊤ . +With these definitions, we can write the following assumptions. +Assumption 1. Let d = 2, 3 and the spatial domain Ω ⊂ Rd and supp(σ) ⊂ Rd, σ : Ω → +Rd×d, be given such that +• Ω and supp(σ) ⊂ Ω are Lipschitz domains, +• and Q = (0, T) × Ω is a star-like domain with respect to a ball B. +Further, let σ, j, ǫ and µ be given functions, which fulfill: +• The conductivity σ ∈ L∞(supp(σ); Rd×d), supp(σ) ⊂ Ω, is uniformly positive definite, +i.e. +σmin := ess inf +x∈supp(σ) +inf +0̸=ξ∈Rd +ξ⊤σ(x)ξ +ξ⊤ξ +> 0. +• The applied current density ja : Q → Rd satisfies ja ∈ L1(0, T; L2(Ω; Rd)). +• The permittivity ǫ: Ω → Rd×d is symmetric, bounded, i.e., ǫ ∈ L∞(Ω; Rd×d), and uni- +formly positive definite, i.e., +ǫmin := ess inf +x∈Ω +inf +0̸=ξ∈Rd +ξ⊤ǫ(x)ξ +ξ⊤ξ +> 0 +and ǫmax := ∥ǫ∥L∞(Ω). +5 + +• For d = 2, the permeability µ: Ω → R satisfies µ ∈ L∞(Ω; R), µmax := ∥µ∥L∞(Ω), and +µmin := ess inf +x∈Ω µ(x) > 0. +For d = 3, the permeability µ: Ω → R3×3 is symmetric, bounded, i.e., µ ∈ L∞(Ω; R3×3), +µmax := ∥µ∥L∞(Ω), and uniformly positive definite, i.e., +µmin := ess inf +x∈Ω +inf +0̸=ξ∈R3 +ξ⊤µ(x)ξ +ξ⊤ξ +> 0. +For the initial data φ and ψ we assume that: +• The initial data φ : Ω → Rd satisfies φ ∈ H0(curlx; Ω). +• The initial data ψ ∈ H(divx; Ω) satisfies divx(εψ) = −ρ(0) in Ω. +After all the assumptions are stated and the vectorial wave equation is introduced, let us +now take a look at the structure of the rest of the paper. In the remainder of this section, we +will introduce the functional spaces that are needed to formulate the space-time variational +formulation of the vectorial wave equation. +We will discuss that they are indeed Hilbert +spaces and go over possible basis representations. At the end of this section, we will derive the +variational formulation. The second section of this paper is dedicated to the proof of unique +solvability for the variational formulation and norm estimates of the solution. n this proof we +will use a Galerkin method and use the previously developed basis representation. In the third +section of this paper, we will discuss the space-time finite element spaces and discretization. +We will derive a CFL condition and take a look at examples that show this CFL condition. +In the end, we will sum up the conclusions and give an outlook. +1.1 +The space-time Sobolev spaces +To derive a variational formulation, we need to consider the appropriate functional space for +A. In this section, we will derive them and show the properties of them that we need for the +proofs later in this paper. +To derive the functional spaces let us assume for a moment that d = 3 then Q ⊂ R4. From +the derivation of the magnetic vector potential A we know that A is a 1-form in R4. For +1-forms the corresponding functional space in the L2-Hilbert complex in R4 is H(Curl, Q; R4), +see e.g. [36]. The space H(Curl, Q; R4) is defined by +H(Curl, Q; R4) := { u ∈ L2(Q; R4) | Curl u ∈ L2(Ω; K) }, +[Curl u]ij := +4 +� +k,l=1 +εijkl∂kul, +see e.g. [36]. Note, that the operator Curl is an operator in both space and time. Additionally, +we use the Weyl gauge, the temporal gauge which implies A0 = 0, to make the potential A +unique in the derivation of the vectorial wave equation. Hence, we are interested in a function +A = (0, A1, A2, A3)T that is in H(Curl, Q; R4). Rewriting the operator Curl for A0 = 0 and +A = (A0, A)T ∈ H(Curl, Q; R4) leads to the conditions +A ∈ L2(Q, R3), +curlx A ∈ L2(Q, R3), +∂tA ∈ L2(Q, R3). +6 + +Before we define the appropriate function space, we take a quick look at the coefficients +in the vectorial wave equation (6). When we will formulate a variational formulation for the +equation (6), we will use the weighted L2(Ω; Rd)- and H0(curl; Ω)-inner products +(u, v)L2ǫ(Ω) := (ǫu, v)L2(Ω) , +(u, v)L2µ(Ω) := +� +µ−1u, v +� +L2(Ω) , +for v, w ∈ L2(Ω; Rd) and +(u, v)H0,ǫ,µ(curl;Ω) := (u, v)L2ǫ(Ω) + (curlx v, curlx w)L2µ(Ω), +for v, w ∈ H0(curl; Ω). Note, that due to the Assumptions 1 the norms ∥·∥L2ǫ(Ω) and ∥·∥H0,ǫ,µ(curl;Ω), +induced by (·, ·)L2ǫ(Ω) and (·, ·)H0,ǫ,µ(curl;Ω), are equivalent to the standard norms ∥·∥L2(Ω) and +∥·∥H0(curl;Ω), respectively. +Now let us define the function spaces which we will be using from this point on. From (6) +we learn that we need to have a trace in time that is well defined in order to incorporate zero +initial or end condition. Hence, we define the following spaces under the Assumption 1 +Hcurl;1 +0;0, +(Q) := { u ∈ L2(Q, Rd) | ∂tu ∈ L2 +ǫ(Q), curlx u ∈ L2 +µ(Q), u(0, x) = 0 for x ∈ Ω, +γtu = 0 on Σ = (0, T) × ∂Ω } +(7) += L2(0, T; H0(curl; Ω)) ∩ H1 +0,(0, T; L2(Q, Rd)), +Hcurl;1 +0;,0 +(Q) := { u ∈ L2(Q, Rd) | ∂tu ∈ L2 +ǫ(Q), curlx u ∈ L2 +µ(Q), u(T, x) = 0 for x ∈ Ω, +γtu = 0 on Σ } +(8) += L2(0, T; H0(curl; Ω)) ∩ H1 +,0(0, T; L2(Q, Rd)). +The subscript ’0,’ and ’, 0’ stands for zero initial and zero end conditions. In Lemma 1.2 we +will see that they are well defined. +It is quite natural to assume that Hcurl;1 +0; +(Q) is a Hilbert space. We will quickly discuss +whether this is indeed true. Afterward, we state in which concept C∞(Q)-functions are dense +in our space. We end up looking at the half norm in this setting which is a good tool for +numerical analysis. +So let us start with showing that Hcurl;1 +0;0, +(Q) and Hcurl;1 +0;,0 +(Q) are Hilbert spaces. +Lemma 1.1. Let d = 2, 3. The space H1(0, T; L2(Ω; Rd)) is isometric to the Hilbert tensor +product H1(0, T)ˆ⊗L2(Ω; Rd) and the space L2(0, T; H(curl; Ω)) is isometric to the Hilbert ten- +sor product L2(0, T)ˆ⊗H(curl; Ω). +Addtionally the space C∞(0, T)ˆ⊗[C∞(Ω))]d is dense in +H1(0, T)ˆ⊗L2(Ω; Rd) and L2(0, T)ˆ⊗H(curl; Ω) +and C∞([0, T])ˆ⊗[C∞ +0 (Ω))]d is dense in L2(0, T)ˆ⊗H0(curl; Ω). +See Aubin [11], Thm. 12.7.1 and Thm. 12.6.1. . The second result follows with [28, Ch +1.6.]. Hence we get that C∞(0, T; [C∞(Ω)]d) is included in both spaces H1(0, T; L2(Ω; Rd)) +and L2(0, T; H(curl; Ω) and therefore is dense in the intersections Hcurl;1 +0;0, +(Q) and Hcurl;1 +0;,0 +(Q). +7 + +Lemma 1.2. The spaces Hcurl;1 +0;0, +(Q) and Hcurl;1 +0;,0 +(Q) are Hilbert spaces equipped with the inner +product +(u, v)Hcurl;1(Q) := (u, v)L2(Q) + (∂tu, ∂tv)L2ǫ(Q) + (curlx u, curlx v)L2µ(Q) +(9) +for all u, v ∈ Hcurl;1 +0; +(Q). +Additionally the initial and end conditions are well defined in +Hcurl;1 +0; +(Q) +Proof. Since the embedding H1(0, T; L2(Ω; Rd)) ⊂ C([0, T], L2(Ω; Rd)) is continuous, see [22, +Prop 23.23], we get that H1(0, T; L2(Ω; Rd)) ⊂ L2(Q; Rd) is continuously embedded, see +[22, Prop 23.2], and since H0(curl; Ω) ⊂ L2(Q; Rd) that L2(0, T; H0(curl; Ω)) ⊂ L2(Q; Rd) +is continuously embedded, see [22, Prop 23.2]. +We know that L2(Q; Rd) is continuously +embedded in the Hausdorff space M0 of measurable functions that are finite a.e., see [23, +Thm. I.1.4]. Therefore the pair (H1(0, T; L2(Ω; Rd)), L2(0, T; H0(curl; Ω))) is a compatible +couple and H1(0, T; L2(Ω; Rd)) ∩ L2(0, T; H0(curl; Ω)) is a Banach space with the norm +∥u∥H1(0,T;L2(Ω;Rd))∩L2(0,T;H0(curl;Ω)) = max{∥u∥H1(0,T;L2(Ω;Rd)), ∥u∥L2(0,T;H0(curl;Ω))}, +see [23, Thm. III.1.3]. Now, (9) defines a norm that is induced by the inner product and it is +equivalent to the above norm. Hence Hcurl;1 +0; +(Q) is a Hilbert space. +Because the embedding H1(0, T; L2(Ω; Rd)) ⊂ C([0, T], L2(Ω; Rd)) is continuous we get well +defined and continuous traces to the boundaries {t = 0} × Ω and {t = T} × Ω for Hcurl;1 +0; +(Q). +Since the spaces Hcurl;1 +0;0, +(Q) and Hcurl;1 +0;,0 +(Q) are the kernels of these traces, we derive that they +are closed subsets of the Hilbert space Hcurl;1 +0; +(Q). +Lemma 1.3. For the spaces Hcurl;1 +0;0, +(Q) and Hcurl;1 +0;,0 +(Q) there exists a cf > 0 such that +∥u∥L2(Q) ≤ cf∥∂tu∥L2(Q) +for all u ∈ Hcurl;1 +0;0, +(Q) or u ∈ Hcurl;1 +0;,0 +(Q), d = 2, 3. Therefore the half norm +|u|2 +Hcurl;1(Q) := ∥∂tu∥2 +L2ǫ(Q) + ∥ curlx u∥2 +L2µ(Q), +is an equivalent norm in Hcurl;1 +0;0, +(Q) and Hcurl;1 +0;,0 +(Q). +This can be proving simply by using the Poincaré-inequality in H1(0, T) and the structure +of the norm ∥.∥H1(0,T;X). See [20, Sec. 4.1] for more details. +1.1.1 +Basis representations +To derive a basis representation in Hcurl;1 +0; +(Q) we first have to state how we decompose +H0(curl; Ω). +The basis representation of Hcurl;1 +0; +(Q) will be used in the proof of the main +theorem on uniqueness and solvability. To understand the proof better, we will quickly derive +the decomposition. +Let us define +X0,ǫ := { u ∈ H0(curl; Ω) | (εu, ∇p)L2(Ω) = 0, ∀p ∈ H1 +0(Ω) }. +8 + +Then we know the Helmholz-Weyl decomposition +H0(curl; Ω) = ∇H1 +0(Ω) ⊕ X0,ǫ +(10) +where the orthogonality holds true with respect to (·, ·)L2ǫ(Ω), (·, ·)H0,ǫ,µ(curl;Ω) and hence, +(curlx ·, curlx ·)L2µ(Ω), see e.g. [6, Lem. 4.5] for d = 3 or for d = 2, and d = 3 as well, we +can use the functional analysis toolbox from [10]. +Lemma 1.4. The space H0(curl; Ω) has a fundamental system {ϕi}i∈Z which is orthonormal +in the L2 +ǫ(Ω)-product. Additionally {ϕi}i∈Z is constructed in such a way that for every i ∈ N0 +exists a λi > 0 such that +(curlx ϕi, curlx v)L2µ(Ω) = λi(ϕi, v)L2ǫ(Ω) +for all v ∈ X0,ǫ(Ω) and for i ∈ Z\N0 there exists a λi > 0 and φi such that ϕi = ∇xφi and +(∇xφi, ∇xv)L2ǫ(Ω) = λi(φi, v)L2ǫ(Ω). +for all v ∈ H1 +0(Ω). +Proof. For ∇H1 +0(Ω) we get an orthonormal basis from the Laplace eigenvalue problem: +Find (λi, φi) ∈ (R, H1 +0(Ω)), i ∈ Z\N0, such that for all v ∈ H1 +0(Ω) +(∇xφi, ∇xv)L2ǫ(Ω) = λi(φi, v)L2(Ω) +and +∥∇xφi∥L2ǫ(Ω) = 1. +The solution to the eigenvalue problem is a non-decreasing sequence of related eigenvalues +λi > 0, satisfying λi → ∞ as i → −∞, see [35, Section 4 in Chapter 4]. +Next, we investigate the eigenvalue problem: +Find (λi, ϕi) ∈ (R, X0,ǫ(Ω)), i ∈ N0, such that for all v ∈ X0,ǫ(Ω) +(ϕi, v)H0,ǫ,µ(curl;Ω) = (1 + λi)(ϕi, v)L2ǫ(Ω) +and +���ϕi +��� +L2ǫ(Ω) = 1. +(11) +The set of eigenfunctions {ϕi ∈ X0,ǫ(Ω) : i ∈ N0} form an orthonormal basis of H(div ǫ0; Ω) +with respect to (·, ·)L2ǫ (Ω). +Additionally the nondecreasing sequence of related eigenvalues +(1+ λi), satisfying λi → ∞ as i → ∞, see [15, Theorem 8.2.4]. Note that λi > 0, i ∈ N0. This +can be shown by estimating +0 < cP (ϕi, ϕi)L2ǫ(Ω) ≤ (curlx ϕi, curlx ϕi)L2µ(Ω) = (ϕi, ϕi)H0,ǫ,µ(curl;Ω) − (ϕi, ϕi)L2ǫ(Ω) = λi +using the Poincaré-Steklov inequality, see e.g. [5, Lem. 44.4], and then the variational formu- +lation (11) for v = ϕi to get the desired result. +Moreover, the set {(1 + λi)−1/2ϕi ∈ X0,ǫ(Ω) : i ∈ N0} is an orthonormal basis of X0,ǫ(Ω) +with respect to (·, ·)H0,ǫ,µ(curl;Ω) by construction, see (11), and since X0,ǫ(Ω) ⊂ H(div ǫ0; Ω). +Additionally, we see that the set {ej ∈ X0,ǫ(Ω) : j ∈ N0} is also orthogonal with respect to +(curlx ·, curlx ·)L2µ(Ω) since it is an equivalent norm in X0,ǫ(Ω) because of the Poincaré-Steklov +inequality, see [5, Lem. 44.4]. +Then, by using the decomposition (10) we arrive at the desired orthonormal basis of H0(curl; Ω) +with the set {∇φi}i∈Z\N0∪{(1+λi)−1/2ϕi}i∈N0, which is orthogonal with respect to (·, ·)H0,ǫ,µ(curl;Ω). +9 + +Now that we know the fundamental system of H0(curl; Ω) we can write w ∈ H0(curl; Ω) +as +w(x) = +∞ +� +i=−∞ +wiϕi(x), +x ∈ Ω, +with the coefficients wi = (w, ϕi)L2ǫ(Ω), i ∈ Z. This basis representation converges in H0(curl; Ω). +Then the seminorm |·|H0,µ(curl;Ω) and the norm ∥·∥H0,ǫ,µ(curl;Ω) admit the representations +|w|2 +H0,µ(curl;Ω) = +∞ +� +i=0 +λi +���� +>0 +|wi|2 , +∥w∥2 +H0,ǫ,µ(curl;Ω) = +∞ +� +i=0 +(1 + λi) |wi|2 + +∞ +� +i=1 +|w−i|2 . +(12) +Let v ∈ Hcurl;1 +0; +(Q). Then we learn from [22, Prop. 23.23] that v coincides on [0, T] with a +continuous mapping v : [0, T] → H0(curl; Ω) up to a subset of measure zero. Hence we write +v(t) = +∞ +� +i=−∞ +ci(t)ϕi, +(13) +for some coefficient functions ci : [0, T] → R. +1.2 +The variational formulation +To derive a suitable variational formulation for (6) we multiply the partial differential equation +with a test function. By using partial integration both in time and space we end up with the +following variational formulation: +Find A ∈ Hcurl;1 +0; +(Q) with A(0, .) = φ, such that +− (ε∂tA, ∂tv)L2(Q) + (σ∂tA, v)L2(Q) + +� +µ−1 curlx A, curlx v +� +L2(Q) +(14) += +� +ja, v +� +L2(Q) − +� +εψ, v(0, ·) +� +L2(Ω) +for all v ∈ Hcurl;1 +0;.,0 (Q). +Note, that the initial condition ∂tA(0) = ψ is incorporated into the variational formulation in +a weak sense while the other conditions A(0) = φ and γtA = 0 are in the ansatz spaces and +therefore are satisfied in a strong sense. In (14) we also see the main problem of the equation, +the different signs in front of the first term and the spatial differential operator. Hence, this +is not equivalent to any norm. However, if σ is large it acts as a stabilization to the equation. +This we will see also in the numerical analysis. +In this paper, we will take a look at the bilinear form +a(A, v) := −(ε∂tA, ∂tv)L2(Q) + (σ∂tA, v)L2(Q) + (µ−1 curlx A, curlx v)L2(Q), +(15) +for A ∈ Hcurl;1 +0; +(Q) and φ ∈ Hcurl;1 +0;.,0 (Q). Additionally, we write right-hand side as the linear +form +F(v) := +� +ja, v +� +L2(Q) − +� +εψ, v(0, ·) +� +L2(Ω) . +(16) +10 + +2 +Existence and Uniqueness +Let us now state the main existence and uniqueness result for the variational formulation (14). +Theorem 2. Let the assumptions 1 hold true. Then there exists a unique solution of the +variational formulation: +Find A ∈ Hcurl;1 +0; +(Q) with A(0, .) = φ, such that +− (ε∂tA, ∂tv)L2(Q) + (σ∂tA, v)L2(Q) + +� +µ−1 curlx A, curlx v +� +L2(Q) = +� +ja, v +� +L2(Q) − +� +εψ, v(0, ·) +� +L2(Ω) +for all v ∈ Hcurl;1 +0;.,0 (Q). +If additionally ja ∈ L2(Q; Rd) then there exist positive constants cφ, cc +φ, cψ, and cf such that +|A|2 +Hcurl;1(Q) ≤ cφ∥φ∥2 +L2ǫ(Ω) + cc +φT∥ curlx φ∥2 +L2µ(Ω) + cψT∥ψ∥2 +L2ǫ(Ω) + cf max{T, T 2}∥ja∥2 +L2ǫ(Q). +To prove this theorem we will use a Galerkin method and split the proof in three different +steps. +First, we prove the existence in Proposition 1. +Then, we show the uniqueness in +Proposition 2. In the end, we derive the norm estimates in Proposition 3 and discuss the +dependencies of the coefficients cφ, cc +φ, cψ, and cf. +2.1 +Existence +We will start with proving the existence of the variational formulation (14). For that purpose, +we have to state two small results that we will be using in the existence proof in Prop. 1. +Lemma 2.1. Let the Assumption 1 hold true and u ∈ H2(0, T; L2(Ω; Rd))∩L2(0, T; H0(curl; Ω)), +d = 2, 3, with u|t=0 = φ and ∂tu|t=0 = ψ be the solution of +2 (ε∂ttu, ∂tu)L2(Q) + 2 +� +µ−1 curlx u, curlx ∂tu +� +L2(Q) ≤ 2 +� +j, ∂tu +� +L2(Q) , +(17) +for j ∈ L1(0, T; L2(Ω; Rd)). Then +z1/2(t) := + + +� +Ω +� +|u|2 + |∂tu|2 + |curlx u|2� +(t, x) dx + + +1/2 +≤ c2(T)z1/2(0) + c3(T)∥j∥2,1,Q, +holds true for all t ∈ [0, T]. The constants c2 and c3 are depend on εmin, (µmax)−1, εmax and +(µmin)−1. +Proof. Following the ideas of [35, Ch. 4.2] we can transfer the results from the scalar to the +vectorial wave equation. Detailed proof can be found in [21]. Here, we will only state the +main idea: First, we rewrite the inequality 17 to get for +z(t) := +� +Ω +(u2 + (∂tu)2 + (curlx u)2)(t, x) dx +the inequality +cǫ,µz(t) ≤ ˆcǫ,µz(0) + 2 +t +� +0 +∥j∥L2(Ω)z1/2(s) ds + 2t +t +� +0 +z(s) ds, +11 + +for all t ∈ [0, T]. Then we write ˆz(t) = max +0≤ξ≤t z(ξ) and solve the inequality on the interval +(0, +√cǫ,µ +4 +). In the end, we use iteration to get the desired result. +Corollary 1. Let the assumptions 1 hold true. Let A ∈ H2(0, T; H(curl; Ω)) with A|t=0 = φ +and ∂tA|t=0 = ψ be the solution of +(ε∂ttA, ∂tA)L2(Q) + (σ∂tA, ∂tA)L2(Q) + (µ−1 curlx A, curlx ∂tA)L2(Q) = (ja, ∂tA)L2(Q). +for ja ∈ L1(0, T; L2(Ω; Rd)). Then we derive that +z1/2(t) := + + +� +Ω +� +|A|2 + |∂tA|2 + |curlx A|2� +(t, x) dx + + +1/2 +≤ c2(t)z1/2(0) + c3(t)∥ja∥2,1,Q, +for all t ∈ [0, T]. The constants c2 and c3 are depend on εmin, (µmax)−1, εmax and (µmin)−1. +Proof. Using the positive semi-definiteness of σ we compute +(ε∂ttA, ∂tA)L2(Q) + (µ−1 curlx A, curlx ∂tA)L2(Q) ≤ (ja, ∂tA)L2(Q). +Therefore we can use Lem. 2.1 to prove the desired estimate. +Now, we are ready to prove the existence of the solution of the variational formulation +(14). +Proposition 1 (Existence). Let Assumption 1 hold true. Then there exists a solution of the +variational formulation (14). +Proof. Let us denote Ωσ := Ω ∩ supp(σ) which is a Lipschitz domain by Assumption 1. The +main idea of this proof is to split the differential equation into two equations over different +domains, namely one domain is Ωσ and the other Ω\Ωσ. Then we add them together to show +existence. The produced solution is the solution to an interface problem with zero tangential +traces on the interface. Hence, let us take a look at the resulting equations on the domain Ωσ. +1. On the domain Ωσ we now use a Galerkin method. Let us define the bilinear +aΩσ(A(t), φ) := (ε∂ttA(t), φ)L2(Ωσ) + (σ∂tA(t), φ)L2(Ωσ) + (µ−1 curlx A(t), curlx φ)L2(Ωσ), +for φ ∈ H(curl; Ωσ), t ∈ (0, T). We consider {ϕi}i∈Z, the fundamental system of H0(curl; Ωσ), +see Lem. 1.4. Let N ∈ N. Using the basis representation (13) we then search for a +AN(t) = +N +� +k=−N +cN +k (t)ϕk(x) +which solves +aΩσ(AN(t), ϕl) = (ja, ϕl)L2(Ωσ), +d +dtcN +k (t) = (ψ, ϕk)L2(Ωσ), +(18) +cN +k (0) = αN +k , +12 + +for all l, k ∈ ZN := {z ∈ Z : |z| ≤ N}, where αN +k are the coefficients of +φN(x) = +N +� +k=−N +αN +k ϕk(x), +and φN → φ in H0(curl; Ωσ) for N → ∞. We have AN(0, x) = φN(x) and define fk := +(ja, ϕk)L2(Ωσ). +Since σ is uniformly positive definite and bounded over Ωσ, the induced weighted scalar +product (σ., .)L2(Ωσ) is equivalent to (ε., .)L2(Ωσ). Hence, there exists a βk ∈ R+ such that +(σϕk, ϕl)L2(Ωσ) = βkδkl +for k, l ∈ ZN. These βk are bounded from below by σmin. +Next, we combine everything to arrive at +c′′N +k (t) + βkc′N +k (t) + λkcN +k (t) = fk(t), +c′N +k (0) = (ψ, ϕk)L2(Ωσ), +cN +k (0) = αN +k +for t ∈ (0, T) and k = 0, . . . , N, i.e. ϕk is an eigenfunction of curlx curlx, and +c′′N +k (t) + βkc′N +k (t) = fk(t), +c′N +k (0) = (ψ, ϕk)L2(Ωσ), +cN +k (0) = αk +if ϕk is part of the kernel of curlx curlx, k = −N, . . . , −1. The solutions can be computed +using standard techniques for ordinary differential equations such as [32, L. 20, L. 21]. The +solutions are well defined for fk ∈ L1(0, T). +Therefore we get for ja ∈ L1(0, T; L2(Ωσ)) +a solution AN ∈ C2(0, T; H(curl; Ωσ)). +If we multiply (18) with c′N +l (t) and sum up over +l = −N, . . . , N, we compute +aΩσ(A(t), ∂tA(t)) = (ja(t), ∂tAN(t))L2(Ωσ). +Now we can apply the result from Cor. 1 and get + + +� +Ωσ +��AN(t) +��2 + +��∂tAN(t) +��2 + +��curlx AN(t) +��2 dx + + +1/2 +≤ c2(t)(zN)1/2(0) + c3(t)∥ja∥2,1,Q +for every t ∈ (0, T). +By construction c2 and c3 are monotonically increasing and there- +fore bounded by c2(T) respectively c3(T). Additionally we get with Bessel’s inequality in +H(curl; Ωσ) and L2(Ω) +zN(0) = +� +Ωσ +��φN��2 + +��ψN��2 + +��curlx φN��2 dx ≤ c∥φ∥2 +H(curl;Ωσ) + c∥ψ∥2 +L2(Ωσ) +Therefore +∥AN∥Hcurl;1(Q) ≤ c∥φ∥2 +H(curl;Ωσ) + ˜c∥ψ∥2 +L2(Ωσ) + ˆc∥j∥2 +2,1,Q < C, +13 + +where C is independent of N. Now we split AN(t, x) = AN +0 (t, x) + φN(x). Since ∥φN∥H(curl;Ω) +is bounded by ∥φ∥H(curl;Ω) because of Bessel’s inequality, the sequence (AN +0 )N∈N is bounded +as well. The space Hcurl;1 +0; +(Q) = H1(0, T; L2(Ω)) � L2(0, T; H0(curl; Ω)) is a Hilbert space, see +Lem. 1.2. Hence there exists a weakly convergent subsequence of (AN +0 )N∈N. We write this +subsequence as {AN +0 }N for convenience. Then there exists a A0 ∈ Hcurl;1 +0;0, +(Q) with +AN +0 ⇀ A0 +in L2(Q), +∂tAN +0 ⇀ ∂tA0 +in L2(Q), +curlx AN +0 ⇀ curlx A0 +in L2(Q). +Then we compute +aΩσ(AN +0 (t), ϕl) = (ja(t), ϕl)L2(Ω) − (µ−1 curlx φN, curlx ϕl)L2(Ω) +for all l = −N, .., N, t ∈ (0, T). Let M ∈ N. Choose N > M and dl ∈ H1(0, T) where +dl(T) = 0, l = −M, . . . , M. Multiply the equation with dl and sum up over l = −M, . . . , M. +Moreover, we get for η(t, x) := +M +� +l=−M +dl(t)ϕl(x) and t ∈ (0, T) the equation +aΩσ(AN +0 (t), η) = (ja(t), η)L2(Ω) − (µ−1 curlx φN, curlx η)L2(Ω). +By integration over (0, T) and using integration by parts for the first term we get to the +bilinear form (15) and +a(AN +0 , η) = (ja, η)L2(Q) − (ε∂tAN|t=0, η|t=0)L2(Ω) − (µ−1 curlx φN, curlx η)L2(Q). +Next, we take the limit N → ∞. Since η, ∂tη and curlx η are in L2(Q; Rd) we get with the +weak convergence that +a(A0, η) = (ja, η)L2(Q) − (εψ(x), η(0, .))L2(Ω) − (µ−1 curlx φ, curlx η)L2(Q). +This equation holds for all η with the representation +M +� +l=−M +dl(t)ϕl(x). Let MM be the space +of such functions. For every M ∈ N we can repeat this argumentation. The space +∞� +M=1 +MM is +dense in L2(0, T; H0(curl; Ω)) because we can approximate every element with such a sum, see +[22, Prop. 23.2d]. Hence the equation above holds true for every η ∈ Hcurl;1 +0;,0 +(Q) and therefore +A is the weak solution of our differential equation in Hcurl;1 +0;0, +(Q). +2. Let us now consider the domain Ω\Ωσ where σ is zero. We use the same technique as +above to get existence, see [21] for detailed proof. Here, we will only state the main ideas. We +consider the equations +c′′N +k (t) + λkcN +k (t) = fk(t), +(19) +c′N +k (0) = (ψ, ϕk)L2(Ωσ), +cN +k (0) = αN +k +14 + +for t ∈ (0, T) and k = 0, . . . , N, i.e. ϕk is an eigenfunction of curlx curlx, and +c′′N +k (t) = fk(t), +(20) +c′N +k (0) = (ψ, ϕk)L2(Ωσ), +cN +k (0) = αk +if ϕk is part of the kernel of curlx curlx, k = −N, . . . , −1. By going through the same steps as +above, we end up with a bounded solution +∥AN∥Hcurl;1(Q) ≤ c∥φ∥2 +H(curl;Ωσ) + ˜c∥ψ∥2 +L2(Ωσ) + ˆc∥ja∥2 +2,1,Q < C, +where C is independent of N. Using the same definitions and arguments as in step 1 we get a +weak solution of our differential equation in L2(0, T; H0(curl; Ω)) ∩ H1 +0,.(0, T; L2(Ω; Rd)). +2.2 +Uniqueness +Next, we take a look at the uniqueness of the variational formulation (14) with the bilinear +form (15) and right-hand side (16). +Proposition 2 (Uniqueness). Let Assumption 1 hold true. Then there exists a unique solution +of the variational formulation (14) with the bilinear form (15) and linear form (16): +Find A ∈ Hcurl;1 +0; +(Q) with A(0, .) = φ, such that +a(A, v) = F(v) +for all v ∈ Hcurl;1 +0;.,0 (Q). +Proof. In Proposition 1 we have already shown existence. What is left to be proven is the +uniqueness. Assume that there are two solutions A′ and A′′, then w := A′ − A′′ satisfies the +variational formulation +−(ε∂tw, ∂tv)L2(Q) + (σ∂tw, v)L2(Q) + (µ−1 curlx w, curlx v)L2(Q) = 0 +for all v ∈ Hcurl;1 +0;,0 +(Q). Additionally w(0, x) = 0 holds true for x ∈ Ω. Choose b ∈ [0, T] +arbitrary and consider +η(t, x) := + + + + + +t� +b +w(τ, x) dτ, +for 0 ≤ t ≤ b +0, +for b ≤ t ≤ T +. +Then η ∈ Hcurl;1 +0;,0 +(Q) with η(t, x) = 0 for t ≥ b and we get for v = η: +−(ε∂tw, ∂tη)L2(Q(0,b)) + (σ∂tw, η)L2(Q(0,b)) + (µ−1 curlx w, curlx η)L2(Q(0,b)) = 0, +where Q(0,b) is the intersection of Q with the half space t < b. Since ∂tη(t, x) = w(t, x) for +(t, x) ∈ Q(0,b), we compute +(ε∂ttη, ∂tη)L2(Q(0,b)) − (σ∂ttη, η)L2(Q(0,b)) − (µ−1 curlx ∂tη, curlx η)L2(Q(0,b)) = 0. +15 + +Through integration by parts we get +−(σ∂ttη, η)L2(Q(0,b)) = (σ∂tη, ∂tη)L2(Q(0,b)) +≥ 0, +because ∂tη(0, x) = w(0, x) = 0 for x ∈ Ω, η(b, x) = 0 by definition and σ is positive semi- +definite. Therefore +(ε∂ttη, ∂tη)L2(Q(0,b)) − (µ−1 curlx ∂tη, curlx η)L2(Q(0,b)) ≤ 0 +holds true. Hence +1 +2 +� +Q(0,b) +∂t(ε∂tη · ∂tη) dx dt − 1 +2 +� +Q(0,b) +∂t(µ−1 curlx η · curlx η) dx dt ≤ 0. +holds true because ε and µ−1 are symmetric. From the definition of η we compute η(b, x) = 0 +and therefore (curlx η)(b, x) = 0 for x ∈ Ω. Additionally it holds true that ∂tη(0, x) = v(0, x) = +0 for x ∈ Ω and therefore +� +Ω +(ε∂tη · ∂tη)(b, x) dx + +� +Ω +(µ−1 curlx η · curlx η)(0, x) dx ≤ 0. +From this we derive by ∂tη(b, x) = w(b, x) that +� +Ω +w2(b, x) dx = 0 +and +� +Ω +�� b +0 +curlx w +�2 +(x) dx = 0 +holds true for any b ∈ (0, T), since ∂tη(t, x) = w(t, x). With this, we can deduce that w(t, x) +vanishes almost everywhere. +2.3 +Norm estimate +Now that we know that the variational formulation (14) is uniquely solvable, we take a look +at the norm estimate. In this part of the section, we consider ja ∈ L2(Q; Rd). We will derive +the norm estimate of 2 and take a closer look at the dependencies of the coefficients cφ, cc +φ, +cψ, and cf. +Lemma 2.2. Let Assumption 1 hold, ja ∈ L2(Q; Rd) and ck, k ∈ N0, be the solution of the +ordinary differential equation +c′′ +k(t) + βkc′ +k(t) + λkck(t) = fk(t), +(21) +ck(0) = αN +k , +c′ +k(0) = (ψ, ϕk)L2(Ωσ) +for t ∈ (0, T), where λk > 0 are the nonzero eigenvalues of the curlx curlx-operator from +Lemma 1.4 and fk := (ja, ϕk)L2(Ω). Then there exist positive constants cκ +α, cα, cψ and cf such +that +� +k∈N0 +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ +� +k∈I +cψ(ψ, ϕk)2 +L2(Ωσ) + (cκ +αλk + cα)(αN +k )2 + cf∥fk∥2 +L2(0,T). +16 + +Proof. For these estimates we need to consider three cases, namely β2 +k −4λk > 0, β2 +k −4λk < 0 +and β2 +k − 4λk = 0. +1. Let β2 +k − 4λk > 0. We define λ1 := +−βk+√ +β2 +k−4λk +2 +, λ2 := +−βk−√ +β2 +k−4λk +2 +, γk = +� +β2 +k − 4λk. +Then the solution of (21) is given by +ck(t) =(ψ, ϕk)L2(Ωσ) +eλ1t − eλ2t +γk ++ αN +k +λ1eλ2t − λ2eλ1t +γk ++ 1 +γk +t +� +0 +(eλ1(t−s) − eλ2(t−s))fk(s) ds. +We know that λ1, λ2 < 0, because β2 +k − 4λk > 0 and so −βk − +� +β2 +k − 4λk < 0. Additionally +we get −(λ1 + λ2) = βk as well as λ1λ2 = λk. With these facts and using (a + b)2 ≤ 2a2 + 2b2 +we compute +T +� +0 +� +eλ1t − eλ2t�2 +dt ≤ 1 +λ1 +(e2λ1T − 1) + 1 +λ2 +(e2λ2T − 1) ≤ +1 +−λ1 ++ +1 +−λ2 += βk +λk +. +In the same way, we estimate +T +� +0 +� +λ1eλ1t − λ2eλ2t�2 +dt ≤ λ1(e2λ1t − 1) + λ2(e2λ2t − 1) ≤ βk. +With this, we can compute an estimate for the desired norms again using (a + b)2 ≤ 2a2 + 2b2 +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤2(ψ, ϕk)2 +L2(Ωσ) +4βk +β2 +k − 4λk ++ 2(αN +k )2λk +4βk +β2 +k − 4λk ++ 2 +2βk +β2 +k − 4λk +T +T +� +0 +f 2 +k(s) ds. +Applying the estimate +1 +(β2 +k−4λk) ≤ +1 +√ +2bk2√λk ≤ +1 +2bk +√λk we arrive at +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ (ψ, ϕk)2 +L2(Ωσ) +4 +√κ0 ++ (αN +k )2λk +4 +√κ0 ++ +2 +√κ0 +T∥fk∥2 +L2(0,T) +for κ0 ≤ κ1 ≤ . . . . +2. Let β2 +k − 4λk < 0 and define γk = +� +4λk − β2 +k. Then we write the solution of (21) as +ck(t) = αN +k e− βk +2 t +� +cos(γkt) + βk +2γk +sin(γkt) +� ++ 1 +γk +(ψ, ϕk)L2(Ωσ)e− βk +2 t sin(γkt) ++ 1 +γk +t +� +0 +e +βk +2 (s−t) sin(γk(t − s))f(s) ds +17 + +Since βk is positive since it is bounded from below by infx∈Ωσ σ(x). Hence we can estimate +T +� +0 +e−βkt dt ≤ 1 +βk +. +Next, we use the fact that sin2(γkt) ≤ 1 and cos2(γkt) ≤ 1. Then we derive by using (a+b)2 ≤ +2a2 + 2b2 and Cauchy Schwarz yet again the following estimate +T +� +0 +λk(ck)2(t) dt ≤ λk(ψ, ϕk)2 +L2(Ωσ) +4 +βkγ2 +k ++ 4(αN +k )2λk +� 1 +βk ++ βk +4γ2 +k +� ++ 2λk +βkγ2 +k +T +T +� +0 +f 2 +k(s) ds. +In the same way, we estimate +T +� +0 +(c′ +k)2(t) dt ≤ (ψ, ϕk)2 +L2(Ωσ) +4 +βkγ2 +k +�β2 +k +2 + 2γ2 +k +� ++ 4(αN +k )2 1 +βk +� +γ2 +k + +β4 +k +16γ2 +k +� ++ +2 +βkγ2 +k +� +2β2 +k +4 + 2γ2 +k +� +T +T +� +0 +f 2 +k(s) ds. +By inserting γ2 +k = 4λk − β2 +k we get +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ (ψ, ϕk)2 +L2(Ωσ) +� +36λk +βk(4λk − β2 +k) + +6βk +4λk − β2 +k +� ++ 4(αN +k )2 +�5λk +βk ++ +λkβk +4(4λk − β2 +k) + +β3 +k +16(4λk − β2 +k) +� ++ +� +2λk +βk(4λk − β2 +k) + +βk +4λk − β2 +k ++ 4 +βk +� +T +T +� +0 +f 2 +k(s) ds. +Note that the term +λk +4λk−β2 +k only shows up when β2 +k < 4λk. It is bounded since βk is bounded +by βmax, which is bounded by supx∈Ωσ σ(x), but λk is increasing monotonically. Therefore +the maximum will be reached for smaller k. +Next we use the estimate +1 +(β2 +k−4λk) ≤ +1 +2bk +√λk to derive +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ (ψ, ϕk)2 +L2(Ωσ) +� +36λk +βmin(4λk − β2 +k) + +3 +√κ0 +� ++ 4(αN +k )2 +� 5λk +βmin ++ +λkβmax +4(4λk − β2 +k) + β2 +max +32√κ0 +� ++ +� +2λk +βk(4λk − β2 +k) + +1 +8√κ0 ++ +4 +βmin +� +T∥fk∥2 +L2(0,T), +18 + +where βmax := maxk βk and βmin := mink βk > 0 3. Let us consider the last case β2 +k −4λk = 0. +Then the solution to (21) is given by +ck(t) = αN +k (1 + βk +2 t)e− βk +2 t + (ψ, ϕk)L2(Ωσ)te− βk +2 t + +t +� +0 +(t − s)e +βk +2 (s−t)f(s) ds. +With the estimate +� t +0 +t2e−βkt dt = 1 +β3 +k +(2 − e−βkt(β2 +kt2 + 2βkt + 2)) ≤ 2 +β3 +k +we then compute by using (a + b)2 ≤ 2a2 + 2b2 and Cauchy Schwarz that +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ (ψ, ϕk)2 +L2(Ωσ) +� +λk +8 +β3 +k ++ 12 +βk +� ++ 4(αN +k )2 +� +λk +3 +βk ++ +1 +2βk +� ++ +�2λk +β3 +k ++ 3 +βk +� +2T +T +� +0 +f 2 +k(s) ds. +At last, we use 4λk = β2 +k to derive +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ (ψ, ϕk)2 +L2(Ωσ) +14 +βmin ++ (αN +k )2λk +5 +4β3 +min ++ 14 +β3 +min +T∥fk∥2 +L2(0,T). +Adding all three cases will give the desired estimate. +Using the above lemma we can finally prove the last statement of Theorem 2. +Proposition 3. Let the assumptions 1 hold true, ja ∈ L2(Q; Rd), and A be the unique solu- +tion of (14). Then there exists positive constants cφ, cc +φ, cψ, and cf such that the following +inequality holds true +|A|2 +Hcurl;1(Q) ≤ cφ∥φ∥2 +L2ǫ(Ω) + cc +φT∥ curlx φ∥L2µ(Ω) + cψT∥ψ∥2 +L2ǫ(Ω) + cf max{T, T 2}∥ja∥2 +L2ǫ(Q). +(22) +The constants cφ, cc +φ, cψ, and cf are dependent on sup σ, σ−1 +min, +1 +√λ0 and max +k∈N0 +β2 +k−4λk<0 +(4 − +β2 +k +λk )−1 which is bounded since βk is bounded by sup σ and the non-zero eigenvalues λk of +curlx µ−1 curlx increase monotonically for k → ∞. The βk ∈ R+ are defined by σ such that +(σϕk, ϕl)L2(Ωσ) = βkδkl +for the fundamental system {ϕk} of Lem. 1.4. +Proof. Again we split the domain into Ωσ = Ω ∩ supp(σ) and Ω0 := Ω\Ωσ. Now we take a +look at Ωσ and ˆQ := (0, T) × Ωσ. To show the inequality we consider +AN(t, x) = +� +k∈Z +cN +k (t)ϕk(x). +19 + +If we insert the representation into the energy norm we get +(ε∂tAN,∂tAN)L2( ˆQ) + (µ−1 curlx AN, curlx AN)L2( ˆQ) += +� +k,j +T +� +0 +∂tcN +k ∂tcN +j dt +� +Ωσ +(εϕk, ϕj) dx + +T +� +0 +cN +k cN +j dt +� +Ωσ +(µ−1 curlx ϕk, curlx ϕj) dx += +� +k∈N0 +T +� +0 +(∂tcN +k )2 dt + +T +� +0 +λk(cN +k )2 dt + +� +k∈Z\N0 +T +� +0 +(∂tcN +k )2 dt. +(23) +For k ∈ I we have to distinguish between the three cases β2 +k − 4λk > 0, β2 +k − 4λk = 0, +β2 +k − 4λk < 0. We know that 0 < λk and λk → ∞ for k → ∞. Since σ is bounded, we know +that the βk = (σϕk, ϕk)L2(Ωσ) are also bounded and since σ is positive semi-definite βk > 0. +So we consider the two ordinary differential equations. The first is +c′′N +k (t) + βkc′N +k (t) = fk(t), +(24) +c′N +k (0) = (ψ, ϕk)L2(Ωσ), +cN +k (0) = αN +k , +and the second is +c′′N +k (t) + βkc′N +k (t) + λkcN +k (t) = fk(t), +(25) +c′N +k (0) = (ψ, ϕk)L2(Ωσ), +cN +k (0) = αN +k , +for t ∈ (0, T), where fk(t) = (ja, ϕk)L2(Ωσ) and λk are the non-zero eigenvalues of the +curlx curlx operator. +For the first equation (24), where k ∈ Z\N0, we compute the solution +ck(t) = αN +k + 1 +βk +(ψ, ϕk)L2(Ωσ)(1 − e−βkt) + 1 +βk +t +� +0 +(1 − eβk(s−t))fk(s) ds. +Now we need to estimate the L2-norm of c′N +k . Using (a + b)2 ≤ 2a2 + 2b2, we arrive with the +Cauchy-Schwarz inequality at +T +� +0 +(c′N +k )2(t) dt ≤ 2(ψ, ϕk)2 +L2(Ωσ) +T +� +0 +(e−2βkt) dt + 2 +T +� +0 +t +� +0 +e2βk(s−t) ds +t +� +0 +f 2 +k(s) ds dt +≤ (ψ, ϕk)2 +L2(Ωσ) +1 − e−2βkT +βk ++ 21 − e−2βkT +2βk +T∥fk∥2 +L2(0,T) +≤ (ψ, ϕk)2 +L2(Ωσ) +1 +βmin ++ +1 +βmin +T∥fk∥2 +L2(0,T) +(26) +20 + +For the second equation (25) and k ∈ N0 we consider the result of Lem. 2.2 +� +k∈I +T +� +0 +λk(ck)2(t) dt + +T +� +0 +(c′ +k)2(t) dt ≤ +� +k∈I +cψ(ψ, ϕk)2 +L2(Ωσ) + (cκ +αλk + cα)(αN +k )2 + cf∥fk∥2 +L2(0,T). +(27) +Since +1 +βmin < cφ and +1 +βmin < c1 +f we can combine (27) and (26) in (23) to arrive at +∞ +� +k=1 +T +� +0 +(∂tcN +k )2 dt + +T +� +0 +λk(cN +k )2 dt ≤ +∞ +� +k=1 +� +c1 +φ + c2 +φλk +� +α2 +k + cψ(ψ, ϕk)2 +L2(Ωσ) +(28) ++ (c1 +f + c2 +fλk)T∥fk∥2 +L2(0,T). +As λk were eigenvalues of the curlx µ−1 curlx-opertor, we want to eliminate the κ from our +right hand side. +Considering φ = �∞ +k=1 αkϕk we compute +∞ +� +k=1 +λk(αk)2 = +∞ +� +k=1 +∞ +� +l=1 +αkαl +� +Ωσ +λk(εϕk, ϕl) dx += +∞ +� +k=1 +∞ +� +l=1 +αkαl +� +Ωσ +(µ−1 curlx ϕk, curlx ϕl) dx += +� +Ωσ +(µ−1 curlx φ, curlx φ) dx. +Using the same consideration for φ and ja, where fk = (ja, ϕk)L2(Ωσ), respectively in (28) we +arrive at +(ε∂tA, ∂tA)L2( ˆQ)+(µ−1 curlx A, curlx A)L2( ˆQ) +≤cψ∥ψ∥2 +L2ǫ(Ωσ) + c1 +φ∥φ∥2 +L2ǫ(Ωσ) + c2 +φ∥ curlx φ∥2 +L2µ(Ωσ) + c1 +fT∥ja∥2 +L2ǫ( ˆQ). +For the solution over Ω0 = Ω\Ωσ and Q0 := (0, T) × Ω0, where σ ≡ 0, we get the following +two ordinary equations. The first equation is given by (20) for k ∈ Z\N0 and the second is +(19) for k ∈ N0 with the solution +cN +k (t) = +1 +√λk +(ψ, ϕk)L2(Ω0) sin( +� +λkt) + αN +k cos( +� +λkt) + +1 +√λk +t +� +0 +fk(s) sin( +� +λk(t − s)) ds. +Then, we follow the same steps as above to compute the estimate +(ε∂tA, ∂tA)L2(Q0)+(µ−1 curlx A, curlx A)L2(Q0) +(29) +≤4T∥ψ∥2 +L2ǫ(Ω0) + 4T∥ curlx φ∥2 +L2µ(Ω0) + T 2∥ja∥2 +L2ǫ(Q0). +Note that we get a T in our estimates by estimating +T� +0 +sin2(√λkt) dt ≤ T and the same integral +with cos. Adding the inequality (29) brings us to the desired inequality (22) +|A|Hcurl;1(Q) ≤ ˜cψT∥ψ∥2 +L2ǫ(Ω) + c1 +φ∥φ∥2 +L2ǫ(Ω) + ˜c2 +φT∥ curlx φ∥2 +L2µ(Ω) + c1 +f max{T, T 2}∥ja∥2 +L2ǫ(Q). +21 + +Remark +Note that with the tools of Ladyzhenskaya [35, Thm. 3.2, p. 160] one can translate +the results of the scalar wave equation to the vectorial wave equation using the same technique +as the above theorem, see [21] for more details. Then we would derive the estimate +(ε∂tA, ∂tA)L2(Q) + (µ−1 curlxA, curlx A)L2(Q) +(30) +≤4T∥ψ∥2 +[L2(Ω)]d + 4T∥ curlx φ∥2 +L2µ(Ω) + T 2∥ja∥2 +[L2(Q)]d. +for the solution of the variational formulation: +Find A ∈ Hcurl;1 +0; +(Q) with A(0, .) = φ such that +−(ε∂tA, ∂tv)L2(Q) + (µ−1 curlx A, curlx v)L2(Q) = (ja, v)L2(Q) − (εψ, v(0, ·))L2(Ω) +for all v ∈ Hcurl;1 +0;,0 +(Q). +3 +A space-time finite element method in a tensor product +With Theorem 2 we have shown the unique solvability of the variational formulation (14). +Now we take a look at the discrete equivalent of the variational formulation for the vectorial +wave equation. First, we will define the finite element spaces that we will be using. Then +we derive the finite element discretization and analyze the resulting linear equation. We will +derive a CFL condition that we will also see in numerical examples. +3.1 +Space-time finite element spaces +Before we compute any examples, we will recap the main ideas of the space-time discretization +of the vectorial wave equation as was presented in [20]. We will use similar nomenclature to +ease the comparison of the results of both papers. +For the discretization in time we decompose the time interval (0, T) +0 = t0 < t1 < · · · < tNt−1 < tNt = T +into N t subintervals τℓ = (tℓ−1, tℓ) ⊂ R, ℓ = 1, . . . , N t. We define the local mesh size of +each element τl as ht,ℓ = tℓ − tℓ−1, ℓ = 1, . . . , N t, whereas the maximal mesh size as ht := +maxℓ=1,...,Nt ht,ℓ. +For the discretization in space we decompose the spatial domain Ω ⊂ Rd into N x elements +ωi ⊂ Rd for i = 1, . . . , N x, satisfying +Ω = +Nx +� +i=1 +ωi +We define the local mesh size in ωi as hx,i = +�� +ωi 1dx +�1/d +, i = 1, . . . , N x, the maximal mesh +size as hx := hx,max(T x +ν ) := maxi=1,...,Nx hx,i and the minimal mesh size as hx,min(T x +ν ) := +mini=1,...,Nx hx,i. The parameter ν ∈ N is the level of refinement and the spatial mesh the +set T x := T x +ν = {ωi}Nx +i=1 at level α. For simplicity, we only consider triangles for d = 2 and +tetrahedra for d = 3 as the spatial elements ωi ⊂ Rd, i = 1, . . . , N x. +Assumption 3. For the spatial mesh, we assume that we have a shape-regular, globally quasi- +uniform sequence (T x +ν )ν∈N of admissible spatial meshes T x +ν . +22 + +From the shape-regularity of the mesh sequence (T x +ν )ν∈N we get a constant cF > 0 such +that +∀ν ∈ N : ∀ω ∈ T x +ν : +sup +x,y∈ω +∥x − y∥2 ≤ cF rω, +(31) +where ∥·∥2 is the Euclidean norm in Rd, and rω > 0 is the radius of the largest ball that can be +inscribed in the element ω. From the global quasi-uniformity of the mesh sequence (T x +ν )ν∈N +we get a constant cG ≥ 1 such that +∀ν ∈ N: +hx,max(T x +ν ) +hx,min(T x +ν ) ≤ cG. +With the discretization of the time interval and spatial domain in mind, we now define +finite element spaces. For the test and ansatz space in time, we define the space of piecewise +quadratic, continuous functions +S2(T t +α) := +� +vht ∈ C[0, T] | ∀ℓ ∈ {1, . . . , N t} : vht|τℓ ∈ P2(τℓ) +� += span{ϕ2 +ℓ}N2 +t +ℓ=0 +with the usual nodal basis functions ϕ2 +ℓ, satisfying ϕ2 +ℓ(tκ) = δℓκ for ℓ, κ = 0, . . . , N 2 +t . The +space Pp(B) is the space of polynomials on a subset B ⊂ R of global degree at most p ∈ N0, +and δℓκ is the Kronecker delta. Then, we introduce the subspaces +S2 +0,(T t +α) := S2(T t +α) ∩ H1 +0,(0, T) = span{ϕ2 +ℓ}N2 +t +ℓ=1, +S2 +,0(T t +α) := S2(T t +α) ∩ H1 +,0(0, T) = span{ϕ2 +ℓ}N2 +t −1 +ℓ=0 +. +In space, we introduce the vector-valued finite element spaces of Nédélec and Raviart– +Thomas finite elements. For a spatial element ω ∈ T x +ν , we define the polynomial spaces +RT 0(ω) := +� +v ∈ [P 1(ω)]d | ∀x ∈ ω : v(x) = a + bx with a ∈ Rd, b ∈ R +� +and +N 0 +I (ω) := +� +v ∈ [P 1(ω)]2 | ∀(x1, x2) ∈ ω : v(x1, x2) = a + b · (−x2, x1)⊤ with a ∈ R2, b ∈ R +� +for d = 2, and +N 0 +I (ω) := +� +v ∈ [P 1(ω)]3 | ∀x ∈ ω : v(x) = a + b × x with a ∈ R3, b ∈ R3� +for d = 3, see [4, Section 14.1, Section 15.1]. With this notation, we define the space of the +lowest-order Raviart–Thomas finite element space and the lowest-order Nédélec finite element +space of the first kind +RT 0(T x +ν ) := +� +vhx ∈ H(div; Ω) | ∀ω ∈ T x +ν : vhx|ω ∈ RT 0(ω) +� += span{ψRT +k +}NRT +x +k=1 , +N 0 +I (T x +ν ) := +� +vhx ∈ H(curl; Ω) | ∀ω ∈ T x +ν : vhx|ω ∈ N 0 +I (ω) +� +, +see [4, 6] for more details on these spaces. Further, we consider the subspace +N 0 +I,0(T x +ν ) := N 0 +I (T x +ν ) ∩ H0(curl; Ω) = span{ψN +k }NN +x +k=1. +23 + +Last, the temporal and spatial meshes T t +α = {τℓ}Nt +ℓ=1 and T x +ν = {ωi}Nx +i=1 lead to a decom- +position +Q = [0, T] × Ω = +Nt +� +ℓ=1 +τℓ × +Nx +� +i=1 +ωi +of the space-time cylinder Q ⊂ Rd+1 with N t ·N x space-time elements. Therefore T t +α ×T x +ν is a +space-time tensor product mesh. To this space-time mesh, we relate space-time finite element +spaces of tensor-product type using ⊗ as the Hilbert tensor-product. These spaces are +S2 +0,(T t +α) ⊗ N 0 +I,0(T x +ν ) +and +S2 +,0(T t +α) ⊗ N 0 +I,0(T x +ν ). +(32) +Then, any function Ah ∈ S2 +0,(T t +α) ⊗ N 0 +I,0(T x +ν ) admits the representation +Ah(t, x) = +N2 +t +� +κ=1 +NN +x +� +k=1 +Aκ +kϕ2 +κ(t)ψN +k (x) +(33) +for (t, x) ∈ Q with coefficients Aκ +k ∈ R. Further, for a given function f ∈ L2(Q; Rd), we +introduce the L2(Q) projection ΠRT ,1 +h +: L2(Q; Rd) → S1(T t +α) ⊗ RT 0(T x +ν ) to find ΠRT ,1 +h +f ∈ +S1(T t +α) ⊗ RT 0(T x +ν ) such that +(ΠRT ,1 +h +f, wh)L2(Q) = (f, wh)L2(Q). +(34) +for all wh ∈ S1(T t +α) ⊗ RT 0(T x +ν ). +With these definitions in mind, we can now formulate a conforming finite element approach +for the variational formulation (14). We use as trial space S2(T t +α) ⊗ N 0 +I,0(T x +ν ) and S2 +,0(T t +α) ⊗ +N 0 +I,0(T x +ν ) as test space. Then we consider the discrete variational formulation: +Find Ah ∈ S2(T t +α) ⊗ N 0 +I,0(T x +ν ), with Ah(0, x) = φ, such that +− (ε∂tAh, ∂tvh)L2(Q) + (σ∂tAh, vh)L2(Q) + +� +µ−1curlxAh, curlxvh +� +L2(Q) +(35) += +� +ΠRT ,1 +h +ja, vh +� +L2(Q) − +� +εψ, vh(0, .) +� +L2(Ω) +for all vh ∈ S2 +,0(T t +α) ⊗ N 0 +I,0(T x +ν ). +3.2 +The finite element discretization +Next, we follow the same steps as [20] to derive the finite element discretization. Let, for +a moment, the initial condition φ be zero. Then we insert (33) into the discrete variational +formulation (35) the integrals split into temporal and spatial matrices. Hence, the discrete +variational formulation (35) is equivalent to the linear system +(−Att ⊗ Mx + At ⊗ Mσ +x + Mt ⊗ Axx)A = J +(36) +with the temporal matrices +Att[ℓ, κ] = (∂tϕ2 +κ, ∂tϕ2 +ℓ)L2(0,T), +Mt[ℓ, κ] = (ϕ2 +κ, ϕ2 +ℓ)L2(0,T), +(37a) +At[ℓ, κ] = (∂tϕ2 +κ, ϕ2 +ℓ)L2(0,T) +(37b) +24 + +for ℓ = 0, . . . , N 2 +t − 1, κ = 1, . . . , N 2 +t and the spatial matrices +Ax[l, k] = (µ−1 curlx ψN +k , curlx ψN +l )L2(Ω), +Mx[l, k] = (ǫψN +k , ψN +l )L2(Ω), +(38a) +Mσ +x [l, k] = (σψN +k , ψN +l )L2(Ω) +(38b) +for k, l = 1, . . . , N N +x . The degrees of freedom are ordered such that the vector of coefficients +in (33) reads as +A = (A1, A2, . . . , AN2 +t )⊤ ∈ RN2 +t NN +x +(39) +with +Aκ = (Aκ +1, Aκ +2, . . . , Aκ +NN +x )⊤ ∈ RNN +x +for κ ∈ {1, . . . , N 2 +t }. +The right-hand side in (36) is given in the same way by +J = (f 0, f 1, . . . , f N2 +t −1)⊤ ∈ RN2 +t NN +x , +where we define +fℓ = (f ℓ +1, f ℓ +2, . . . , f ℓ +NN +x )⊤ ∈ RNN +x +for ℓ = 0, . . . , N 2 +t − 1 +with +f ℓ +l = (ΠRT ,1 +h +j, ϕ1 +ℓψN +l )L2(Q) − ϕ1 +ℓ(0) +� +εψ, ψN +l +� +L2(Ω) +(40) +for ℓ = 0, . . . , N 2 +t − 1, l = 1, . . . , N N +x . +Remark +If we have non-zero initial condition φ(x), then we consider a continuous extension +in time ˜φ that is zero on all other degrees of freedom in time. Hence, for our application we +use the representation (33) to derive for t = 0 +˜φh(0, x) = +NN +x +� +k=1 +A0 +kψN +k (x) ≈ ˜φ(0, x). +Then, using the ordering of the degrees of freedom from Section 3.1,´ we end up with the +system +(−Att ⊗ Mx + At ⊗ Mσ +x + Mt ⊗ Axx)A = J − AIni, +(41) +where +AIni := (A0 +Ini, . . . , AN2 +t −1 +Ini +), +Aℓ +Ini := (Aℓ +Ini;1, . . . , Aℓ +Ini;NN +x ), +Aℓ +Ini;l := +NN +x +� +k=1 +(−Att[ℓ, 0]Mx[l, k] + At[ℓ, 0]Mσ +x [l, k] + Mt[ℓ, 0]Axx[l, k])A0 +k +for ℓ = 0, . . . , N 2 +t − 1 and l = 1, . . . , N N +x with the matrices defined in (38) and (37). +25 + +3.3 +The CFL Condition +If we solve the equation (36) for simple examples, we see conditional stability in our results +which hints at a CFL condition. To compute this CFL condition for functions with second- +order elements in time use a similar tactic as [31] did for linear elements. To simplify the +calculation we assume that we have an equidistant discretization in time with step size ht. +Let λi, i ∈ N0, be the eigenvalues of the curlµ−1curl operator in the weighted L2 +ǫ(Ω)-norm +from Lemma 1.4. To analyze the stability of the discrete system we choose initial data φ = 0, +ψ = 0 as well as the right-hand side ja = 0. +First let σ = 0. Note that, since σ acts like a stabilizer to our system, the case σ ≡ 0 is the +hardest case. Now, we consider the basis ansatz from Section 1.1.1 for A ∈ Hcurl;1 +0;0, +and write +AN = +N +� +k=−N +˜Ak(t)ϕk(x) +Then we can rewrite the variational formulation (14) into: Find ˜Ak ∈ H1 +0,(0, T) such that +−(∂t ˜Ak, ∂tv)(0,T) + λk( ˜Ak, v)(0,T) = 0 +(42) +for all v ∈ H1 +,0(0, T) and all eigenvalues λk > 0, k ∈ N0. On the other hand, for k ∈ Z\N0, we +end up with the formulation: Find ˜Ak ∈ H1 +0,(0, T) such that +(∂t ˜Ak, ∂tv)(0,T) = 0 +for all v ∈ H1 +,0(0, T). This is equivalent to the Laplace equation and does not add to our +instabilities. +To derive the appropriate CFL condition, we need to analyze the stability of (42) in the +discrete. +Using again the ansatz and test spaces S2 +0,(T t) and S2 +,0(T t) we get the discrete +formulation: +Find ˜Ah +k ∈ S2 +0,(T t) such that +−(∂t ˜Ah +k, ∂tvh)(0,T) + λk( ˜Ah +k, vh)(0,T) = 0 +for all vh ∈ S2 +,0(T t) and all eigenvalues λk, k ∈ Z. This is equivalent to analyzing the linear +system +(−Att + λkMt) ˜A = 0 +(43) +for its stability. +In equation (43) we have the temporal matrices as defined in (37) and +the solution vector ˜A = ( ˜Ak(t1), . . . , ˜Ak(T))T ∈ RN2 +t which are the coefficients of the basis +representation ˜Ah +k(t) = �N2 +t +κ=1 ˜Ak(tκ)ϕ2 +κ(t). Let us write λ = λk and h = ht. For the second- +order ansatz and test functions, we compute the element matrices +Me +t = h +30 + + +4 +2 +−1 +2 +16 +2 +−1 +2 +4 + + , +Ae +tt = 1 +3h + + +7 +−8 +1 +−8 +16 +−8 +1 +−8 +7 + + . +26 + +This results in the system matrix +Ksys = + + + + + + + + +8 +3h + λh +15 +− 1 +3h − λh +30 +−16 +3h + 8λh +15 +8 +3h + λh +15 +8 +3h + λh +15 +− 14 +3h + 4λh +15 +8 +3h + λh +15 +− 1 +3h − λh +30 +8 +3h + λh +15 +−16 +3h + 8λh +15 +8 +3h + λh +15 +− 1 +3h − λh +30 +8 +3h + λh +15 +− 14 +3h + 4λh +15 +8 +3h + λh +15 +− 1 +3h + λh +30 +. . . +. . . +. . . +. . . +. . . + + + + + + + + +. +By multiplying with 3h we get the following formula +� +8 + λh2 +5 +−1 − λh2 +10 +−16 + 8λh2 +5 +8 + λh2 +5 +� �u2k−1 +u2k +� += +� +−8 − λh2 +5 +14 − 4λh2 +5 +0 +−8 − λh2 +5 +� �u2k−3 +u2k−2 +� ++ +� +0 +1 + λh3 +10 +0 +0 +� �u2k−5 +u2k−4 +� +. +This can be understood as a two-step method +Azk = B1zk−1 + B2zk−2 +for the vector zk = (u2k−1, u2k)T with +A := +� +8 + λh2 +5 +−1 − λh2 +10 +−16 + 8λh2 +5 +8 + λh2 +5 +� +, +B1 := +� +−8 − λh2 +5 +14 − 4λh2 +5 +0 +−8 − λh2 +5 +� +, +B2 := +� +0 +1 + λh3 +10 +0 +0 +� +. +Further, we can rewrite the two-step method as the system +Yk = AsysYk−1 +(45) +with Yk = (u2k−3, u2k−2, u2k−1, u2k)T and +Asys := +� +0 +I +A−1B2 +A−1B1 +� +. +Next, we solve (45) by iterate in k and arrive at the formula +Yk = Ak−1 +sys Y1 +for the solution Yk with the initial values +�u1 +u2 +� += A−1 +� +7 − 2λh2 +5 +−8 − λh2 +5 +� +u0, +where u−1 = 0. Therefore, if we want our system to be stable, we need the real parts of the +eigenvalues of Asys to be less than one. Hence we analyze the eigenvalues of Asys and see that +two of the eigenvalues λA +1 and λA +2 are zero. The other two are given by +λA +3,4 = −2a2 − bd ± +� +b(2c + d)(4a2 − 2bc + bd) +2(a2 − bc) +27 + +where +a = 8 + λh2 +5 , +b = −16 + 8λh2 +5 +, +c = −1 − λh2 +10 , +d = 14 − 4λh2 +5 +. +The absolute value of real part of λa +3,4, namely |ℜ(λA +3,4)| is smaller than one if λih2 +t ≤ 60 and +λih2 +t /∈ [10, 12], for i ∈ N0. If we introduce σ(x) > 0, x ∈ Ω large enough the small instability +region λih2 +t ∈ [10, 12] would vanish and we are left with the condition λih2 +t ≤ 60, for i ∈ N0 . +In computational examples, we often observe only the bound λh2 +t ≤ 60. However, there +might be unstable examples where λh2 +t ∈ [10, 12]. Note that these estimates can also be used +for the scalar wave equation. +To use these insights in computational examples, we need to estimate the eigenvalues λ. +For this purpose, we use an inverse inequality for the curlxµ−1curlx operator in the weighted +L2 +ǫ(Ω)-norm. +For lowest order Nédélec elements, we have the inverse inequality +∥ curlx uh∥2 +L2(T x) ≤ cIh−2 +max∥uh∥2 +L2(T x) +(46) +for all uh ∈ span{φN I +0 +E }E∈E, where +cI := +�18c2 +F +π +n = 2, +80c4 +F ( +9 +16π2 )2/3 +n = 3, +hmax := max +ωl∈T x(|τl|−d). +For the proof consider [19, Lem. A.2] or [21]. The proof is done by computing each norm on +each element and showing the inequality there, then summing everything up. For complete- +ness, we quickly state where the constants come from. The constant cF is the shape regularity +constant defined in (31). The coefficient cIh−2 +max comes from estimating 18λmax(JlJT +l ) +2∆l +h−2 +l +for +d = 2 and for d = 3 estimating the term 1604λmax(JlJT +l )2 +3(6∆l)2 +, where Jl = +�∂Fl,i +∂ˆxj (ˆxj) +� +1≤i,j,≤d is the +derivative of transformation Fl : ˆω → ωl which maps the reference element ˆτ to the current +element ωl. +Now, we apply the inverse inequality to estimate the eigenvalues λi, i ∈ N0, from Lem. 1.4. +We rewrite the inverse inequality (46) in the weighted norms to get +(µ−1 curlx uh, curlx uh)L2(T x) ≤ cI(µminǫmin)−1h−2 +max∥uh∥2 +L2ǫ(T x). +Then we derive the following CFL condition for the vectorial wave equation +cI h−2 +x +≤ 10 h−2 +t , +and therefore +ht ≤ +� +10 +cI +hx, +(47) +28 + +where ht and hx are the respective maximal temporal and spatial step sizes from Section 3.1. +In case σ is big enough, we even arrive at the CFL condition +ht ≤ +� +60 +cI +hx, +(48) +If we consider Ω = (0, 1)2 and use a shape regular triangulation with isosceles rectangular +triangles as in Fig. 1 then we compute, by estimating the eigenvalues of JlJT +l +directly, that +cI = 18 as in [20]. Hence, in this case, we arrive at the CFL conditions +ht < +� +60 +18hx ≈ 1.825741858 hx +(49) +and the stricter condition +ht < +� +10 +18hx ≈ 0.74535599 hx +(50) +Figure 1: Triangulation with isosceles rectangular triangles +3.4 +Numerical Results +We will first take a look at the results for σ = 0 to have a baseline from which we can judge +other results. +To compute the linear system (36), we first have to assemble the right-hand side. The pro- +jection ΠRT ,1 +h +ja of the right-hand side ja in (34) are calculated by using-high-order quadrature +rules for the integrals, see [20] for more details. The calculation of all spatial and temporal +matrices (38) is done with the help of the finite element library NGSolve, see www.ngsolve.org +and [27]. Finally, the linear system (36) is solved by the sparse direct solver UMFPACK 5.7.1 +[12]. +3.5 +Testing the convergence for σ ≡ 0 +We will first take a look at examples for σ ≡ 0 to investigate the CFL condition. For our +examples in this section we consider the domain Q = (0, 2) × (0, 1)2, with T = 2 and Ω = +[0, 1] × [0, 1], and ǫ ≡ +�1 +0 +0 +1 +� +and µ ≡ 1. We will consider two constructed solutions. The +first is given by +A1(t, x1, x2) = t3x1(1 − x1)x2(1 − x2) +� x2 +−x1 +� +. +(51) +29 + +If we insert A1 into the vectorial wave equation, we compute +j1(t, x1, x2) := +� +2x1(1 − x1)x2(1 − x2)x2 +(2x1(1 − x1)x2(1 − x2)(−x1) +� ++ t2 +� x1(x1(5 − 12x2) + 10x2 − 4) +−x2(−2x1(6x2 − 5) + 5x2 − 4) +� +. +Next, we solve the discrete system (36) for the above configuration and take a look at the +convergence rates. For that purpose, we compute the experimental order of convergence (EOC) +with +EOC = ln(errL−1) − ln(errL) +ln(hL−1) − ln(hL) +, +where errL is the L2-error, or the error in the Hcurl;1-half norm respectively, at level L. +Additionally we use bisection in the refinement, hence ln(hL−1) − ln(hL) = ln(2). +If we then solve the discrete linear system described in (36) for second-order elements in time +and lowest order Nédélec elements in space, we compute +L +hx +ht +#fdofs +∥A − Ah∥L2(Q) +EOC +|A − Ah|Hcurl;1(Q) +EOC +1 +0.5000 +0.5000 +80 +5.33706e-02 +- +3.05563e-01 +- +2 +0.2500 +0.2500 +896 +2.16399e-02 +1.30 +2.12075e-01 +0.53 +3 +0.1250 +0.1250 +8192 +5.02257e-03 +2.11 +1.00680e-01 +1.07 +4 +0.0625 +0.0625 +69632 +1.23691e-03 +2.02 +4.94712e-02 +1.03 +5 +0.0312 +0.0312 +573440 +3.06510e-04 +2.01 +2.46265e-02 +1.01 +Table 1: Error table for the Galerkin–Petrov FEM (35) for the unit square Ω and T = 2 and +the solution A1 in (51) using a uniform refinement strategy. +Here we see second order convergence in the ∥.∥L2(Q)-norm and first order convergence in the +|.|Hcurl;1(Q)-halfnorm. This is the same result as we would get for linear elements in time and +ΠRT ,1 +h +ja as the projection of the right-hand side, see [20]. +Next, we want to test the sharpness of the CFL condition (49). To that purpose, we use +a more complicated artificial solution A1, which was also used in [20]. We define +A2(t, x1, x2) = +�−5t2x2(1 − x2) +t2x1(1 − x1) +� ++ t3 +�sin(πx1)x2(1 − x2) +0 +� +(52) +for (t, x1, x2) ∈ Q. The function A fulfills the homogeneous boundary condition γtA = 0 and +has homogeneous initial conditions. The related right-hand side j is given by +j2(t, x1, x2) = +�−10(t2 − x2 +2 + x2) +2(t2 − x2 +1 + x1) +� ++ +�2t3 sin(πx1) + 6t sin(πx1)x2(1 − x2) +πt3(1 − 2x2) cos(πx1) +� +for (t, x1, x2) ∈ Q and divxja ̸= 0. +To show the behavior of the CFL condition we take a look at the final time T = +√ +10.4. +For these values, we see, e.g., in the last row and second to last column that the ratio +ht +hx += 0.0806 +0.0442 ≈ 1.82352941 +is below the CFL condition in both Tables 2 and 3. +30 + +hx +ht +0.6450 +0.3225 +0.1612 +0.0806 +0.0403 +0.1768 +3.54e-01 +3.54e-01 +3.54e-01 +3.54e-01 +3.54e-01 +0.0884 +8.90e-02 +1.04e+00 +8.85e-02 +8.85e-02 +8.85e-02 +0.0442 +2.29e-02 +4.87e-01 +1.95e+05 +2.21e-02 +2.21e-02 +Table 2: Errors in ∥·∥L2(Q) for the Galerkin–Petrov FEM (35) for the unit square Ω and +T = +√ +10.4 and the solution A2 in (52) using a uniform refinement strategy. +hx +ht +00.6450 +0.3225 +0.1612 +0.0806 +0.0403 +0.1768 +4.29e+00 +4.28e+00 +4.28e+00 +4.28e+00 +4.28e+00 +0.0884 +2.15e+00 +5.25e+01 +2.14e+00 +2.14e+00 +2.14e+00 +0.0442 +1.08e+00 +4.57e+01 +2.00e+07 +1.07e+00 +1.07e+00 +Table 3: Errors in |·|Hcurl;1(Q) for the Galerkin–Petrov FEM (35) for the unit square Ω and +T = +√ +10.4 and the solution A2 in (52) using a uniform refinement strategy. +3.6 +The convergence for σ ̸≡ 0 +Now, we consider Q = (0, 2) × [0, 1]2 again with ǫ ≡ +�1 +0 +0 +1 +� +, µ ≡ 1 and the constructed +solution +A3(t, x) = t2x1(1 − x1)x2(1 − x2) +� x2 +−x1 +� +. +(53) +However, we choose σ = 1 over conv{(0.5, 0.35), (0.65, 0.5), (0.5, 0.65), (0.35, 0.5)} and zero +elsewhere. Then we compute ja = ∂ttA3 + σ∂tA3 + curlx curlx A3. When we solve (36) we get +the following L2-error table for piecewise quadratic functions in time. +hx +ht +0.2500 +0.1250 +0.0625 +0.0312 +0.0156 +0.2500 +1.135e-02 +1.135e-02 +1.135e-02 +1.135e-02 +1.135e-02 +0.1250 +3.986e-03 +3.981e-03 +3.981e-03 +3.981e-03 +3.981e-03 +0.0625 +2.501e-03 +4.490e-01 +2.462e-03 +2.462e-03 +2.462e-03 +Table 4: Errors in ∥·∥L2(Q) for the Galerkin–Petrov FEM (35) for the unit square Ω and T = 2 +and the solution A3 in (53) using a uniform refinement strategy. +Again we see the CFL condition as in the case of σ ≡ 0. Indeed, in the last line we see from +ht/hx = 4 to ht/hx = 2 an increase in the error which stops after the CFL condition (49) is +full filled. Note, that the stricter CFL condition (50) does not apply here. +Remark +In this case of ε = ε0, µ−1 = µ−1 +0 +we get +ht < +� +ε0µ0 +2π +3c2 +F +hx ≈ 1.165410−8 hx +(54) +31 + +since +µ0 = 1.256637 10−6, +ε0 = 8.854188 10−12. +In this case, since µ−1 and σ are at least of order 1018 greater, it is reasonable to go to the +Eddy Current problem instead of solving (35). This will be the topic of future work. +4 +Conclusion +In this paper, we have investigated the unique solvability for the variational formulation of +the vectorial wave equation considering Ohm’s law. First, we have taken a look at the trial +and test spaces and showed properties of the functional spaces that we needed to prove the +solvability. Then we proved unique solvability for the variational formula (14), where ja ∈ +L1(0, T; L2(Ω; Rd)). For electromagnetic problems, the variational formula (14) applies to a +variety of electromagnetic examples, since it is posed for Ohm’s law and anisotropic material. +Having proven the unique solvability, we turned to computational examples in the tensor +product. +We used piecewise quadratic ansatz functions in time and lowest order Nédélec +elements in space. Here, as in the case of linear ansatz functions in time, [20], we realized that +there is a CFL condition. We calculated the CFL condition and gave examples. Using the +reasoning from Section 3.3, it is also possible to derive a CFL condition for other higher-order +elements in time. We thus learn that simply increasing the order of finite elements in time +does not lead to an unconditionally stable method. In the case of the tensor product structure, +we can always expect a CFL condition for the space-time discretization of the vectorial wave +equation. A possible solution is the use of the modified Hilbert transform as performed in +[20]. +The results of this work form the basis for more complicated electromagnetic problems. +We learned the main difficulties of the vectorial wave equation and what it implies. This will +be a good starting point for future work on the calculation of eddy current problems and +further applied calculations. +References +[1] Jin, J. The finite element method in electromagnetics, second ed. Wiley-Interscience +[John Wiley & Sons], New York, 2002. +[2] Bai, X., and Rui, H. A second-order space-time accurate scheme for Maxwell’s equa- +tions in a Cole–Cole dispersive medium. Engineering with Computers 38, 6 (2022), 5153– +5172. +[3] Xie, J., Liang, D., and Zhang, Z. Energy-preserving local mesh-refined splitting +FDTD schemes for two dimensional Maxwell’s equations. J. Comput. Phys. 425 (2021), +Paper No. 109896, 29. +[4] Ern, A., and Guermond, J.-L. 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Anal. 56, 6 (2018), 3196–3218. +34 + diff --git a/ONE1T4oBgHgl3EQftwWv/content/tmp_files/load_file.txt b/ONE1T4oBgHgl3EQftwWv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9eb5c115996ef46cf0606ff4128943d676ce8f52 --- /dev/null +++ b/ONE1T4oBgHgl3EQftwWv/content/tmp_files/load_file.txt @@ -0,0 +1,1370 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf,len=1369 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='03381v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='NA] 9 Jan 2023 Space-Time FEM for the Vectorial Wave Equation under Consideration of Ohm’s Law Julia I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hauser1 1Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, Zellescher Weg 25, 01217 Dresden, Germany julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='hauser@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='de Abstract The ability to deal with complex geometries and to go to higher orders is the main advantage of space-time finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore, we want to develop a solid background from which we can construct appropriate space-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this pa- per, we will treat time as another space direction, which is the main idea of space-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we will briefly discuss how exactly the vectorial wave equation is derived from Maxwell’s equations in a space-time structure, taking into account Ohm’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we will derive a space-time variational formulation for the vectorial wave equation using different trial and test spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This paper has two main goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we prove unique solvability for the resulting Galerkin–Petrov variational formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Second, we analyze the discrete equivalent of the equation in a tensor product and show conditional stability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' a CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Understanding the vectorial wave equation and the corresponding space-time finite ele- ment methods is crucial for improving the existing theory of Maxwell’s equations and paves the way to computations of more complicated electromagnetic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1 Modeling and Introduction Maxwell’s equations are used to model electromagnetic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' They are the foundation of classical electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Maxwell’s equations are given by curlx E = −∂tB, (1a) curlx H = ∂tD + j, (1b) divxB = 0, (1c) divxD = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (1d) In equation (1), there are four unknowns: The electric field E, the magnetic field H, the electric flux density D and the magnetic flux density B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The variable j is the given electric current density and ρ is the given charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additional to the Maxwell system (1) we use constitutive relations, or material laws, relating the electric flux density D with the electric 1 field E and the magnetic field H with the magnetic flux density B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this paper, we will consider the constitutive relations D = εE, (2a) H = µB, (2b) where ε is the permittivity and µ the permeability, which will be defined in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' How to solve the Maxwell system (1) has been an active field of study in the last century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' One possible solution is going into the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For that purpose, we assume that the solutions of Maxwell’s equations look like waves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' There exists several introductions to Maxwell’s equations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will not discuss the modeling background in the frequency domain, but simply refer to some of these works such as [33, 18] or, from a mathematical and numerical point of view [6, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' A good introduction to the computational theory of the frequency domain would be [6, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the last century, the time-harmonic Maxwell’s equations were often applied to scattering problems, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We find analytical solutions to scattering problems from 1970 in [17] and theory on inverse scattering and optimal control in scattering by Colton and Kress in 1983, [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' An introduction to inverse scattering problems can be found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Another possibility to solve Maxwell’s equations (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For an introduction we refer to [1, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2] and references there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' However, the time-stepping approach observes instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' A possibility to deal with these complications is to stabilize the system which is created by the time-stepping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Examples can be found in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', [2, 3, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' However, since the time- stepping method is a type of finite difference method, it cannot deal with complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The possibility to deal with complex geometries as well as going into higher order is the main advantage of space-time finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this paper, we will set the theoretical background that is needed for such a method and explore the possibilities and restrictions of the vectorial wave equation that is derived from Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will also study elements of second order in time when we go over to the study of the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The goal of this paper is to derive a space-time variational formulation for the vectorial wave equation, show its unique solvability and analyze its numerical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we will start by deriving the vectorial wave equation from Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the modeling, we will assume that all functions in (1) are smooth and we consider a space-time domain Q in R4 which is star-like with respect to a ball B and Lipschitz in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The star-like property implies the convex hull of any x ∈ Ω and B is contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us not treat time differently from space, but rewrite the equations such that the partial derivative in time is simply a partial derivative of a space-time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This can be naturally done in terms of differential forms for which we use the exterior derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, we need a form for which the exterior derivative of that form gives us one or more of Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Following the derivation of [9] we combine the 1-form ˜E corresponding to the electric field E with the 2-form ˜B corresponding to the magnetic flux density B to get the Faraday 2-form F = ˜E ∧ dt + ˜B, where dt ∈ Λ1(Q) is the basis element for 1-forms in the direction of time, for more details see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Second, we combine the 2-form ˜D corresponding to the electric flux density D and the 1-form ˜H corresponding to the magnetic field H into the Maxwell 2-form G = ˜H ∧ dt − ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2 Additionally, we need the source 3-form J = ˜j ∧ dt − ˜ρ, where ˜j is the 2-form corresponding to j and ˜ρ is the corresponding 3-form to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By inserting these forms into the equations we end up with a representation in four dimensions d F = 0, d G = J , (3) G = ⋆ε,−µ−1F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the derivation of the equations consider [9] or [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The operator ⋆ε,−µ−1 is a weighted Hodge star operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For R4 with the euclidean metric ǫ is the weight in the direction ⋆(dx01, dx02, dx03)T and (−µ−1) in the direction ⋆(dx23, dx31, dx12)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us take a closer look at the equations in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The third equation in (1) includes the constitutive equations (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The second equation, d G = J , includes the second and fourth equation of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The first equation, d F = 0, includes the first and third equations of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Moreover, the first equation in (3) implies that the form F is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since the domain Q is starlike, the Poincaré-Lemma, [7, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1], can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The Poincaré-Lemma tells us that if the form F is exact then there is a potential A such that dA = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' If we insert F = dA into the second equation combined with the third equation of (3), we derive the following wave-type equation d ⋆ε,−µ−1 dA = J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (4) Additionally, we have the following two relations E = −∂tA + ∇xA0, B = ∇ × A, where A0 is the time component and A := (A1, A2, A3)T the spatial component of A, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 389].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that (4) will result in the scalar wave equation if we use a differential form of order zero instead of one for A, while the equation (4) as such will end up in the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, in terms of differential forms in 4D, the scalar and the vectorial wave equation are closely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Moreover, we know that A is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' If we take a 0-form ˜ A adding it to A then F = dA = dA+dd ˜ A = d(A+d ˜ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore we can add the gradient of any H1 0(Ω)-function to A and get another viable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence we need a gauge to fix the potential A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' There are many gauges, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' a paper from 2001 on the history of gauge invariance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Depending on the gauge we can derive different equations from (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To derive the vectorial wave equation, we use the Weyl gauge, which is the temporal gauge A0 = 0, to arrive in euclidean space at ∂t(ε(∂tA)) + curlx � µ−1 curlx A � = j, (5a) divx(ε(∂tA)) = −ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (5b) Note that the equation (5a) is the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In addition to these two equations, we assume that the charge is preserved and therefore the continuity equation ∂tρ + divxj = 0 3 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The continuity equation is, however, already included in the combination of both equations of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' On the other hand, we can rewrite the second equation (5b) into initial conditions for ∂tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Indeed, if we assume enough regularity of A, ρ, and j, we can take the spatial divergence of the first equation (5a) and use the continuity equation to derive ∂t(divx(ε∂tA(t)) + ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence divx(ε∂tA(t))(t)+ρ(t) = divx(ε∂tA(t))(0)+ρ(0), for all t ∈ (0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore, if ∂tA(0) satisfies divx(ε∂tA(0)) = −ρ(0) then we satisfy (5b) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now that we derived the vectorial wave equation, let us take a look at Ohm’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the case of a conducting material, the electromagnetic field itself induces currents and in the easiest case this can be modeled by Ohm’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, we want to include Ohm’s law into our equation which is given by j = σE + ja, where σ is the conductivity and ja the applied current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The relationship between E and the magnetic vector potential A was given by E = −∂tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore, in this paper, we consider the following equation ε∂ttA + σ∂tA + curlx(µ−1 curlx A) = ja in Q = (0, T) × Ω, (6) ∂tA(0, x) = ψ(x) in Ω, A(0, x) = φ(x) in Ω, γtA = 0 on Σ = {0, T} × ∂Ω, where γt is the tangential trace of A on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Before we state the general assumption of this paper on the functions in (6), we have to define a few functional spaces that will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we define for Q ⊂ Rd, d ∈ N, the space L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) as the usual Lebesgue space for vector-valued functions v: Q → Rd with the inner product (v, w)L2(Ω) := (v, w)L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Rd) := � Q(v(x), w(x))Rddx for v, w ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) and the induced norm ∥·∥L2(Q) := ∥·∥L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Rd) := � (·, ·)L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Second, we define L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R) := L∞(Ω) for Ω ⊂ Rd, d ∈ N, as the space of measurable functions bounded almost everywhere and equipped with the usual norm ∥·∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Third, we define the space L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd×d), d ∈ N, as the space of matrix-valued measurable functions bounded almost everywhere with the norm ∥w∥L∞(Ω) := ∥w∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Rd×d) := ess sup x∈Ω sup 0̸=ξ∈Rd ξ⊤ǫ(x)ξ ξ⊤ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Finally, we define space-time spaces L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' X) with the inner product (v, w)L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='X) := � T 0 (v(x), w(x))Xdx and L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) with the norm ∥v∥2,1,Q := ∥v∥L1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Rd)) := � T 0 ���∥v∥L2(Ω) ��� dt in the same way as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally, we can define the space H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' X) as the Hilbert space H1(0, T) over the Hilbert space X, see [22] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 4 With these definitions we define the tangential trace operator γt for d = 3 as the contin- uous mapping γt : H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) → H−1/2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The tangential trace operator is the unique extension of the vector-valued function γtv = v|∂Ω × nx for v ∈ H1(Ω)3 that is defined by the Green’s identity for the curl operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For d = 2 we define the tangential trace operator γt : H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) → H−1/2(∂Ω) as the unique extension of the scalar function γtv = v|∂Ω · τ x for v ∈ H1(Ω)2, where τ x is the unit tangent vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', τ x · nx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that γv := (v)|∂Ω is the usual trace operator γ : H1(Ω, Rd) → H1/2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) for d ∈ N and H−1/2(∂Ω) is the dual space of the fractional-order Sobolev space H1/2(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For more details on the tangential trace operator γt consider [15, 4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For detailed definitions of the spaces H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) := {v ∈ H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) | γtv = 0} and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω), we reference [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' However, we quickly note how the curl operator behaves dif- ferently in two and three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For d = 2, we set the curl of a sufficiently smooth vector-valued function v: Ω → R2 with v = (v1, v2)⊤ as the scalar-valued curl operator curlx v = ∂x1v2 − ∂x2v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Sometimes this curl operator is called rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally, for a sufficiently smooth scalar function w: Ω → R, we define the vector-valued curl operator curlx w = (∂x2w, −∂x1w)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that the vector-valued curl operator is the adjoint operator of the scalar-valued curl operator for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the case of d = 3, the curl of a sufficiently smooth vector-valued function v: Ω → R3 with v = (v1, v2, v3)⊤ is given by the vector-valued function curlx v = (∂x2v3 − ∂x3v2, ∂x3v1 − ∂x1v3, ∂x1v2 − ∂x2v1)⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With these definitions, we can write the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let d = 2, 3 and the spatial domain Ω ⊂ Rd and supp(σ) ⊂ Rd, σ : Ω → Rd×d, be given such that Ω and supp(σ) ⊂ Ω are Lipschitz domains, and Q = (0, T) × Ω is a star-like domain with respect to a ball B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Further, let σ, j, ǫ and µ be given functions, which fulfill: The conductivity σ ∈ L∞(supp(σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd×d), supp(σ) ⊂ Ω, is uniformly positive definite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' σmin := ess inf x∈supp(σ) inf 0̸=ξ∈Rd ξ⊤σ(x)ξ ξ⊤ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The applied current density ja : Q → Rd satisfies ja ∈ L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The permittivity ǫ: Ω → Rd×d is symmetric, bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', ǫ ∈ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd×d), and uni- formly positive definite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', ǫmin := ess inf x∈Ω inf 0̸=ξ∈Rd ξ⊤ǫ(x)ξ ξ⊤ξ > 0 and ǫmax := ∥ǫ∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 5 For d = 2, the permeability µ: Ω → R satisfies µ ∈ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R), µmax := ∥µ∥L∞(Ω), and µmin := ess inf x∈Ω µ(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For d = 3, the permeability µ: Ω → R3×3 is symmetric, bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', µ ∈ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R3×3), µmax := ∥µ∥L∞(Ω), and uniformly positive definite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', µmin := ess inf x∈Ω inf 0̸=ξ∈R3 ξ⊤µ(x)ξ ξ⊤ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the initial data φ and ψ we assume that: The initial data φ : Ω → Rd satisfies φ ∈ H0(curlx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The initial data ψ ∈ H(divx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) satisfies divx(εψ) = −ρ(0) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' After all the assumptions are stated and the vectorial wave equation is introduced, let us now take a look at the structure of the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the remainder of this section, we will introduce the functional spaces that are needed to formulate the space-time variational formulation of the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will discuss that they are indeed Hilbert spaces and go over possible basis representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' At the end of this section, we will derive the variational formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The second section of this paper is dedicated to the proof of unique solvability for the variational formulation and norm estimates of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' n this proof we will use a Galerkin method and use the previously developed basis representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the third section of this paper, we will discuss the space-time finite element spaces and discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will derive a CFL condition and take a look at examples that show this CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the end, we will sum up the conclusions and give an outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 The space-time Sobolev spaces To derive a variational formulation, we need to consider the appropriate functional space for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this section, we will derive them and show the properties of them that we need for the proofs later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To derive the functional spaces let us assume for a moment that d = 3 then Q ⊂ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' From the derivation of the magnetic vector potential A we know that A is a 1-form in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For 1-forms the corresponding functional space in the L2-Hilbert complex in R4 is H(Curl, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R4), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The space H(Curl, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R4) is defined by H(Curl, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R4) := { u ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R4) | Curl u ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' K) }, [Curl u]ij := 4 � k,l=1 εijkl∂kul, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note, that the operator Curl is an operator in both space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally, we use the Weyl gauge, the temporal gauge which implies A0 = 0, to make the potential A unique in the derivation of the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, we are interested in a function A = (0, A1, A2, A3)T that is in H(Curl, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rewriting the operator Curl for A0 = 0 and A = (A0, A)T ∈ H(Curl, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' R4) leads to the conditions A ∈ L2(Q, R3), curlx A ∈ L2(Q, R3), ∂tA ∈ L2(Q, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 6 Before we define the appropriate function space, we take a quick look at the coefficients in the vectorial wave equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' When we will formulate a variational formulation for the equation (6), we will use the weighted L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)- and H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)-inner products (u, v)L2ǫ(Ω) := (ǫu, v)L2(Ω) , (u, v)L2µ(Ω) := � µ−1u, v � L2(Ω) , for v, w ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) and (u, v)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) := (u, v)L2ǫ(Ω) + (curlx v, curlx w)L2µ(Ω), for v, w ∈ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note, that due to the Assumptions 1 the norms ∥·∥L2ǫ(Ω) and ∥·∥H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω), induced by (·, ·)L2ǫ(Ω) and (·, ·)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω), are equivalent to the standard norms ∥·∥L2(Ω) and ∥·∥H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now let us define the function spaces which we will be using from this point on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' From (6) we learn that we need to have a trace in time that is well defined in order to incorporate zero initial or end condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, we define the following spaces under the Assumption 1 Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) := { u ∈ L2(Q, Rd) | ∂tu ∈ L2 ǫ(Q), curlx u ∈ L2 µ(Q), u(0, x) = 0 for x ∈ Ω, γtu = 0 on Σ = (0, T) × ∂Ω } (7) = L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) ∩ H1 0,(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Q, Rd)), Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) := { u ∈ L2(Q, Rd) | ∂tu ∈ L2 ǫ(Q), curlx u ∈ L2 µ(Q), u(T, x) = 0 for x ∈ Ω, γtu = 0 on Σ } (8) = L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) ∩ H1 ,0(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Q, Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The subscript ’0,’ and ’, 0’ stands for zero initial and zero end conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2 we will see that they are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' It is quite natural to assume that Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will quickly discuss whether this is indeed true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Afterward, we state in which concept C∞(Q)-functions are dense in our space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We end up looking at the half norm in this setting which is a good tool for numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' So let us start with showing that Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) and Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) are Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let d = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The space H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) is isometric to the Hilbert tensor product H1(0, T)ˆ⊗L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) and the space L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) is isometric to the Hilbert ten- sor product L2(0, T)ˆ⊗H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Addtionally the space C∞(0, T)ˆ⊗[C∞(Ω))]d is dense in H1(0, T)ˆ⊗L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) and L2(0, T)ˆ⊗H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) and C∞([0, T])ˆ⊗[C∞ 0 (Ω))]d is dense in L2(0, T)ˆ⊗H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' See Aubin [11], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The second result follows with [28, Ch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence we get that C∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [C∞(Ω)]d) is included in both spaces H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) and L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) and therefore is dense in the intersections Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) and Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 7 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The spaces Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) and Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) are Hilbert spaces equipped with the inner product (u, v)Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) := (u, v)L2(Q) + (∂tu, ∂tv)L2ǫ(Q) + (curlx u, curlx v)L2µ(Q) (9) for all u, v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally the initial and end conditions are well defined in Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since the embedding H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) ⊂ C([0, T], L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) is continuous, see [22, Prop 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='23], we get that H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) ⊂ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) is continuously embedded, see [22, Prop 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2], and since H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) ⊂ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) that L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) ⊂ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) is continuously embedded, see [22, Prop 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We know that L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) is continuously embedded in the Hausdorff space M0 of measurable functions that are finite a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', see [23, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore the pair (H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)), L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω))) is a compatible couple and H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) is a Banach space with the norm ∥u∥H1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Rd))∩L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω)) = max{∥u∥H1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Rd)), ∥u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω))}, see [23, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now, (9) defines a norm that is induced by the inner product and it is equivalent to the above norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Because the embedding H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) ⊂ C([0, T], L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)) is continuous we get well defined and continuous traces to the boundaries {t = 0} × Ω and {t = T} × Ω for Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since the spaces Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) and Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) are the kernels of these traces, we derive that they are closed subsets of the Hilbert space Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the spaces Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) and Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) there exists a cf > 0 such that ∥u∥L2(Q) ≤ cf∥∂tu∥L2(Q) for all u ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) or u ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q), d = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore the half norm |u|2 Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) := ∥∂tu∥2 L2ǫ(Q) + ∥ curlx u∥2 L2µ(Q), is an equivalent norm in Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) and Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This can be proving simply by using the Poincaré-inequality in H1(0, T) and the structure of the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='∥H1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' See [20, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 Basis representations To derive a basis representation in Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) we first have to state how we decompose H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The basis representation of Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) will be used in the proof of the main theorem on uniqueness and solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To understand the proof better, we will quickly derive the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us define X0,ǫ := { u ∈ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) | (εu, ∇p)L2(Ω) = 0, ∀p ∈ H1 0(Ω) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 8 Then we know the Helmholz-Weyl decomposition H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) = ∇H1 0(Ω) ⊕ X0,ǫ (10) where the orthogonality holds true with respect to (·, ·)L2ǫ(Ω), (·, ·)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) and hence, (curlx ·, curlx ·)L2µ(Ω), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [6, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5] for d = 3 or for d = 2, and d = 3 as well, we can use the functional analysis toolbox from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The space H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) has a fundamental system {ϕi}i∈Z which is orthonormal in the L2 ǫ(Ω)-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally {ϕi}i∈Z is constructed in such a way that for every i ∈ N0 exists a λi > 0 such that (curlx ϕi, curlx v)L2µ(Ω) = λi(ϕi, v)L2ǫ(Ω) for all v ∈ X0,ǫ(Ω) and for i ∈ Z\\N0 there exists a λi > 0 and φi such that ϕi = ∇xφi and (∇xφi, ∇xv)L2ǫ(Ω) = λi(φi, v)L2ǫ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' for all v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For ∇H1 0(Ω) we get an orthonormal basis from the Laplace eigenvalue problem: Find (λi, φi) ∈ (R, H1 0(Ω)), i ∈ Z\\N0, such that for all v ∈ H1 0(Ω) (∇xφi, ∇xv)L2ǫ(Ω) = λi(φi, v)L2(Ω) and ∥∇xφi∥L2ǫ(Ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The solution to the eigenvalue problem is a non-decreasing sequence of related eigenvalues λi > 0, satisfying λi → ∞ as i → −∞, see [35, Section 4 in Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we investigate the eigenvalue problem: Find (λi, ϕi) ∈ (R, X0,ǫ(Ω)), i ∈ N0, such that for all v ∈ X0,ǫ(Ω) (ϕi, v)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) = (1 + λi)(ϕi, v)L2ǫ(Ω) and ���ϕi ��� L2ǫ(Ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (11) The set of eigenfunctions {ϕi ∈ X0,ǫ(Ω) : i ∈ N0} form an orthonormal basis of H(div ǫ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) with respect to (·, ·)L2ǫ (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally the nondecreasing sequence of related eigenvalues (1+ λi), satisfying λi → ∞ as i → ∞, see [15, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that λi > 0, i ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This can be shown by estimating 0 < cP (ϕi, ϕi)L2ǫ(Ω) ≤ (curlx ϕi, curlx ϕi)L2µ(Ω) = (ϕi, ϕi)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) − (ϕi, ϕi)L2ǫ(Ω) = λi using the Poincaré-Steklov inequality, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [5, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4], and then the variational formu- lation (11) for v = ϕi to get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Moreover, the set {(1 + λi)−1/2ϕi ∈ X0,ǫ(Ω) : i ∈ N0} is an orthonormal basis of X0,ǫ(Ω) with respect to (·, ·)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) by construction, see (11), and since X0,ǫ(Ω) ⊂ H(div ǫ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally, we see that the set {ej ∈ X0,ǫ(Ω) : j ∈ N0} is also orthogonal with respect to (curlx ·, curlx ·)L2µ(Ω) since it is an equivalent norm in X0,ǫ(Ω) because of the Poincaré-Steklov inequality, see [5, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then, by using the decomposition (10) we arrive at the desired orthonormal basis of H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) with the set {∇φi}i∈Z\\N0∪{(1+λi)−1/2ϕi}i∈N0, which is orthogonal with respect to (·, ·)H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 9 Now that we know the fundamental system of H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) we can write w ∈ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) as w(x) = ∞ � i=−∞ wiϕi(x), x ∈ Ω, with the coefficients wi = (w, ϕi)L2ǫ(Ω), i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This basis representation converges in H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then the seminorm |·|H0,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) and the norm ∥·∥H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) admit the representations |w|2 H0,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) = ∞ � i=0 λi ���� >0 |wi|2 , ∥w∥2 H0,ǫ,µ(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) = ∞ � i=0 (1 + λi) |wi|2 + ∞ � i=1 |w−i|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (12) Let v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we learn from [22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='23] that v coincides on [0, T] with a continuous mapping v : [0, T] → H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) up to a subset of measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence we write v(t) = ∞ � i=−∞ ci(t)ϕi, (13) for some coefficient functions ci : [0, T] → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2 The variational formulation To derive a suitable variational formulation for (6) we multiply the partial differential equation with a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By using partial integration both in time and space we end up with the following variational formulation: Find A ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) with A(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=') = φ, such that − (ε∂tA, ∂tv)L2(Q) + (σ∂tA, v)L2(Q) + � µ−1 curlx A, curlx v � L2(Q) (14) = � ja, v � L2(Q) − � εψ, v(0, ·) � L2(Ω) for all v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='.,0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note, that the initial condition ∂tA(0) = ψ is incorporated into the variational formulation in a weak sense while the other conditions A(0) = φ and γtA = 0 are in the ansatz spaces and therefore are satisfied in a strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In (14) we also see the main problem of the equation, the different signs in front of the first term and the spatial differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, this is not equivalent to any norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' However, if σ is large it acts as a stabilization to the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This we will see also in the numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this paper, we will take a look at the bilinear form a(A, v) := −(ε∂tA, ∂tv)L2(Q) + (σ∂tA, v)L2(Q) + (µ−1 curlx A, curlx v)L2(Q), (15) for A ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) and φ ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='.,0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally, we write right-hand side as the linear form F(v) := � ja, v � L2(Q) − � εψ, v(0, ·) � L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (16) 10 2 Existence and Uniqueness Let us now state the main existence and uniqueness result for the variational formulation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let the assumptions 1 hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then there exists a unique solution of the variational formulation: Find A ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) with A(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=') = φ, such that − (ε∂tA, ∂tv)L2(Q) + (σ∂tA, v)L2(Q) + � µ−1 curlx A, curlx v � L2(Q) = � ja, v � L2(Q) − � εψ, v(0, ·) � L2(Ω) for all v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='.,0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' If additionally ja ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) then there exist positive constants cφ, cc φ, cψ, and cf such that |A|2 Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) ≤ cφ∥φ∥2 L2ǫ(Ω) + cc φT∥ curlx φ∥2 L2µ(Ω) + cψT∥ψ∥2 L2ǫ(Ω) + cf max{T, T 2}∥ja∥2 L2ǫ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To prove this theorem we will use a Galerkin method and split the proof in three different steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we prove the existence in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then, we show the uniqueness in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the end, we derive the norm estimates in Proposition 3 and discuss the dependencies of the coefficients cφ, cc φ, cψ, and cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 Existence We will start with proving the existence of the variational formulation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For that purpose, we have to state two small results that we will be using in the existence proof in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let the Assumption 1 hold true and u ∈ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd))∩L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)), d = 2, 3, with u|t=0 = φ and ∂tu|t=0 = ψ be the solution of 2 (ε∂ttu, ∂tu)L2(Q) + 2 � µ−1 curlx u, curlx ∂tu � L2(Q) ≤ 2 � j, ∂tu � L2(Q) , (17) for j ∈ L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then z1/2(t) := \uf8eb \uf8ed � Ω � |u|2 + |∂tu|2 + |curlx u|2� (t, x) dx \uf8f6 \uf8f8 1/2 ≤ c2(T)z1/2(0) + c3(T)∥j∥2,1,Q, holds true for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The constants c2 and c3 are depend on εmin, (µmax)−1, εmax and (µmin)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Following the ideas of [35, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2] we can transfer the results from the scalar to the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Detailed proof can be found in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Here, we will only state the main idea: First, we rewrite the inequality 17 to get for z(t) := � Ω (u2 + (∂tu)2 + (curlx u)2)(t, x) dx the inequality cǫ,µz(t) ≤ ˆcǫ,µz(0) + 2 t � 0 ∥j∥L2(Ω)z1/2(s) ds + 2t t � 0 z(s) ds, 11 for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we write ˆz(t) = max 0≤ξ≤t z(ξ) and solve the inequality on the interval (0, √cǫ,µ 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the end, we use iteration to get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let the assumptions 1 hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let A ∈ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) with A|t=0 = φ and ∂tA|t=0 = ψ be the solution of (ε∂ttA, ∂tA)L2(Q) + (σ∂tA, ∂tA)L2(Q) + (µ−1 curlx A, curlx ∂tA)L2(Q) = (ja, ∂tA)L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' for ja ∈ L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we derive that z1/2(t) := \uf8eb \uf8ed � Ω � |A|2 + |∂tA|2 + |curlx A|2� (t, x) dx \uf8f6 \uf8f8 1/2 ≤ c2(t)z1/2(0) + c3(t)∥ja∥2,1,Q, for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The constants c2 and c3 are depend on εmin, (µmax)−1, εmax and (µmin)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using the positive semi-definiteness of σ we compute (ε∂ttA, ∂tA)L2(Q) + (µ−1 curlx A, curlx ∂tA)L2(Q) ≤ (ja, ∂tA)L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore we can use Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 to prove the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now, we are ready to prove the existence of the solution of the variational formulation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proposition 1 (Existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let Assumption 1 hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then there exists a solution of the variational formulation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us denote Ωσ := Ω ∩ supp(σ) which is a Lipschitz domain by Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The main idea of this proof is to split the differential equation into two equations over different domains, namely one domain is Ωσ and the other Ω\\Ωσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we add them together to show existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The produced solution is the solution to an interface problem with zero tangential traces on the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, let us take a look at the resulting equations on the domain Ωσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' On the domain Ωσ we now use a Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us define the bilinear aΩσ(A(t), φ) := (ε∂ttA(t), φ)L2(Ωσ) + (σ∂tA(t), φ)L2(Ωσ) + (µ−1 curlx A(t), curlx φ)L2(Ωσ), for φ ∈ H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ωσ), t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We consider {ϕi}i∈Z, the fundamental system of H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ωσ), see Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using the basis representation (13) we then search for a AN(t) = N � k=−N cN k (t)ϕk(x) which solves aΩσ(AN(t), ϕl) = (ja, ϕl)L2(Ωσ), d dtcN k (t) = (ψ, ϕk)L2(Ωσ), (18) cN k (0) = αN k , 12 for all l, k ∈ ZN := {z ∈ Z : |z| ≤ N}, where αN k are the coefficients of φN(x) = N � k=−N αN k ϕk(x), and φN → φ in H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ωσ) for N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We have AN(0, x) = φN(x) and define fk := (ja, ϕk)L2(Ωσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since σ is uniformly positive definite and bounded over Ωσ, the induced weighted scalar product (σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' )L2(Ωσ) is equivalent to (ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' )L2(Ωσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, there exists a βk ∈ R+ such that (σϕk, ϕl)L2(Ωσ) = βkδkl for k, l ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' These βk are bounded from below by σmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we combine everything to arrive at c′′N k (t) + βkc′N k (t) + λkcN k (t) = fk(t), c′N k (0) = (ψ, ϕk)L2(Ωσ), cN k (0) = αN k for t ∈ (0, T) and k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' ϕk is an eigenfunction of curlx curlx, and c′′N k (t) + βkc′N k (t) = fk(t), c′N k (0) = (ψ, ϕk)L2(Ωσ), cN k (0) = αk if ϕk is part of the kernel of curlx curlx, k = −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The solutions can be computed using standard techniques for ordinary differential equations such as [32, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 20, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The solutions are well defined for fk ∈ L1(0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore we get for ja ∈ L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ωσ)) a solution AN ∈ C2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ωσ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' If we multiply (18) with c′N l (t) and sum up over l = −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N, we compute aΩσ(A(t), ∂tA(t)) = (ja(t), ∂tAN(t))L2(Ωσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now we can apply the result from Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1 and get \uf8eb \uf8ed � Ωσ ��AN(t) ��2 + ��∂tAN(t) ��2 + ��curlx AN(t) ��2 dx \uf8f6 \uf8f8 1/2 ≤ c2(t)(zN)1/2(0) + c3(t)∥ja∥2,1,Q for every t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By construction c2 and c3 are monotonically increasing and there- fore bounded by c2(T) respectively c3(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally we get with Bessel’s inequality in H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ωσ) and L2(Ω) zN(0) = � Ωσ ��φN��2 + ��ψN��2 + ��curlx φN��2 dx ≤ c∥φ∥2 H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ωσ) + c∥ψ∥2 L2(Ωσ) Therefore ∥AN∥Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) ≤ c∥φ∥2 H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ωσ) + ˜c∥ψ∥2 L2(Ωσ) + ˆc∥j∥2 2,1,Q < C, 13 where C is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now we split AN(t, x) = AN 0 (t, x) + φN(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since ∥φN∥H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) is bounded by ∥φ∥H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ω) because of Bessel’s inequality, the sequence (AN 0 )N∈N is bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The space Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) = H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω)) � L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) is a Hilbert space, see Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence there exists a weakly convergent subsequence of (AN 0 )N∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We write this subsequence as {AN 0 }N for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then there exists a A0 ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q) with AN 0 ⇀ A0 in L2(Q), ∂tAN 0 ⇀ ∂tA0 in L2(Q), curlx AN 0 ⇀ curlx A0 in L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we compute aΩσ(AN 0 (t), ϕl) = (ja(t), ϕl)L2(Ω) − (µ−1 curlx φN, curlx ϕl)L2(Ω) for all l = −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='., N, t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let M ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Choose N > M and dl ∈ H1(0, T) where dl(T) = 0, l = −M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Multiply the equation with dl and sum up over l = −M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Moreover, we get for η(t, x) := M � l=−M dl(t)ϕl(x) and t ∈ (0, T) the equation aΩσ(AN 0 (t), η) = (ja(t), η)L2(Ω) − (µ−1 curlx φN, curlx η)L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By integration over (0, T) and using integration by parts for the first term we get to the bilinear form (15) and a(AN 0 , η) = (ja, η)L2(Q) − (ε∂tAN|t=0, η|t=0)L2(Ω) − (µ−1 curlx φN, curlx η)L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we take the limit N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since η, ∂tη and curlx η are in L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) we get with the weak convergence that a(A0, η) = (ja, η)L2(Q) − (εψ(x), η(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' ))L2(Ω) − (µ−1 curlx φ, curlx η)L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This equation holds for all η with the representation M � l=−M dl(t)ϕl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let MM be the space of such functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For every M ∈ N we can repeat this argumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The space ∞� M=1 MM is dense in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) because we can approximate every element with such a sum, see [22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence the equation above holds true for every η ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) and therefore A is the weak solution of our differential equation in Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us now consider the domain Ω\\Ωσ where σ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We use the same technique as above to get existence, see [21] for detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Here, we will only state the main ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We consider the equations c′′N k (t) + λkcN k (t) = fk(t), (19) c′N k (0) = (ψ, ϕk)L2(Ωσ), cN k (0) = αN k 14 for t ∈ (0, T) and k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' ϕk is an eigenfunction of curlx curlx, and c′′N k (t) = fk(t), (20) c′N k (0) = (ψ, ϕk)L2(Ωσ), cN k (0) = αk if ϕk is part of the kernel of curlx curlx, k = −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By going through the same steps as above, we end up with a bounded solution ∥AN∥Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) ≤ c∥φ∥2 H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='Ωσ) + ˜c∥ψ∥2 L2(Ωσ) + ˆc∥ja∥2 2,1,Q < C, where C is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using the same definitions and arguments as in step 1 we get a weak solution of our differential equation in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω)) ∩ H1 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2 Uniqueness Next, we take a look at the uniqueness of the variational formulation (14) with the bilinear form (15) and right-hand side (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proposition 2 (Uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let Assumption 1 hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then there exists a unique solution of the variational formulation (14) with the bilinear form (15) and linear form (16): Find A ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) with A(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=') = φ, such that a(A, v) = F(v) for all v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='.,0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In Proposition 1 we have already shown existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' What is left to be proven is the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Assume that there are two solutions A′ and A′′, then w := A′ − A′′ satisfies the variational formulation −(ε∂tw, ∂tv)L2(Q) + (σ∂tw, v)L2(Q) + (µ−1 curlx w, curlx v)L2(Q) = 0 for all v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally w(0, x) = 0 holds true for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Choose b ∈ [0, T] arbitrary and consider η(t, x) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 t� b w(τ, x) dτ, for 0 ≤ t ≤ b 0, for b ≤ t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then η ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q) with η(t, x) = 0 for t ≥ b and we get for v = η: −(ε∂tw, ∂tη)L2(Q(0,b)) + (σ∂tw, η)L2(Q(0,b)) + (µ−1 curlx w, curlx η)L2(Q(0,b)) = 0, where Q(0,b) is the intersection of Q with the half space t < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since ∂tη(t, x) = w(t, x) for (t, x) ∈ Q(0,b), we compute (ε∂ttη, ∂tη)L2(Q(0,b)) − (σ∂ttη, η)L2(Q(0,b)) − (µ−1 curlx ∂tη, curlx η)L2(Q(0,b)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 15 Through integration by parts we get −(σ∂ttη, η)L2(Q(0,b)) = (σ∂tη, ∂tη)L2(Q(0,b)) ≥ 0, because ∂tη(0, x) = w(0, x) = 0 for x ∈ Ω, η(b, x) = 0 by definition and σ is positive semi- definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore (ε∂ttη, ∂tη)L2(Q(0,b)) − (µ−1 curlx ∂tη, curlx η)L2(Q(0,b)) ≤ 0 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence 1 2 � Q(0,b) ∂t(ε∂tη · ∂tη) dx dt − 1 2 � Q(0,b) ∂t(µ−1 curlx η · curlx η) dx dt ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' holds true because ε and µ−1 are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' From the definition of η we compute η(b, x) = 0 and therefore (curlx η)(b, x) = 0 for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally it holds true that ∂tη(0, x) = v(0, x) = 0 for x ∈ Ω and therefore � Ω (ε∂tη · ∂tη)(b, x) dx + � Ω (µ−1 curlx η · curlx η)(0, x) dx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' From this we derive by ∂tη(b, x) = w(b, x) that � Ω w2(b, x) dx = 0 and � Ω �� b 0 curlx w �2 (x) dx = 0 holds true for any b ∈ (0, T), since ∂tη(t, x) = w(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With this, we can deduce that w(t, x) vanishes almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3 Norm estimate Now that we know that the variational formulation (14) is uniquely solvable, we take a look at the norm estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this part of the section, we consider ja ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will derive the norm estimate of 2 and take a closer look at the dependencies of the coefficients cφ, cc φ, cψ, and cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let Assumption 1 hold, ja ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) and ck, k ∈ N0, be the solution of the ordinary differential equation c′′ k(t) + βkc′ k(t) + λkck(t) = fk(t), (21) ck(0) = αN k , c′ k(0) = (ψ, ϕk)L2(Ωσ) for t ∈ (0, T), where λk > 0 are the nonzero eigenvalues of the curlx curlx-operator from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4 and fk := (ja, ϕk)L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then there exist positive constants cκ α, cα, cψ and cf such that � k∈N0 T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ � k∈I cψ(ψ, ϕk)2 L2(Ωσ) + (cκ αλk + cα)(αN k )2 + cf∥fk∥2 L2(0,T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For these estimates we need to consider three cases, namely β2 k −4λk > 0, β2 k −4λk < 0 and β2 k − 4λk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let β2 k − 4λk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We define λ1 := −βk+√ β2 k−4λk 2 , λ2 := −βk−√ β2 k−4λk 2 , γk = � β2 k − 4λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then the solution of (21) is given by ck(t) =(ψ, ϕk)L2(Ωσ) eλ1t − eλ2t γk + αN k λ1eλ2t − λ2eλ1t γk + 1 γk t � 0 (eλ1(t−s) − eλ2(t−s))fk(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We know that λ1, λ2 < 0, because β2 k − 4λk > 0 and so −βk − � β2 k − 4λk < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally we get −(λ1 + λ2) = βk as well as λ1λ2 = λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With these facts and using (a + b)2 ≤ 2a2 + 2b2 we compute T � 0 � eλ1t − eλ2t�2 dt ≤ 1 λ1 (e2λ1T − 1) + 1 λ2 (e2λ2T − 1) ≤ 1 −λ1 + 1 −λ2 = βk λk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the same way, we estimate T � 0 � λ1eλ1t − λ2eλ2t�2 dt ≤ λ1(e2λ1t − 1) + λ2(e2λ2t − 1) ≤ βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With this, we can compute an estimate for the desired norms again using (a + b)2 ≤ 2a2 + 2b2 T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤2(ψ, ϕk)2 L2(Ωσ) 4βk β2 k − 4λk + 2(αN k )2λk 4βk β2 k − 4λk + 2 2βk β2 k − 4λk T T � 0 f 2 k(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Applying the estimate 1 (β2 k−4λk) ≤ 1 √ 2bk2√λk ≤ 1 2bk √λk we arrive at T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ (ψ, ϕk)2 L2(Ωσ) 4 √κ0 + (αN k )2λk 4 √κ0 + 2 √κ0 T∥fk∥2 L2(0,T) for κ0 ≤ κ1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let β2 k − 4λk < 0 and define γk = � 4λk − β2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we write the solution of (21) as ck(t) = αN k e− βk 2 t � cos(γkt) + βk 2γk sin(γkt) � + 1 γk (ψ, ϕk)L2(Ωσ)e− βk 2 t sin(γkt) + 1 γk t � 0 e βk 2 (s−t) sin(γk(t − s))f(s) ds 17 Since βk is positive since it is bounded from below by infx∈Ωσ σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence we can estimate T � 0 e−βkt dt ≤ 1 βk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we use the fact that sin2(γkt) ≤ 1 and cos2(γkt) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we derive by using (a+b)2 ≤ 2a2 + 2b2 and Cauchy Schwarz yet again the following estimate T � 0 λk(ck)2(t) dt ≤ λk(ψ, ϕk)2 L2(Ωσ) 4 βkγ2 k + 4(αN k )2λk � 1 βk + βk 4γ2 k � + 2λk βkγ2 k T T � 0 f 2 k(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the same way, we estimate T � 0 (c′ k)2(t) dt ≤ (ψ, ϕk)2 L2(Ωσ) 4 βkγ2 k �β2 k 2 + 2γ2 k � + 4(αN k )2 1 βk � γ2 k + β4 k 16γ2 k � + 2 βkγ2 k � 2β2 k 4 + 2γ2 k � T T � 0 f 2 k(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By inserting γ2 k = 4λk − β2 k we get T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ (ψ, ϕk)2 L2(Ωσ) � 36λk βk(4λk − β2 k) + 6βk 4λk − β2 k � + 4(αN k )2 �5λk βk + λkβk 4(4λk − β2 k) + β3 k 16(4λk − β2 k) � + � 2λk βk(4λk − β2 k) + βk 4λk − β2 k + 4 βk � T T � 0 f 2 k(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that the term λk 4λk−β2 k only shows up when β2 k < 4λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' It is bounded since βk is bounded by βmax, which is bounded by supx∈Ωσ σ(x), but λk is increasing monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore the maximum will be reached for smaller k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next we use the estimate 1 (β2 k−4λk) ≤ 1 2bk √λk to derive T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ (ψ, ϕk)2 L2(Ωσ) � 36λk βmin(4λk − β2 k) + 3 √κ0 � + 4(αN k )2 � 5λk βmin + λkβmax 4(4λk − β2 k) + β2 max 32√κ0 � + � 2λk βk(4λk − β2 k) + 1 8√κ0 + 4 βmin � T∥fk∥2 L2(0,T), 18 where βmax := maxk βk and βmin := mink βk > 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us consider the last case β2 k −4λk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then the solution to (21) is given by ck(t) = αN k (1 + βk 2 t)e− βk 2 t + (ψ, ϕk)L2(Ωσ)te− βk 2 t + t � 0 (t − s)e βk 2 (s−t)f(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With the estimate � t 0 t2e−βkt dt = 1 β3 k (2 − e−βkt(β2 kt2 + 2βkt + 2)) ≤ 2 β3 k we then compute by using (a + b)2 ≤ 2a2 + 2b2 and Cauchy Schwarz that T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ (ψ, ϕk)2 L2(Ωσ) � λk 8 β3 k + 12 βk � + 4(αN k )2 � λk 3 βk + 1 2βk � + �2λk β3 k + 3 βk � 2T T � 0 f 2 k(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' At last, we use 4λk = β2 k to derive T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ (ψ, ϕk)2 L2(Ωσ) 14 βmin + (αN k )2λk 5 4β3 min + 14 β3 min T∥fk∥2 L2(0,T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Adding all three cases will give the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using the above lemma we can finally prove the last statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let the assumptions 1 hold true, ja ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd), and A be the unique solu- tion of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then there exists positive constants cφ, cc φ, cψ, and cf such that the following inequality holds true |A|2 Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) ≤ cφ∥φ∥2 L2ǫ(Ω) + cc φT∥ curlx φ∥L2µ(Ω) + cψT∥ψ∥2 L2ǫ(Ω) + cf max{T, T 2}∥ja∥2 L2ǫ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (22) The constants cφ, cc φ, cψ, and cf are dependent on sup σ, σ−1 min, 1 √λ0 and max k∈N0 β2 k−4λk<0 (4 − β2 k λk )−1 which is bounded since βk is bounded by sup σ and the non-zero eigenvalues λk of curlx µ−1 curlx increase monotonically for k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The βk ∈ R+ are defined by σ such that (σϕk, ϕl)L2(Ωσ) = βkδkl for the fundamental system {ϕk} of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Again we split the domain into Ωσ = Ω ∩ supp(σ) and Ω0 := Ω\\Ωσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now we take a look at Ωσ and ˆQ := (0, T) × Ωσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To show the inequality we consider AN(t, x) = � k∈Z cN k (t)ϕk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 19 If we insert the representation into the energy norm we get (ε∂tAN,∂tAN)L2( ˆQ) + (µ−1 curlx AN, curlx AN)L2( ˆQ) = � k,j T � 0 ∂tcN k ∂tcN j dt � Ωσ (εϕk, ϕj) dx + T � 0 cN k cN j dt � Ωσ (µ−1 curlx ϕk, curlx ϕj) dx = � k∈N0 T � 0 (∂tcN k )2 dt + T � 0 λk(cN k )2 dt + � k∈Z\\N0 T � 0 (∂tcN k )2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (23) For k ∈ I we have to distinguish between the three cases β2 k − 4λk > 0, β2 k − 4λk = 0, β2 k − 4λk < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We know that 0 < λk and λk → ∞ for k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Since σ is bounded, we know that the βk = (σϕk, ϕk)L2(Ωσ) are also bounded and since σ is positive semi-definite βk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' So we consider the two ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The first is c′′N k (t) + βkc′N k (t) = fk(t), (24) c′N k (0) = (ψ, ϕk)L2(Ωσ), cN k (0) = αN k , and the second is c′′N k (t) + βkc′N k (t) + λkcN k (t) = fk(t), (25) c′N k (0) = (ψ, ϕk)L2(Ωσ), cN k (0) = αN k , for t ∈ (0, T), where fk(t) = (ja, ϕk)L2(Ωσ) and λk are the non-zero eigenvalues of the curlx curlx operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the first equation (24), where k ∈ Z\\N0, we compute the solution ck(t) = αN k + 1 βk (ψ, ϕk)L2(Ωσ)(1 − e−βkt) + 1 βk t � 0 (1 − eβk(s−t))fk(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now we need to estimate the L2-norm of c′N k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using (a + b)2 ≤ 2a2 + 2b2, we arrive with the Cauchy-Schwarz inequality at T � 0 (c′N k )2(t) dt ≤ 2(ψ, ϕk)2 L2(Ωσ) T � 0 (e−2βkt) dt + 2 T � 0 t � 0 e2βk(s−t) ds t � 0 f 2 k(s) ds dt ≤ (ψ, ϕk)2 L2(Ωσ) 1 − e−2βkT βk + 21 − e−2βkT 2βk T∥fk∥2 L2(0,T) ≤ (ψ, ϕk)2 L2(Ωσ) 1 βmin + 1 βmin T∥fk∥2 L2(0,T) (26) 20 For the second equation (25) and k ∈ N0 we consider the result of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2 � k∈I T � 0 λk(ck)2(t) dt + T � 0 (c′ k)2(t) dt ≤ � k∈I cψ(ψ, ϕk)2 L2(Ωσ) + (cκ αλk + cα)(αN k )2 + cf∥fk∥2 L2(0,T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (27) Since 1 βmin < cφ and 1 βmin < c1 f we can combine (27) and (26) in (23) to arrive at ∞ � k=1 T � 0 (∂tcN k )2 dt + T � 0 λk(cN k )2 dt ≤ ∞ � k=1 � c1 φ + c2 φλk � α2 k + cψ(ψ, ϕk)2 L2(Ωσ) (28) + (c1 f + c2 fλk)T∥fk∥2 L2(0,T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' As λk were eigenvalues of the curlx µ−1 curlx-opertor, we want to eliminate the κ from our right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Considering φ = �∞ k=1 αkϕk we compute ∞ � k=1 λk(αk)2 = ∞ � k=1 ∞ � l=1 αkαl � Ωσ λk(εϕk, ϕl) dx = ∞ � k=1 ∞ � l=1 αkαl � Ωσ (µ−1 curlx ϕk, curlx ϕl) dx = � Ωσ (µ−1 curlx φ, curlx φ) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using the same consideration for φ and ja, where fk = (ja, ϕk)L2(Ωσ), respectively in (28) we arrive at (ε∂tA, ∂tA)L2( ˆQ)+(µ−1 curlx A, curlx A)L2( ˆQ) ≤cψ∥ψ∥2 L2ǫ(Ωσ) + c1 φ∥φ∥2 L2ǫ(Ωσ) + c2 φ∥ curlx φ∥2 L2µ(Ωσ) + c1 fT∥ja∥2 L2ǫ( ˆQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the solution over Ω0 = Ω\\Ωσ and Q0 := (0, T) × Ω0, where σ ≡ 0, we get the following two ordinary equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The first equation is given by (20) for k ∈ Z\\N0 and the second is (19) for k ∈ N0 with the solution cN k (t) = 1 √λk (ψ, ϕk)L2(Ω0) sin( � λkt) + αN k cos( � λkt) + 1 √λk t � 0 fk(s) sin( � λk(t − s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then, we follow the same steps as above to compute the estimate (ε∂tA, ∂tA)L2(Q0)+(µ−1 curlx A, curlx A)L2(Q0) (29) ≤4T∥ψ∥2 L2ǫ(Ω0) + 4T∥ curlx φ∥2 L2µ(Ω0) + T 2∥ja∥2 L2ǫ(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that we get a T in our estimates by estimating T� 0 sin2(√λkt) dt ≤ T and the same integral with cos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Adding the inequality (29) brings us to the desired inequality (22) |A|Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) ≤ ˜cψT∥ψ∥2 L2ǫ(Ω) + c1 φ∥φ∥2 L2ǫ(Ω) + ˜c2 φT∥ curlx φ∥2 L2µ(Ω) + c1 f max{T, T 2}∥ja∥2 L2ǫ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 21 Remark Note that with the tools of Ladyzhenskaya [35, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 160] one can translate the results of the scalar wave equation to the vectorial wave equation using the same technique as the above theorem, see [21] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we would derive the estimate (ε∂tA, ∂tA)L2(Q) + (µ−1 curlxA, curlx A)L2(Q) (30) ≤4T∥ψ∥2 [L2(Ω)]d + 4T∥ curlx φ∥2 L2µ(Ω) + T 2∥ja∥2 [L2(Q)]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' for the solution of the variational formulation: Find A ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (Q) with A(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=') = φ such that −(ε∂tA, ∂tv)L2(Q) + (µ−1 curlx A, curlx v)L2(Q) = (ja, v)L2(Q) − (εψ, v(0, ·))L2(Ω) for all v ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',0 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 3 A space-time finite element method in a tensor product With Theorem 2 we have shown the unique solvability of the variational formulation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now we take a look at the discrete equivalent of the variational formulation for the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we will define the finite element spaces that we will be using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we derive the finite element discretization and analyze the resulting linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will derive a CFL condition that we will also see in numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 Space-time finite element spaces Before we compute any examples, we will recap the main ideas of the space-time discretization of the vectorial wave equation as was presented in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will use similar nomenclature to ease the comparison of the results of both papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the discretization in time we decompose the time interval (0, T) 0 = t0 < t1 < · · · < tNt−1 < tNt = T into N t subintervals τℓ = (tℓ−1, tℓ) ⊂ R, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We define the local mesh size of each element τl as ht,ℓ = tℓ − tℓ−1, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N t, whereas the maximal mesh size as ht := maxℓ=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',Nt ht,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the discretization in space we decompose the spatial domain Ω ⊂ Rd into N x elements ωi ⊂ Rd for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N x, satisfying Ω = Nx � i=1 ωi We define the local mesh size in ωi as hx,i = �� ωi 1dx �1/d , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N x, the maximal mesh size as hx := hx,max(T x ν ) := maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',Nx hx,i and the minimal mesh size as hx,min(T x ν ) := mini=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=',Nx hx,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The parameter ν ∈ N is the level of refinement and the spatial mesh the set T x := T x ν = {ωi}Nx i=1 at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For simplicity, we only consider triangles for d = 2 and tetrahedra for d = 3 as the spatial elements ωi ⊂ Rd, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the spatial mesh, we assume that we have a shape-regular, globally quasi- uniform sequence (T x ν )ν∈N of admissible spatial meshes T x ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 22 From the shape-regularity of the mesh sequence (T x ν )ν∈N we get a constant cF > 0 such that ∀ν ∈ N : ∀ω ∈ T x ν : sup x,y∈ω ∥x − y∥2 ≤ cF rω, (31) where ∥·∥2 is the Euclidean norm in Rd, and rω > 0 is the radius of the largest ball that can be inscribed in the element ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' From the global quasi-uniformity of the mesh sequence (T x ν )ν∈N we get a constant cG ≥ 1 such that ∀ν ∈ N: hx,max(T x ν ) hx,min(T x ν ) ≤ cG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With the discretization of the time interval and spatial domain in mind, we now define finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the test and ansatz space in time, we define the space of piecewise quadratic, continuous functions S2(T t α) := � vht ∈ C[0, T] | ∀ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N t} : vht|τℓ ∈ P2(τℓ) � = span{ϕ2 ℓ}N2 t ℓ=0 with the usual nodal basis functions ϕ2 ℓ, satisfying ϕ2 ℓ(tκ) = δℓκ for ℓ, κ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The space Pp(B) is the space of polynomials on a subset B ⊂ R of global degree at most p ∈ N0, and δℓκ is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then, we introduce the subspaces S2 0,(T t α) := S2(T t α) ∩ H1 0,(0, T) = span{ϕ2 ℓ}N2 t ℓ=1, S2 ,0(T t α) := S2(T t α) ∩ H1 ,0(0, T) = span{ϕ2 ℓ}N2 t −1 ℓ=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In space, we introduce the vector-valued finite element spaces of Nédélec and Raviart– Thomas finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For a spatial element ω ∈ T x ν , we define the polynomial spaces RT 0(ω) := � v ∈ [P 1(ω)]d | ∀x ∈ ω : v(x) = a + bx with a ∈ Rd, b ∈ R � and N 0 I (ω) := � v ∈ [P 1(ω)]2 | ∀(x1, x2) ∈ ω : v(x1, x2) = a + b · (−x2, x1)⊤ with a ∈ R2, b ∈ R � for d = 2, and N 0 I (ω) := � v ∈ [P 1(ω)]3 | ∀x ∈ ω : v(x) = a + b × x with a ∈ R3, b ∈ R3� for d = 3, see [4, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With this notation, we define the space of the lowest-order Raviart–Thomas finite element space and the lowest-order Nédélec finite element space of the first kind RT 0(T x ν ) := � vhx ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) | ∀ω ∈ T x ν : vhx|ω ∈ RT 0(ω) � = span{ψRT k }NRT x k=1 , N 0 I (T x ν ) := � vhx ∈ H(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) | ∀ω ∈ T x ν : vhx|ω ∈ N 0 I (ω) � , see [4, 6] for more details on these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Further, we consider the subspace N 0 I,0(T x ν ) := N 0 I (T x ν ) ∩ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Ω) = span{ψN k }NN x k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 23 Last, the temporal and spatial meshes T t α = {τℓ}Nt ℓ=1 and T x ν = {ωi}Nx i=1 lead to a decom- position Q = [0, T] × Ω = Nt � ℓ=1 τℓ × Nx � i=1 ωi of the space-time cylinder Q ⊂ Rd+1 with N t ·N x space-time elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore T t α ×T x ν is a space-time tensor product mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To this space-time mesh, we relate space-time finite element spaces of tensor-product type using ⊗ as the Hilbert tensor-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' These spaces are S2 0,(T t α) ⊗ N 0 I,0(T x ν ) and S2 ,0(T t α) ⊗ N 0 I,0(T x ν ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (32) Then, any function Ah ∈ S2 0,(T t α) ⊗ N 0 I,0(T x ν ) admits the representation Ah(t, x) = N2 t � κ=1 NN x � k=1 Aκ kϕ2 κ(t)ψN k (x) (33) for (t, x) ∈ Q with coefficients Aκ k ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Further, for a given function f ∈ L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd), we introduce the L2(Q) projection ΠRT ,1 h : L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd) → S1(T t α) ⊗ RT 0(T x ν ) to find ΠRT ,1 h f ∈ S1(T t α) ⊗ RT 0(T x ν ) such that (ΠRT ,1 h f, wh)L2(Q) = (f, wh)L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (34) for all wh ∈ S1(T t α) ⊗ RT 0(T x ν ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' With these definitions in mind, we can now formulate a conforming finite element approach for the variational formulation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We use as trial space S2(T t α) ⊗ N 0 I,0(T x ν ) and S2 ,0(T t α) ⊗ N 0 I,0(T x ν ) as test space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we consider the discrete variational formulation: Find Ah ∈ S2(T t α) ⊗ N 0 I,0(T x ν ), with Ah(0, x) = φ, such that − (ε∂tAh, ∂tvh)L2(Q) + (σ∂tAh, vh)L2(Q) + � µ−1curlxAh, curlxvh � L2(Q) (35) = � ΠRT ,1 h ja, vh � L2(Q) − � εψ, vh(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=') � L2(Ω) for all vh ∈ S2 ,0(T t α) ⊗ N 0 I,0(T x ν ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2 The finite element discretization Next, we follow the same steps as [20] to derive the finite element discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let, for a moment, the initial condition φ be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we insert (33) into the discrete variational formulation (35) the integrals split into temporal and spatial matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, the discrete variational formulation (35) is equivalent to the linear system (−Att ⊗ Mx + At ⊗ Mσ x + Mt ⊗ Axx)A = J (36) with the temporal matrices Att[ℓ, κ] = (∂tϕ2 κ, ∂tϕ2 ℓ)L2(0,T), Mt[ℓ, κ] = (ϕ2 κ, ϕ2 ℓ)L2(0,T), (37a) At[ℓ, κ] = (∂tϕ2 κ, ϕ2 ℓ)L2(0,T) (37b) 24 for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t − 1, κ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t and the spatial matrices Ax[l, k] = (µ−1 curlx ψN k , curlx ψN l )L2(Ω), Mx[l, k] = (ǫψN k , ψN l )L2(Ω), (38a) Mσ x [l, k] = (σψN k , ψN l )L2(Ω) (38b) for k, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N N x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The degrees of freedom are ordered such that the vector of coefficients in (33) reads as A = (A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , AN2 t )⊤ ∈ RN2 t NN x (39) with Aκ = (Aκ 1, Aκ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , Aκ NN x )⊤ ∈ RNN x for κ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The right-hand side in (36) is given in the same way by J = (f 0, f 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , f N2 t −1)⊤ ∈ RN2 t NN x , where we define fℓ = (f ℓ 1, f ℓ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , f ℓ NN x )⊤ ∈ RNN x for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t − 1 with f ℓ l = (ΠRT ,1 h j, ϕ1 ℓψN l )L2(Q) − ϕ1 ℓ(0) � εψ, ψN l � L2(Ω) (40) for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t − 1, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N N x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Remark If we have non-zero initial condition φ(x), then we consider a continuous extension in time ˜φ that is zero on all other degrees of freedom in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, for our application we use the representation (33) to derive for t = 0 ˜φh(0, x) = NN x � k=1 A0 kψN k (x) ≈ ˜φ(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then, using the ordering of the degrees of freedom from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1,´ we end up with the system (−Att ⊗ Mx + At ⊗ Mσ x + Mt ⊗ Axx)A = J − AIni, (41) where AIni := (A0 Ini, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , AN2 t −1 Ini ), Aℓ Ini := (Aℓ Ini;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , Aℓ Ini;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='NN x ), Aℓ Ini;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='l := NN x � k=1 (−Att[ℓ, 0]Mx[l, k] + At[ℓ, 0]Mσ x [l, k] + Mt[ℓ, 0]Axx[l, k])A0 k for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N 2 t − 1 and l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , N N x with the matrices defined in (38) and (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3 The CFL Condition If we solve the equation (36) for simple examples, we see conditional stability in our results which hints at a CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To compute this CFL condition for functions with second- order elements in time use a similar tactic as [31] did for linear elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To simplify the calculation we assume that we have an equidistant discretization in time with step size ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let λi, i ∈ N0, be the eigenvalues of the curlµ−1curl operator in the weighted L2 ǫ(Ω)-norm from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To analyze the stability of the discrete system we choose initial data φ = 0, ψ = 0 as well as the right-hand side ja = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First let σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that, since σ acts like a stabilizer to our system, the case σ ≡ 0 is the hardest case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now, we consider the basis ansatz from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 for A ∈ Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0, and write AN = N � k=−N ˜Ak(t)ϕk(x) Then we can rewrite the variational formulation (14) into: Find ˜Ak ∈ H1 0,(0, T) such that −(∂t ˜Ak, ∂tv)(0,T) + λk( ˜Ak, v)(0,T) = 0 (42) for all v ∈ H1 ,0(0, T) and all eigenvalues λk > 0, k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' On the other hand, for k ∈ Z\\N0, we end up with the formulation: Find ˜Ak ∈ H1 0,(0, T) such that (∂t ˜Ak, ∂tv)(0,T) = 0 for all v ∈ H1 ,0(0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This is equivalent to the Laplace equation and does not add to our instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To derive the appropriate CFL condition, we need to analyze the stability of (42) in the discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using again the ansatz and test spaces S2 0,(T t) and S2 ,0(T t) we get the discrete formulation: Find ˜Ah k ∈ S2 0,(T t) such that −(∂t ˜Ah k, ∂tvh)(0,T) + λk( ˜Ah k, vh)(0,T) = 0 for all vh ∈ S2 ,0(T t) and all eigenvalues λk, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This is equivalent to analyzing the linear system (−Att + λkMt) ˜A = 0 (43) for its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In equation (43) we have the temporal matrices as defined in (37) and the solution vector ˜A = ( ˜Ak(t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' , ˜Ak(T))T ∈ RN2 t which are the coefficients of the basis representation ˜Ah k(t) = �N2 t κ=1 ˜Ak(tκ)ϕ2 κ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Let us write λ = λk and h = ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the second- order ansatz and test functions, we compute the element matrices Me t = h 30 \uf8eb \uf8ed 4 2 −1 2 16 2 −1 2 4 \uf8f6 \uf8f8 , Ae tt = 1 3h \uf8eb \uf8ed 7 −8 1 −8 16 −8 1 −8 7 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 26 This results in the system matrix Ksys = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 8 3h + λh 15 − 1 3h − λh 30 −16 3h + 8λh 15 8 3h + λh 15 8 3h + λh 15 − 14 3h + 4λh 15 8 3h + λh 15 − 1 3h − λh 30 8 3h + λh 15 −16 3h + 8λh 15 8 3h + λh 15 − 1 3h − λh 30 8 3h + λh 15 − 14 3h + 4λh 15 8 3h + λh 15 − 1 3h + λh 30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' By multiplying with 3h we get the following formula � 8 + λh2 5 −1 − λh2 10 −16 + 8λh2 5 8 + λh2 5 � �u2k−1 u2k � = � −8 − λh2 5 14 − 4λh2 5 0 −8 − λh2 5 � �u2k−3 u2k−2 � + � 0 1 + λh3 10 0 0 � �u2k−5 u2k−4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This can be understood as a two-step method Azk = B1zk−1 + B2zk−2 for the vector zk = (u2k−1, u2k)T with A := � 8 + λh2 5 −1 − λh2 10 −16 + 8λh2 5 8 + λh2 5 � , B1 := � −8 − λh2 5 14 − 4λh2 5 0 −8 − λh2 5 � , B2 := � 0 1 + λh3 10 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Further, we can rewrite the two-step method as the system Yk = AsysYk−1 (45) with Yk = (u2k−3, u2k−2, u2k−1, u2k)T and Asys := � 0 I A−1B2 A−1B1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we solve (45) by iterate in k and arrive at the formula Yk = Ak−1 sys Y1 for the solution Yk with the initial values �u1 u2 � = A−1 � 7 − 2λh2 5 −8 − λh2 5 � u0, where u−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Therefore, if we want our system to be stable, we need the real parts of the eigenvalues of Asys to be less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence we analyze the eigenvalues of Asys and see that two of the eigenvalues λA 1 and λA 2 are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The other two are given by λA 3,4 = −2a2 − bd ± � b(2c + d)(4a2 − 2bc + bd) 2(a2 − bc) 27 where a = 8 + λh2 5 , b = −16 + 8λh2 5 , c = −1 − λh2 10 , d = 14 − 4λh2 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The absolute value of real part of λa 3,4, namely |ℜ(λA 3,4)| is smaller than one if λih2 t ≤ 60 and λih2 t /∈ [10, 12], for i ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' If we introduce σ(x) > 0, x ∈ Ω large enough the small instability region λih2 t ∈ [10, 12] would vanish and we are left with the condition λih2 t ≤ 60, for i ∈ N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In computational examples, we often observe only the bound λh2 t ≤ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' However, there might be unstable examples where λh2 t ∈ [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note that these estimates can also be used for the scalar wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To use these insights in computational examples, we need to estimate the eigenvalues λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For this purpose, we use an inverse inequality for the curlxµ−1curlx operator in the weighted L2 ǫ(Ω)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For lowest order Nédélec elements, we have the inverse inequality ∥ curlx uh∥2 L2(T x) ≤ cIh−2 max∥uh∥2 L2(T x) (46) for all uh ∈ span{φN I 0 E }E∈E, where cI := �18c2 F π n = 2, 80c4 F ( 9 16π2 )2/3 n = 3, hmax := max ωl∈T x(|τl|−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For the proof consider [19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2] or [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The proof is done by computing each norm on each element and showing the inequality there, then summing everything up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For complete- ness, we quickly state where the constants come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The constant cF is the shape regularity constant defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The coefficient cIh−2 max comes from estimating 18λmax(JlJT l ) 2∆l h−2 l for d = 2 and for d = 3 estimating the term 1604λmax(JlJT l )2 3(6∆l)2 , where Jl = �∂Fl,i ∂ˆxj (ˆxj) � 1≤i,j,≤d is the derivative of transformation Fl : ˆω → ωl which maps the reference element ˆτ to the current element ωl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Now, we apply the inverse inequality to estimate the eigenvalues λi, i ∈ N0, from Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We rewrite the inverse inequality (46) in the weighted norms to get (µ−1 curlx uh, curlx uh)L2(T x) ≤ cI(µminǫmin)−1h−2 max∥uh∥2 L2ǫ(T x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we derive the following CFL condition for the vectorial wave equation cI h−2 x ≤ 10 h−2 t , and therefore ht ≤ � 10 cI hx, (47) 28 where ht and hx are the respective maximal temporal and spatial step sizes from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In case σ is big enough, we even arrive at the CFL condition ht ≤ � 60 cI hx, (48) If we consider Ω = (0, 1)2 and use a shape regular triangulation with isosceles rectangular triangles as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 1 then we compute, by estimating the eigenvalues of JlJT l directly, that cI = 18 as in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Hence, in this case, we arrive at the CFL conditions ht < � 60 18hx ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='825741858 hx (49) and the stricter condition ht < � 10 18hx ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='74535599 hx (50) Figure 1: Triangulation with isosceles rectangular triangles 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4 Numerical Results We will first take a look at the results for σ = 0 to have a baseline from which we can judge other results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To compute the linear system (36), we first have to assemble the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The pro- jection ΠRT ,1 h ja of the right-hand side ja in (34) are calculated by using-high-order quadrature rules for the integrals, see [20] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The calculation of all spatial and temporal matrices (38) is done with the help of the finite element library NGSolve, see www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='ngsolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='org and [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Finally, the linear system (36) is solved by the sparse direct solver UMFPACK 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5 Testing the convergence for σ ≡ 0 We will first take a look at examples for σ ≡ 0 to investigate the CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For our examples in this section we consider the domain Q = (0, 2) × (0, 1)2, with T = 2 and Ω = [0, 1] × [0, 1], and ǫ ≡ �1 0 0 1 � and µ ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We will consider two constructed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The first is given by A1(t, x1, x2) = t3x1(1 − x1)x2(1 − x2) � x2 −x1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (51) 29 If we insert A1 into the vectorial wave equation, we compute j1(t, x1, x2) := � 2x1(1 − x1)x2(1 − x2)x2 (2x1(1 − x1)x2(1 − x2)(−x1) � + t2 � x1(x1(5 − 12x2) + 10x2 − 4) −x2(−2x1(6x2 − 5) + 5x2 − 4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we solve the discrete system (36) for the above configuration and take a look at the convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For that purpose, we compute the experimental order of convergence (EOC) with EOC = ln(errL−1) − ln(errL) ln(hL−1) − ln(hL) , where errL is the L2-error, or the error in the Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1-half norm respectively, at level L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Additionally we use bisection in the refinement, hence ln(hL−1) − ln(hL) = ln(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' If we then solve the discrete linear system described in (36) for second-order elements in time and lowest order Nédélec elements in space, we compute L hx ht #fdofs ∥A − Ah∥L2(Q) EOC |A − Ah|Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) EOC 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5000 80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='33706e-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='05563e-01 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2500 896 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='16399e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='12075e-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='53 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1250 8192 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='02257e-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='00680e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='07 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0625 69632 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='23691e-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='94712e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='03 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0312 573440 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='06510e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='46265e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='01 Table 1: Error table for the Galerkin–Petrov FEM (35) for the unit square Ω and T = 2 and the solution A1 in (51) using a uniform refinement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Here we see second order convergence in the ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='∥L2(Q)-norm and first order convergence in the |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='|Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q)-halfnorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This is the same result as we would get for linear elements in time and ΠRT ,1 h ja as the projection of the right-hand side, see [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Next, we want to test the sharpness of the CFL condition (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To that purpose, we use a more complicated artificial solution A1, which was also used in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We define A2(t, x1, x2) = �−5t2x2(1 − x2) t2x1(1 − x1) � + t3 �sin(πx1)x2(1 − x2) 0 � (52) for (t, x1, x2) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The function A fulfills the homogeneous boundary condition γtA = 0 and has homogeneous initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The related right-hand side j is given by j2(t, x1, x2) = �−10(t2 − x2 2 + x2) 2(t2 − x2 1 + x1) � + �2t3 sin(πx1) + 6t sin(πx1)x2(1 − x2) πt3(1 − 2x2) cos(πx1) � for (t, x1, x2) ∈ Q and divxja ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' To show the behavior of the CFL condition we take a look at the final time T = √ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For these values, we see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', in the last row and second to last column that the ratio ht hx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0442 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='82352941 is below the CFL condition in both Tables 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 30 hx ht 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='6450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1768 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='54e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='54e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='54e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='54e-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='54e-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0884 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='90e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='04e+00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='85e-02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='85e-02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='85e-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0442 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='29e-02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='87e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='95e+05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='21e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='21e-02 Table 2: Errors in ∥·∥L2(Q) for the Galerkin–Petrov FEM (35) for the unit square Ω and T = √ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4 and the solution A2 in (52) using a uniform refinement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' hx ht 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='6450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1768 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='29e+00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='28e+00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='28e+00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='28e+00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='28e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0884 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='15e+00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='25e+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='14e+00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='14e+00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='14e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0442 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='08e+00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='57e+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='00e+07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='07e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='07e+00 Table 3: Errors in |·|Hcurl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1(Q) for the Galerkin–Petrov FEM (35) for the unit square Ω and T = √ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='4 and the solution A2 in (52) using a uniform refinement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='6 The convergence for σ ̸≡ 0 Now, we consider Q = (0, 2) × [0, 1]2 again with ǫ ≡ �1 0 0 1 � , µ ≡ 1 and the constructed solution A3(t, x) = t2x1(1 − x1)x2(1 − x2) � x2 −x1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' (53) However, we choose σ = 1 over conv{(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='35), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='65), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='5)} and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we compute ja = ∂ttA3 + σ∂tA3 + curlx curlx A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' When we solve (36) we get the following L2-error table for piecewise quadratic functions in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' hx ht 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='2500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='135e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='135e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='135e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='135e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='135e-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='1250 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='986e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='981e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='981e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='981e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='981e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='0625 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='501e-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='490e-01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='462e-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='462e-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='462e-03 Table 4: Errors in ∥·∥L2(Q) for the Galerkin–Petrov FEM (35) for the unit square Ω and T = 2 and the solution A3 in (53) using a uniform refinement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Again we see the CFL condition as in the case of σ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Indeed, in the last line we see from ht/hx = 4 to ht/hx = 2 an increase in the error which stops after the CFL condition (49) is full filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Note, that the stricter CFL condition (50) does not apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Remark In this case of ε = ε0, µ−1 = µ−1 0 we get ht < � ε0µ0 2π 3c2 F hx ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='165410−8 hx (54) 31 since µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='256637 10−6, ε0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='854188 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In this case, since µ−1 and σ are at least of order 1018 greater, it is reasonable to go to the Eddy Current problem instead of solving (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This will be the topic of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 4 Conclusion In this paper, we have investigated the unique solvability for the variational formulation of the vectorial wave equation considering Ohm’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' First, we have taken a look at the trial and test spaces and showed properties of the functional spaces that we needed to prove the solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Then we proved unique solvability for the variational formula (14), where ja ∈ L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' For electromagnetic problems, the variational formula (14) applies to a variety of electromagnetic examples, since it is posed for Ohm’s law and anisotropic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Having proven the unique solvability, we turned to computational examples in the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We used piecewise quadratic ansatz functions in time and lowest order Nédélec elements in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Here, as in the case of linear ansatz functions in time, [20], we realized that there is a CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We calculated the CFL condition and gave examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Using the reasoning from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='3, it is also possible to derive a CFL condition for other higher-order elements in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We thus learn that simply increasing the order of finite elements in time does not lead to an unconditionally stable method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' In the case of the tensor product structure, we can always expect a CFL condition for the space-time discretization of the vectorial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' A possible solution is the use of the modified Hilbert transform as performed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The results of this work form the basis for more complicated electromagnetic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' We learned the main difficulties of the vectorial wave equation and what it implies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' This will be a good starting point for future work on the calculation of eddy current problems and further applied calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' References [1] Jin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' The finite element method in electromagnetics, second ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Wiley-Interscience [John Wiley & Sons], New York, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [2] Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', and Rui, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' A second-order space-time accurate scheme for Maxwell’s equa- tions in a Cole–Cole dispersive medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Engineering with Computers 38, 6 (2022), 5153– 5172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [3] Xie, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 425 (2021), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 109896, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [4] Ern, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=', and Guermond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Finite elements I—Approximation and interpolation, vol.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Teil 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Mathematische Leitfäden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [Mathematical Textbooks].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Teubner, Stuttgart, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Grundlagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [Foundations].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [29] Cessenat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' Mathematical Methods in Electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' WORLD SCIENTIFIC, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' [30] Crawford, Z.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE1T4oBgHgl3EQftwWv/content/2301.03381v1.pdf'} diff --git a/SNE3T4oBgHgl3EQfDglb/vector_store/index.pkl b/SNE3T4oBgHgl3EQfDglb/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ac91303fdeccbf741ee0c476613305ef4ebc3ef3 --- /dev/null +++ b/SNE3T4oBgHgl3EQfDglb/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bec815f338336986c485aab24ac3569fd8182f800e22b684f103ca8c8cbf618e +size 124870 diff --git a/V9E2T4oBgHgl3EQfuAiM/content/tmp_files/2301.04076v1.pdf.txt b/V9E2T4oBgHgl3EQfuAiM/content/tmp_files/2301.04076v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b3f8a649c3f20302bf3ed37293e28eabcf8a08f --- /dev/null +++ b/V9E2T4oBgHgl3EQfuAiM/content/tmp_files/2301.04076v1.pdf.txt @@ -0,0 +1,1642 @@ +Electronic Born–Oppenheimer Approximation in Nuclear-Electronic +Orbital Dynamics +Tao E. Lia) and Sharon Hammes-Schifferb) +Department of Chemistry, Yale University, New Haven, Connecticut, 06520, +USA +Within the nuclear-electronic orbital (NEO) framework, the real-time time-dependent density functional +theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics. In this +approach, the electrons and quantum nuclei are propagated in time on the same footing. A relatively small +time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of +long-time nuclear quantum dynamics. Herein, the electronic Born–Oppenheimer (BO) approximation within +the NEO framework is presented. In this approach, the electronic density is quenched to the ground state +at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic +ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density. +Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of- +magnitude larger time step, thus greatly reducing the computational cost. Moreover, invoking the electronic +BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical +RT-NEO-TDDFT simulations of polaritons under vibrational strong coupling even for small Rabi splitting, +instead yielding a stable, symmetric Rabi splitting. For intramolecular proton transfer in malonaldehyde, +both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the +real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the foundation for a wide +range of chemical and biological applications. +I. +INTRODUCTION +Simulating nuclear quantum dynamics is important +for understanding a wide range of chemical and biolog- +ical processes, such as vibrationally excited chemistry +and proton transfer reactions. +A variety of theoreti- +cal methods for simulating nuclear quantum dynamics +have been developed, including the linearized semiclas- +sical initial value representation (LSC-IVR) method,1,2 +ring polymer molecular dynamics,3 the multiconfigura- +tional time-dependent Hartree (MCTDH) method,4 and +the exact factorization approach.5 The nuclear-electronic +orbital (NEO) method6,7 is another approach for simu- +lating nuclear quantum dynamics.8–14 Within the NEO +framework, both electrons and selected nuclei, usually +the protons, are described by first-principles methods +such as density functional theory (DFT), while the re- +maining heavy nuclei are treated classically. As an exten- +sion of conventional electronic structure theory, the NEO +method can be combined with conventional nonadiabatic +dynamics approaches, such as Ehrenfest dynamics15,16 +and trajectory surface hopping,17 to simulate nonadi- +abatic dynamics on vibronic surfaces rather than elec- +tronic surfaces. +A promising strategy for simulating nuclear quan- +tum dynamics within the NEO framework is real-time +NEO time-dependent density functional theory (RT- +NEO-TDDFT).11 In this approach, the quantum dy- +namics of the electrons and quantum protons are prop- +agated in the time domain. +As a real-time version of +a)Electronic mail: tao.li@yale.edu +b)Electronic mail: sharon.hammes-schiffer@yale.edu +linear-response multicomponent TDDFT,18–21 RT-NEO- +TDDFT can capture the linear-response electronic and +protonic excited-state spectra by Fourier transforming +the real-time dipole signals. +Moreover, this approach +can also directly capture the coupled electron-proton +nonadiabatic dynamics. These calculations can be per- +formed with fixed classical nuclei, or the classical nu- +clei can be propagated on the mean-field potential en- +ergy surface associated with the nonequilibrium elec- +trons and quantum protons using the RT-NEO-Ehrenfest +dynamics approach,12,13 which captures the full nona- +diabatic dynamics of the electrons, quantum protons, +and classical nuclei. +Another important extension of +RT-NEO-TDDFT is semiclassical RT-NEO-TDDFT for +polaritons,14 where the coupled dynamics between the +NEO molecular subsystem and classical photon modes +are propagated in the time domain. +When the light- +matter coupling is large enough, this approach can de- +scribe strong light-matter interactions under both elec- +tronic and vibrational strong couplings, thus potentially +providing a powerful scheme for simulating polariton +chemistry.22–25 +For these RT-NEO dynamics approaches, the electrons +and quantum nuclei are propagated on the same footing. +As a result, a very small time step (≤ 0.01 fs) is re- +quired for the time propagation due to the fast electronic +dynamics.11–14 The requirement of a small time step may +prohibit the study of long-time nuclear dynamics. When +nuclear quantum dynamics on the electronic ground state +is considered, such a small time step can be avoided by +invoking the electronic Born–Oppenheimer (BO) approx- +imation between the electrons and both quantum and +classical nuclei. In this case, the electrons are quenched +to the ground state for each time step of the dynam- +arXiv:2301.04076v1 [physics.chem-ph] 10 Jan 2023 + +2 +ics, i.e., for each classical nuclear geometry and corre- +sponding nonequilibrium quantum proton density. This +electronic BO approximation is different from the con- +ventional BO approximation used in ab initio molecular +dynamics. In the conventional BO approximation, the +electronic ground state is determined solely by the ge- +ometry of the classical nuclei, whereas in the NEO elec- +tronic BO approximation, the electronic ground state +is determined by the nonequilibrium proton density as +well as the geometry of the classical nuclei. For a given +classical nuclear geometry, the nonequilibrium proton +density in a BO-RT-NEO dynamics simulation can be +a mixture of vibrational states, thereby also influenc- +ing the electronic ground state. The BO-RT-NEO dy- +namics approach is also different from the constrained +NEO (cNEO) dynamics method developed by Yang and +coworkers.26 The cNEO approach enables the inclusion of +anharmonicity in vibrational spectra in a computation- +ally efficient manner.9 In addition to this capability, the +BO-RT-NEO approach also provides real-time dynamics +associated with nonequilibrium proton densities. +In this manuscript, we show that invoking the BO ap- +proximation for RT-NEO dynamics allows an order-of- +magnitude larger time step to be used during the time +propagation compared to RT-NEO dynamics without the +BO approximation. Thus, the BO approximation greatly +reduces the computational cost, while producing nearly +identical dynamics for the electronically adiabatic sys- +tems studied. Moreover, invoking the electronic BO ap- +proximation also overcomes a serious drawback in semi- +classical RT-NEO dynamics for polaritons, namely the +previously observed unphysical asymmetirc Rabi split- +ting under vibrational strong coupling even when the +Rabi splitting is small.14 Herein we show that the BO- +RT-NEO method produces symmetric Rabi splittings un- +der these conditions. Lastly, using intramolecular proton +transfer in malonaldehyde27–31 as an example, we show +that BO-RT-NEO-Ehrenfest dynamics can capture the +proton delocalization associated with these types of pro- +ton transfer processes. +II. +THEORY +The equations of motion for RT-NEO dynamics, semi- +classical RT-NEO dynamics for polaritons, and RT-NEO- +Ehrenfest dynamics are summarized in Table I. A more +detailed review of these approaches is given below. +A. +RT-NEO dynamics with fixed classical nuclei +Within the framework of the RT-NEO approach with +fixed classical nuclei,11 the dynamics of both electrons +(assuming closed-shell) and quantum nuclei are propa- +gated by the following von Neumann equations +i ∂ +∂tPe(t) = [Fe(t), Pe(t)] +(1a) +i ∂ +∂tPn(t) = [Fn(t), Pn(t)] +(1b) +Here, the density matrices are defined as Pe = CeCe† +and Pn = CnCn†, where Ce (or Cn) denotes the coeffi- +cient vector of the electronic (or nuclear) wavefunction in +the orthogonal atomic orbital basis. The transformation +between the density matrices in the orthogonal (labeled +without prime) and non-orthogonal (labeled with prime) +atomic basis is governed by +Pe = [Se]1/2Pe′[Se]1/2 +(2) +where Se is the electronic overlap matrix. Similarly, in +Eq. (1a), the Kohn–Sham matrices Fe in the orthogonal +atomic orbital basis are defined as +Fe = [Se]−1/2Fe′[Se]−1/2 +(3) +Here, Fe′ denotes the Kohn–Sham matrix for the elec- +trons in the non-orthogonal atomic orbital basis. +The +analogs to Eqs. (2) and (3) for the quantum nuclei are +identical with the superscript e replaced by n. According +to NEO-DFT,32,33 Fe′ and Fn′ are defined as +Fe′(t) = He′ +core + Jee′(Pe′(t)) + Ve′ +xc(Pe′(t)) +− Jen′(Pn′(t)) + Ven′ +c (Pe′(t), Pn′(t)) +(4a) +Fn′(t) = Hn′ +core + Jnn′(Pn′(t)) + Vn′ +xc(Pn′(t)) +− Jne′(Pe′(t)) + Vne′ +c (Pn′(t), Pe′(t)) +(4b) +In Eq. (4), He′ +core (or Hn′ +core) denotes the core Hamilto- +nian, which includes the kinetic energy and the Coulomb +interaction between the electrons (or quantum nuclei) +and the classical nuclei; +Jee′ (or Jnn′) denotes the +Coulomb interactions among the electrons (or quantum +nuclei); Ve′ +xc (or Vn′ +xc) denotes the exchange-correlation +potential for the electrons (or quantum nuclei); Jne′ (or +Jen′) denotes the Coulomb interaction between the elec- +trons and quantum nuclei; and Ven′ +c +(or Ven′ +c ) denotes the +correlation potential between the electrons and quantum +nuclei. In the Hartree–Fock limit, Vne′ +c += Ven′ +c += 0, and +Ve′ +xc (or Vn′ +xc) becomes the Hartree–Fock exchange term +for electrons (or quantum nuclei). Similar to most RT- +TDDFT implementations,34,35 the adiabatic approxima- +tion is invoked, and the above functionals depend locally +on time. +B. +Semiclassical RT-NEO dynamics for polaritons +Beyond RT-NEO with fixed classical nuclei, the semi- +classical RT-NEO approach for polaritons14 can describe + +3 +TABLE I. Equations of motion for different types of NEO dynamics. +RT-NEO11a +Semiclassical RT-NEO for polaritons14 RT-NEO-Ehrenfest12,13 +electrons/non-BO i ∂ +∂tPe = [Fe, Pe] +i ∂ +∂tPe = +� +Fe + � +k,λ εk,λqk,λˆµe +λ, Pe� +i ∂ +∂tPe = [Fe, Pe] +electrons/BO +Pe′ = SCF[· · · ] +Pe′ = SCF[ℜ(Pn′), {RI}] +Pe′ = SCF[· · · ] +quantum nuclei +i ∂ +∂tPn = [Fn, Pn] i ∂ +∂tPn = +� +Fn + � +k,λ εk,λqk,λˆµn +λ, Pn� +i ∂ +∂tPn = [Fn, Pn] +classical nuclei +fixed +fixed +MI ¨RI = −∇IE +photons +N/A +¨qk,λ = −ω2 +k,λqk,λ − εk,λµλ − γcpk,λ +N/A +a Here, RT-NEO refers to RT-NEO dynamics with fixed classical nuclei. +strong light-matter interactions between cavity photon +modes and molecules36–42. +Within this approach, the +cavity photons are propagated classically: +˙qk,λ = pk,λ +(5a) +˙pk,λ = −ω2 +k,λqk,λ − εk,λµλ − γcpk,λ +(5b) +Here, qk,λ, pk,λ, and ωk,λ denote the position, momen- +tum, and frequency of the cavity photon mode charac- +terized by the wave vector k = |k| and polarization unit +vector ξλ, where k · ξλ = 0 (e.g., if the k direction is +z, λ can be x or y); εk,λ denotes the light-matter cou- +pling; µλ denotes the dipole moment of the molecule +along the direction of ξλ; γc denotes the cavity loss rate. +In practice, when calculating µλ(t), we subtract the per- +manent dipole contribution, i.e., µλ(t) = 2Tr [Pe(t)ˆµe +λ] + +Tr [Pn(t)ˆµn +λ] − 2Tr [Pe(0)ˆµe +λ] − Tr [Pn(0)ˆµn +λ], so at time +t = 0, qk,λ = pk,λ = 0 always represents the photonic +ground state.14 Here, ˆµe +λ (or ˆµn +λ) denotes the dipole ma- +trix of the electrons (or quantum nuclei) projected along +the direction of ξλ in the orthogonal atomic orbital basis, +and the prefactor 2 in the electronic dipole moment is in- +cluded because of the restricted Kohn–Sham calculation. +Due to the interaction with cavity photons, the dy- +namics of the electrons and quantum nuclei become +i ∂ +∂tPe(t) = +� +�Fe(t) + +� +k,λ +εk,λqk,λˆµe +λ, Pe(t) +� +� +(6a) +i ∂ +∂tPn(t) = +� +�Fn(t) + +� +k,λ +εk,λqk,λˆµn +λ, Pn(t) +� +� +(6b) +In Eqs. (5) and (6), although many cavity modes indexed +by k, λ have been considered, in the simulation below, for +simplicity, we will take into account only one cavity mode +polarized along the x-direction. +Similar to RT-NEO dynamics with fixed classical nu- +clei, here the classical nuclei are also assumed to be fixed. +Eqs. (5) and (6) can be further combined with a mean- +field propagation of the classical nuclei via Ehrenfest dy- +namics, thus providing a full dynamics scheme for po- +lariton chemistry applications. Because this extension is +beyond the scope of this manuscript, we will report this +development elsewhere. +C. +RT-NEO-Ehrenfest dynamics +The RT-NEO-Ehrenfest dynamics12,13 combines the +real-time dynamics of the electrons and quantum nuclei +and the mean-field motion of the classical nuclei. In this +approach, the electrons and quantum nuclei are propa- +gated according to Eq. +(1), and the remaining nuclei +are propagated classically by the following equations of +motion: +˙RI = PI +MI +(7a) +˙PI = −∇IE[Pe′(t), Pn′(t), {RI}] +(7b) +Here, RI, PI, and MI denote the position, momentum, +and mass of the I-th classical nucleus; the total energy +of the molecular system E[Pe′(t), Pn′(t), {RI}] is a func- +tion of the nonequilibrium densities of the electrons and +quantum nuclei (Pe′(t) and Pn′(t)) as well as the po- +sitions of all classical nuclei RI. Ref. 12 provides the +explicit form of the Ehrenfest gradients ∇IE. +When using RT-NEO-Ehrenfest dynamics for describ- +ing proton transfer, a reasonable choice for treating the +proton basis function centers is to use a large proton +basis set including several different fixed proton basis +(FPB) function centers spanning the region sampled by +the transferring proton.11 Another choice is the traveling +proton basis (TPB) approach,12,13 in which the proton +basis function centers are allowed to move semiclassically +along the proton transfer trajectory. Because this TPB +approach is a semiclassical approximation of the FPB +approach, we will focus on the FPB approach in this +manuscript. +However, the RT-NEO-Ehrenfest dynam- +ics simulations performed herein can also be performed +with the TPB approach in a straightforward manner. +Moreover, as mentioned above, the RT-NEO-Ehrenfest +approach can also be used in conjunction with semiclas- +sical RT-NEO dynamics for polaritons. +D. +Electronic BO Approximation +When the electronic BO approximation is applied, the +protonic dynamics is still propagated by Eq. (1b) or Eq. + +4 +(6b). For the electrons, at each time step, the electronic +density matrix is quenched to the ground state by solving +the electronic self-consistent field (SCF) equation: +Pe′(t) = SCF[ℜ(Pn′(t)), {RI(t)}] +(8) +Here, ℜ(Pn′(t)) denotes the real component of the pro- +tonic density matrix in the non-orthogonal atomic orbital +basis. Because the converged Pe′ and electronic energy +are real-valued, the imaginary component of Pn′ does not +need to be included in the electronic Kohn-Sham ma- +trix in Eq. +(4a) for this SCF procedure. +Hence, only +ℜ(Pn′(t)) is used to solve the electronic SCF equation. +More specifically, when solving Eq. (8), we iteratively +find the converged electronic density satisfying the fol- +lowing Hartee–Fock–Roothaan equation: +Fe′Ce′ = SeCe′εe +(9) +where εe is the orbital energy matrix and Fe′ and Ce′ +have been defined above. Here, Fe′ is a function of Pe′ +(or Ce′), ℜ(Pn′), and {RI}. +III. +SIMULATION DETAILS +All the above approaches have been implemented in +a developer version of Q-Chem.43 The initial molecular +geometries for the calculations below are given in the +Supplementary Material. +The electrons and quantum +nuclei were propagated by a modified midpoint unitary +transform time-propagation scheme algorithm.34,44 Dur- +ing time propagation, an additional predictor-corrector +procedure45 was used to control the growth of numerical +error in the electronic and nuclear quantum dynamics. +The velocity Verlet algorithm was used to propagate the +classical nuclei and cavity modes. The step-by-step al- +gorithms of the above approaches are provided in the +Supplementary Material. +The RT-NEO method with fixed classical nuclei +was applied to a single HCN molecule. +The B3LYP +functional46–48 was used for electron-electron exchange- +correlation and the epc17-2 functional49,50 was used for +electron-proton correlation. For the electronic basis, the +cc-pVDZ electronic basis set51 was used for the heavy +nuclei and the cc-pV5Z electronic basis set was used for +the proton; for the protonic basis, the PB4-F2 proton +basis set52 was used. The initial densities for the elec- +trons and quantum proton were obtained from the SCF +ground state NEO-DFT solution32,33 with a tight energy +convergence criterion of 10−12 a.u. During the real-time +propagation, at time t = 0 a delta pulse was used to per- +turbed the protonic Fock matrix as Fn′ + E · µn′, where +E = (E0, E0, E0) and µn′ denotes the protonic dipole mo- +ment matrix vector in the non-orthogonal atomic orbital +basis. Because we will compare the performance of calcu- +lations with different time steps ∆tq, E0∆tq = 4 × 10−4 +a.u. +is always assumed. +This restriction ensures that +simulations with different time steps will produce dipole +signals with the same amplitude. As we will compare the +performance with and without the electronic BO approx- +imation, the delta pulse was not applied to the electronic +subsystem. +The same HCN molecule was also used in the semiclas- +sical RT-NEO calculations for polaritons. A single cav- +ity mode polarized along the x direction was resonantly +coupled to the C H stretch mode of the molecule at +ωc = 3685 cm−1 with light-matter coupling ε = 6 × 10−4 +a.u. The initial conditions and computational methods +for HCN were the same as those described for the RT- +NEO calculations in free space. The initial condition for +the cavity mode was set as pc(0) = 0 and qc(0) = 0.1 +a.u., and no delta pulse was applied to the molecular +subsystem. Because the position of the cavity mode was +displaced to 0.1 a.u., in later times the excess energy in +the cavity mode transferred to the C H stretch mode +and generated real-time Rabi oscillations. +The cavity +loss rate was assumed to be γc = 0. +The RT-NEO-Ehrenfest method was applied to in- +tramolecular proton transfer in malonaldehyde with +the transferring proton treated quantum mechanically. +The molecule was described at the B3LPY/epc17-2/cc- +pVDZ/PB4-F2 level. +The initial electronic and quan- +tum protonic densities were obtained from the NEO-DFT +ground state solution with an energy convergence crite- +rion of 10−9 a.u. +The initial velocities of the classical +nuclei were set to zero. +Because these classical nuclei +were chosen to start out in a symmetric configuration, +whereas the equilibrium geometry for the classical nu- +clei is asymmetric, the classical nuclei experienced forces +directed toward the asymmetric relaxed geometry, thus +driving proton transfer. For the quantum proton, three +fixed proton basis function centers were used, and each +center contained a PB4-F2 proton basis set and a cc- +pVDZ electronic basis set. +These three proton basis +function centers were chosen to be near the donor oxy- +gen atom (OD), the acceptor oxygen atom (OA), and the +midpoint between the two centers. The quantum proton +position is defined as the expectation value of the pro- +ton position. For RT-NEO-Ehrenfest dynamics without +the electronic BO approximation, by default we set the +time step for the electronic and protonic quantum dy- +namics as ∆tq = 0.010 fs, and the nuclear gradients were +evaluated every 10 time steps of the quantum dynamics. +Under the electronic BO approximation, because a 10- +fold larger time step was used for the protonic quantum +dynamics, the nuclear gradients were evaluated at every +time step of the quantum dynamics. + +5 +IV. +RESULTS AND DISCUSSION +A. +RT-NEO vs BO-RT-NEO dynamics with fixed classical +nuclei +To compare the performance of RT-NEO and BO-RT- +NEO with fixed classical nuclei, we applied both ap- +proaches to a single HCN molecule oriented along the +x-axis. +Fig. +1a shows the RT-NEO-TDDFT dynam- +ics of the x-component of the HCN nuclear dipole mo- +ment, µn +x(t) = Tr [Pn(t)ˆµn +x], when the ground-state pro- +ton density is perturbed by a delta pulse at time t = 0 fs. +As shown in this figure, the RT-NEO-TDDFT approach +yields the same dipole oscillations for any time step +∆tq ≤ 0.010 fs, whereas a larger time step ∆tq ≥ 0.097 +fs leads to divergence. +Fig. +1b plots the corresponding power spectrum of +the dipole signal in Fig. 1a calculated by the following +Fourier transform: +Pn(ω) = +� +i=x,y,z +��F +� +µn +i (t)e−γt��� +(10) +Here, for a better visualization of the spectrum, a small, +artificial damping term e−γt (with γ = 10−5 a.u.) pro- +vides a small linewidth of 13.8 cm−1 for the peaks in +the frequency domain. The Padé approximation of the +Fourier transform34,53 is used in Eq. (10) for better fre- +quency resolution. Similar to Fig. 1a, consistent spec- +tra are obtained only when ∆tq ≤ 0.010 fs: the peak +at 1787 cm−1 is the C H bending mode, and the peak +at 3685 cm−1 is the C H stretch mode. Note that the +quantitative accuracy of these frequencies can be im- +proved by increasing the size of the electronic and pro- +tonic basis sets.54 When ∆tq ≤ 0.010 fs, the RT-NEO- +TDDFT frequencies agree well with the linear-response +NEO-TDDFT results, as denoted by the vertical gray +dashed lines in the spectrum. Note that the real-time +and linear-response results also predict the same absorp- +tion intensities as the linear-response results, as shown in +Fig. S1 of the Supplementary Material. +Fig. +1c and d show the analogous results obtained +with the BO-RT-NEO-TDDFT approach instead of the +RT-NEO-TDDFT approach. +Here, all the time steps +0.001 ≤ ∆tq ≤ 0.194 fs yield the same real-time dynam- +ics and spectrum. Comparing Fig. 1c and Fig. 1a, we +find that the BO-RT-NEO-TDDFT approach produces +reliable results with a 20-fold larger time step than that +required for the RT-NEO-TDDFT approach. +Because +the electronic density needs to be quenched to the ground +state by solving the electronic SCF equation (see Eq. (8)) +for each time step of the electronic BO dynamics, each +time step for BO-RT-NEO-TDDFT is more expensive +than a time step for RT-NEO-TDDFT. In practice, we +find that for the calculations in Fig. 1, the computational +cost of BO-RT-NEO-TDDFT per time step is less than +three times that of RT-NEO-TDDFT. Hence, overall, in- +voking the electronic BO approximation can accelerate +the calculation of the proton vibrational spectrum by a +factor of > 20/3 ≈ 7. +B. +Semiclassical RT-NEO vs semiclassical BO-RT-NEO +dynamics for polaritons +Beyond the conventional RT-NEO-TDDFT dynamics +with fixed classical nuclei, when the molecular system is +coupled to a classical cavity photon mode, the semiclassi- +cal RT-NEO-TDDFT approach can provide a unified de- +scription of vibrational and electronic strong couplings.14 +Fig. +2a shows µn +x(t) for the x-oriented HCN molecule +under vibrational strong coupling when the molecule is +resonantly coupled to an x-polarized cavity mode with +the cavity frequency the same as the C H stretch mode +(ωc = 3685 cm−1). After an initial perturbation of the +cavity mode, the coherent energy transfer between the +cavity mode and the molecule leads to vibrational Rabi +oscillations in the time domain. However, this behavior +is only reliably captured when the simulation time step is +∆tq = 0.005 fs (solid black line), whereas using a larger +time step ∆tq ≥ 0.010 fs (from red to magenta lines) +leads to divergent time-domain dynamics. +In the frequency domain, the vibrational polariton +spectrum can be calculated from the power spectrum of +µn +x(t) using Eq. +(10). +Although the time-domain dy- +namics is converged only when ∆tq = 0.005 fs (solid +black), in the frequency domain the results with ∆tq = +0.005 fs (solid black) and ∆tq = 0.010 fs (dashed red) +are virtually identical, presumably because the ∆tq = +0.010 fs real-time dipole dynamics is stable for a rela- +tively long time (up to t ∼ 300 fs). For the converged +spectrum (black or red line), the two polariton peaks +are asymmetric with respect to the resonance frequency +(ωc = 3685 cm−1, vertical thick gray line). Because the +Rabi splitting between the two polaritons is very small +(within 100 cm−1) compared to the resonance frequency +ωc, the polariton peaks are expected to be symmetric +with respect to the resonance frequency.55–57 Hence, this +observed asymmetry is unphysical, reflecting a limitation +of the semiclassical RT-NEO approach when describing +vibrational strong coupling. +Figs. 2c and d show the dipole dynamics and spec- +tra obtained with the electronic BO approximation un- +der the same conditions. Comparing Fig. 2c and Fig. +2a, we find that invoking the electronic BO approxi- +mation captures stable real-time Rabi oscillations even +with a 40-fold larger time step (∆tq = 0.194 fs, magenta +line) than that required for the semiclassical RT-NEO +approach (∆tq = 0.005 fs, gray line). Moreover, in the +frequency domain (Fig. +2d), the two polariton peaks +are symmetric with respect to the resonance frequency +(ωc = 3685 cm−1, vertical thick gray line), representing +a significant advantage of the BO-RT-NEO-TDDFT ap- +proach for simulating molecular polaritons. +The comparison between these two approaches allows +us to understand why semiclassical RT-NEO-TDDFT + +6 +0 +20 +40 +60 +80 +100 +−1 +0 +1 +µn +x [×10−5 a.u.] +(a) +RT-NEO fixed classical nuclei +1500 +2000 +2500 +3000 +3500 +4000 +0.0 +0.5 +1.0 +Pn(ω) [arb. units] +(b) +0.001 fs +0.010 fs +0.097 fs +0.194 fs +0 +20 +40 +60 +80 +100 +time [fs] +−1 +0 +1 +µn +x [×10−5 a.u.] +(c) +BO-RT-NEO fixed classical nuclei +1500 +2000 +2500 +3000 +3500 +4000 +frequency [cm−1] +0.0 +0.5 +1.0 +Pn(ω) [arb. units] +(d) +FIG. 1. (a) RT-NEO-TDDFT dynamics of the x-component of the HCN nuclear dipole moment, µn +x(t), in vacuum with fixed +classical nuclei. The quantum proton is perturbed by a delta pulse at t = 0 fs. Simulation results with different time steps are +compared: ∆tq = 0.001 fs (solid black), 0.010 fs (dashed red), 0.097 fs (dash-dotted blue), and 0.194 fs (dash-dotted magenta). +(b) Corresponding nuclear dipole power spectrum, Pn(ω), as well as the linear-response NEO-TDDFT vibrationally excited +state transition frequencies (vertical gray dashed line). (c,d) The same plots as (a,b) except that the BO-RT-NEO-TDDFT +approach is used. +Note that although the RT-NEO approach requires a relatively small ∆tq (0.010 fs), the BO-RT-NEO +approach can still recover accurate nuclear dipole signals using a 20-fold larger ∆tq (0.194 fs). +without the electronic BO approximation predicts asym- +metric polaritons even when the Rabi splitting is very +small. +Comparing Fig. +2a and Fig. +2c, the nuclear +dipole signals without the electronic BO approximation +have smaller oscillation amplitudes, suggesting that the +cavity energy is also transferred to the electronic degrees +of freedom in the molecular system, as the cavity mode is +coupled to both the electronic and nuclear Kohn–Sham +matrices (see Eq. (6)). Due to this energy transfer, the +higher-energy electronic excitations influence the vibra- +tional polariton spectrum, leading to the asymmetry in +the vibrational Rabi splitting. +This argument can also be understood by a simple +three-state model: +H = +� +� +� +ωc gv +ge +gv ωv +0 +ge +0 +ωe +� +� +� +(11) +where gv (ge) denotes the coupling between the cavity +mode and the vibrational (electronic) transition of fre- +quency ωv (ωe). In the absence of the electronic transi- +tion (i.e., let ge = 0), at resonance condition ωc = ωv = +ω0, the two polariton frequencies are +ω± = ω0 ± gv +(12) +which is symmetric with respect to ω0. Due to the exis- +tence of the high-energy electronic transition, according +to second-order perturbation theory, as previously shown +by Shao and coworkers,58 the polariton frequencies are +modified to +ω′ +± = ω± − +g2 +e +ωe − ω± +(13) +Because ωe ≫ ω±, the high-energy electronic transition +would redshift the vibrational polaritons, as observed in +Fig. 2b compared to Fig. 2d. The high-energy electronic +transitions predicted in semiclassical RT-NEO-TDDFT +probably arise from approximations underlying the light- +matter Hamiltonian and are not physically meaningful. +In contrast, if the electronic BO approximation is ap- +plied, because the electrons are always quenched to the +ground state, the influence of the unphysical high-energy +electronic excitations is eliminated, thus preserving the +symmetry of vibrational polaritons. Note that the single- +molecule vibrational strong coupling example described +by Fig. 2 is used to test our methods and should not be +used to interpret polariton experiments in the collective +regime. + +X7 +0 +200 +400 +600 +800 +1000 +−0.005 +0.000 +0.005 +µn +x [a.u.] +(a) +semiclassical RT-NEO for polaritons +3600 +3650 +3700 +3750 +0.0 +0.5 +1.0 +Pn(ω) [arb. units] +(b) +0.005 fs +0.010 fs +0.024 fs +0.194 fs +0 +200 +400 +600 +800 +1000 +time [fs] +−0.005 +0.000 +0.005 +µn +x [a.u.] +(c) +semiclassical BO-RT-NEO for polaritons +3600 +3650 +3700 +3750 +frequency [cm−1] +0.0 +0.5 +1.0 +Pn(ω) [arb. units] +(d) +FIG. 2. +(a) Semiclassical RT-NEO-TDDFT dynamics of µn +x(t) and (b) the corresponding power spectrum when a single HCN +molecule is coupled to an x-polarized cavity mode with ωc = 3685 cm−1 (the vertical thick gray line in (b)) and coupling +strength ε = 6 × 10−4 a.u. Simulation results with different time steps are compared: ∆tq = 0.005 fs (solid black), 0.010 +fs (dashed red), 0.024 fs (dash-dotted blue), and 0.194 fs (dash-dotted magenta). (c,d) The analogous plots as (a,b) except +that the semiclassical BO-RT-NEO-TDDFT approach is used. Note that invoking the BO approximation not only generates +stable long-time simulation results when a larger ∆tq is used, but also removes the unphysical asymmetry in the Rabi splitting +observed by the semiclassiacl RT-NEO approach. +C. +RT-NEO-Ehrenfest vs BO-RT-NEO-Ehrenfest dynamics +Intramolecular proton transfer in malonaldehyde has +been extensively studied both experimentally27,28 and +theoretically.29–31 Within the NEO framework, the trans- +ferring proton is treated quantum mechanically, and the +remaining nuclei are treated classically. +For our sim- +ulation, the classical nuclear geometry is chosen to be +symmetric, obtained by averaging the equilibrium reac- +tant and product geometries. +The quantum proton is +described by three proton basis function centers span- +ning the region between the donor and acceptor oxygen +atoms. As shown in the inset of Fig. 3a, the quantum +proton density is chosen to be the NEO-DFT SCF so- +lution localized near the donor oxygen, OD. +The pro- +ton remains localized at the NEO-DFT level because a +multireference treatment is required to produce a delo- +calized, bilobal proton density.10 Note that this specific +initial nuclear geometry and proton density serve as an +illustration of the method and are not relevant to exper- +imental studies of this molecule. +As shown in Fig. +3a, RT-NEO-Ehrenfest dynam- +ics (black lines) predicts proton transfer from OD to +OA within t = 100 fs. +This fast proton transfer re- +action is induced by the nonequilibrium initial geome- +try. +Specifically, the initial heavy nuclear geometry is +symmetric, whereas the equilibrium geometry is asym- +metric, with the proton bonded to one of the oxygen +atoms. Hence, the classical nuclei experience forces to- +ward the asymmetric equilibrium geometry, accompanied +by nonequilibrium quantum dynamics of the transferring +proton. If the proton transfer time is defined as the time +when the H OD distance (solid lines) is the same as the +H OA distance (dashed lines), RT-NEO-Ehrenfest dy- +namics (black lines) predicts that the proton transfer +occurs at t = 48 fs.59 The BO-RT-NEO-Ehrenfest dy- +namics (red lines) is virtually identical to the RT-NEO- +Ehrenfest dynamics, although the time step for BO-RT- +NEO-Ehrenfest (∆tq = 0.102 fs) is 10-folder larger than +that for RT-NEO-Ehrenfest dynamics (∆tq = 0.010 fs). +For a better understanding of the proton transfer dy- +namics, Fig. 4 further depicts the nonequilibrium proton +density at different times along the trajectory shown in +Fig. 3a. Interestingly, during and after proton transfer, +the proton can become delocalized in the region between +OD and OA. This delocalization is enabled by the use +of three proton basis function centers spanning the re- +gion that is sampled. +A more accurate description of +hydrogen tunneling is expected to require a multiref- +erence NEO approach, such as multistate DFT (NEO- +MSDFT)10 or a complete active space self-consistent-field +(NEO-CASSCF) method.6 +If the quantum proton is replaced by a classical proton +and all the other conditions are the same as in Fig. 3a, + +y +X (cavity polarization)8 +0 +20 +40 +60 +80 +100 +1.0 +1.2 +1.4 +1.6 +H-O distance [Angstrom] +(a) +RT-NEO-Ehrenfest +BO-RT-NEO-Ehrenfest +0 +20 +40 +60 +80 +100 +time [fs] +1.0 +1.2 +1.4 +1.6 +H-O distance [Angstrom] +(b) +Ehrenfest classical proton +BOMD classical proton +FIG. 3. (a) Electronic ground state proton transfer dynam- +ics in malonaldehyde. The initial geometry and proton den- +sity (in blue) are shown in the inset. +Two approaches are +compared: (i) RT-NEO-Ehrenfest dynamics (black, with ∆tq += 0.010 fs) and (ii) BO-RT-NEO-Ehrenfest dynamics (red, +with ∆tq = 0.102 fs). The quantum proton is described by +three fixed proton basis function centers spanning the region +between OD and OA. +The solid (dashed) lines denote the +distance between OD (OA) and the expectation value of the +quantum proton position. (b) The corresponding plot shown +in (a) for the case when the quantum proton is replaced by a +classical proton. In this case, both Ehrenfest dynamics (black) +and BO molecular dynamics (red) predict no proton transfer, +emphasizing the importance of a quantum treatment of the +transferring proton. In both plots, the black and red lines are +virtually indistinguishable. +both Ehrenfest dynamics (black lines) and BO molecu- +lar dynamics (red lines) predict no proton transfer (Fig. +3b). The difference between Fig. 3a and Fig. 3b high- +lights the importance of a quantum mechanical treatment +of the transferring proton for proton transfer reactions, +especially when hydrogen tunneling is significant. Note +that the quantization of other modes for malonaldehyde +has been shown to be important for a quantitatitvely ac- +curate description of hydrogen tunneling30 and will be +investigated within the NEO framework in future stud- +ies. The total energy along the trajectories shown in Fig. +3 is plotted in the Supplementary Material. +V. +CONCLUSION +In this manuscript, we have explored the electronic BO +approximation in three different flavors of NEO dynam- +ics: (i) RT-NEO for proton dynamics with fixed classical +nuclei, (ii) semiclassical RT-NEO for polariton dynamics +with fixed classical nuclei, and (iii) RT-NEO-Ehrenfest +dynamics for full molecular dynamics of molecular sys- +tems. +When the BO approximation between the elec- +trons and quantum protons is not invoked, the electronic +and protonic dynamics are propagated on the same foot- +ing. Because the electronic dynamics is much faster than +the protonic dynamics, a small time step is needed for a +converged result. By invoking the electronic BO approx- +imation, which involves quenching the electronic density +to the SCF ground state at each time step, we can use an +order-of-magnitude larger time step to perform the cal- +culations, thus greatly reducing the computational cost. +We emphasize that with this treatment, because the pro- +ton density is still propagated in real time, the nonequi- +librium quantum dynamics of the proton is preserved. +Moreover, we have also found that under vibrational +strong coupling, the unphysical asymmetic Rabi split- +ting observed in previous semiclassical RT-NEO simu- +lations of polaritonic systems14 can be fixed by invoking +the electronic BO approximation, demonstrating another +significant advantage of this treatment. +The application to intramolecular proton transfer in +malonaldehyde highlights the importance of treating the +transferring proton quantum mechanically. In this sim- +ulation, a quantum treatment leads to proton transfer, +whereas a classical treatment predicts no proton trans- +fer. The ability to capture proton delocalization in this +proton transfer system also demonstrates the capacity of +the (BO)-RT-NEO-Ehrenfest approach for more exciting +applications. We emphasize that the BO-RT-NEO meth- +ods are only applicable for electronically adiabatic sys- +tems. Thus, this approximation should not be used when +there are significant non-BO effects between the electrons +and quantum nuclei, as in some proton-coupled electron +transfer reactions.60 Compared with the widely used ring +polymer molecular dynamics approach3 for adiabatic nu- +clear quantum dynamics, the BO-RT-NEO methods pro- +vide a complementary perspective for describing vibra- +tionally excited dynamics and nuclear quantum coher- +ence, which will be topics of future studies. Overall, this +work lays the groundwork for applying RT-NEO methods +to a wide range of electronically adiabatic and nonadia- +btic chemical and biological systems. +VI. +ACKNOWLEDGMENTS +This material is based upon work supported by the Air +Force Office of Scientific Research under AFOSR Award +No. FA9550-18-1-0134 for the polariton simulations and +by the National Science Foundation Grant No. +CHE- +1954348 for the general NEO method developments. We + +9 +FIG. 4. Quantum proton density dynamics predicted by the BO-RT-NEO-Ehrenfest approach for the trajectory shown in Fig. +3a. Virtually identical proton densities are predicted by the RT-NEO-Ehrenfest approach. The dashed horizontal gray lines +indicate the approximate equilibrium proton positions near OD and OA. During and after proton transfer, the delocalization +of proton density between OD and OA is captured. The quantum proton isosurface is plotted by setting the isovalue as 0.001 +a.u. 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Shao, “Quantum-Electrodynamical Time- +Dependent Density Functional Theory within Gaussian Atomic +Basis,” J. Chem. Phys. 155, 064107 (2021). +59Note that a more general definition of proton transfer time would +require the proton to form a stable bond with the acceptor oxy- +gen. +60S. Hammes-Schiffer, “Proton-Coupled Electron Transfer: Moving +Together and Charging Forward,” J. Am. Chem. Soc. 137, 8860– +8871 (2015). + +Supplementary Material +Electronic Born–Oppenheimer Approximation in +Nuclear-Electronic Orbital Dynamics +Tao E. Lia) and Sharon Hammes-Schifferb) +Department of Chemistry, Yale University, New Haven, Connecticut, 06520, +USA +a)Electronic mail: tao.li@yale.edu +b)Electronic mail: sharon.hammes-schiffer@yale.edu +1 + +S1. +INITIAL GEOMETRIES FOR HCN AND MALONALDEHYDE +HCN (Units: +Angstrom) +N 0.0492158067 0.000 0.000 +C 1.2046693425 0.000 0.000 +H 2.1221148508 0.000 0.000 +Malonaldehyde (Units: +Angstrom; Gh: +additional proton basis centers) +O 0.0000000000 -1.3008730953 2.0456624985 +O 0.0000000000 1.2908329047 2.0456624985 +C 0.0000000000 -1.2160310953 0.7592594985 +C 0.0000000000 1.2059909047 0.7592594985 +C 0.0000000000 -0.0050200953 0.0533444985 +H 0.0000000000 -0.0050200953 -1.0248235015 +H 0.0000000000 -2.1631460953 0.2189674985 +H 0.0000000000 2.1531059047 0.2189674985 +H 0.0000000000 -0.3010120550 2.3554201375 +Gh 0.0000000000 0.0000000000 2.3554165853 +Gh 0.0000000000 0.2553440000 2.3869010000 +S1 + +S2. +ADDITIONAL SIMULATION RESULTS +0 +20 +40 +60 +80 +100 +−1 +0 +1 +µn +x [×10−5 a.u.] +(a) +RT-NEO fixed classical nuclei +1500 +2000 +2500 +3000 +3500 +4000 +0.0 +0.5 +1.0 +An(ω) [arb. units] +(b) +0.001 fs +0.010 fs +0.097 fs +0.194 fs +0 +20 +40 +60 +80 +100 +time [fs] +−1 +0 +1 +µn +x [×10−5 a.u.] +(c) +BO-RT-NEO fixed classical nuclei +1500 +2000 +2500 +3000 +3500 +4000 +frequency [cm−1] +0.0 +0.5 +1.0 +An(ω) [arb. units] +(d) +FIG. S1. The same plots as those shown in Fig. 1 except that here the absorption spectrum is +plotted: An = � +i=x,y,z −ωIm +� +F +� +µn +i (t)e−γt�� +.S1 The linear-response peak heights (vertical gray +solid lines) indicate the absorption intensities. Here, the peak ratios between the real-time signals +and the linear-response signals (vertical gray lines) agree exactly. +S2 + +X0 +20 +40 +60 +80 +100 +−0.0005 +0.0000 +0.0005 +relative energy [a.u.] +(a) +RT-NEO-Ehrenfest (∆tq= 0.010 fs) +BO-RT-NEO-Ehrenfest (∆tq= 0.102 fs) +BO-RT-NEO-Ehrenfest (∆tq= 0.010 fs) +0 +20 +40 +60 +80 +100 +time [fs] +−0.0005 +0.0000 +0.0005 +relative energy [a.u.] +(b) +Ehrenfest classical proton (∆tq= 0.010 fs) +BOMD classical proton (∆tq= 0.102 fs) +FIG. S2. Energy conservation along the trajectories shown in Fig. 3. For both the NEO and +classical simulations, the black lines indicate the trajectories without the BO approximation (with +a time step ∆tq = 0.010 fs), and the red lines indicate the trajectories with the BO approximation +(with a time step ∆tq = 0.102 fs). The additional yellow line in part (a) indicates the BO-RT- +NEO-Ehrenfest trajectory with the same time step as the RT-NEO-Ehrenfest dynamics (black +line), showing that the energy conservation behavior of BO-RT-NEO-Ehrenfest dynamics can be +improved by reducing the time step. +S3 + +S3. +DETAILED ALGORITHMS FOR NEO DYNAMICS +Algorithm S1 Semiclassical BO-RT-NEO dynamics for polaritons with fixed classical nu- +clei. +1: Calculate µλ(τ = 0) = Tr +� +Pn′(τ = 0)ˆµn′ +λ +� ++ 2Tr +� +Pe′(τ = 0)ˆµe′ +λ +� +2: for τ = ∆tq, 2∆tq, · · · do +3: +Calculate µλ(τ) = Tr +� +Pn′(τ)ˆµn′ +λ +� ++ 2Tr +� +Pe′(τ)ˆµe′ +λ +� +− µλ(τ = 0) +4: +pk,λ(τ + 1 +2∆tq) = pk,λ(τ − 1 +2∆tq) − [ω2 +k,λqk,λ(τ) + εk,λµk,λ(τ)]∆tq +5: +qk,λ(τ + ∆tq) = qk,λ(τ) + pk,λ(τ + 1 +2∆tq)∆tq +6: +pk,λ(τ + 1 +2∆tq) ∗= e−γc∆tq +//cavity loss +7: +Pe(n)′(τ) = [Se(n)]−1/2Pe(n)(τ)[Se(n)]−1/2 +8: +Build Fe(n)′ +incav(τ) using Pe(n)′(τ) and qk,λ(τ) +9: +Fe(n) +incav(τ + 1 +2∆tq) = 2Fe(n) +incav(τ) − Fe(n) +incav(τ − 1 +2∆tq) +10: +counter = 1 +11: +while True do +12: +Pn(τ + ∆tq) = e−i∆tqFn +incav(τ+ 1 +2 ∆tq)Pn(τ)ei∆tqFn +incav(τ+ 1 +2 ∆tq) +13: +if scf_e AND counter == 1 then +14: +Converge Pe′(τ + ∆tq) to ground state given Pn′(τ + ∆tq) +15: +else +16: +Pe(τ + ∆tq) = e−i∆tqFe +incav(τ+ 1 +2 ∆tq)Pe(τ)ei∆tqFe +incav(τ+ 1 +2 ∆tq) +17: +end if +18: +Build Fe(n)′ +incav(τ + ∆tq) using Pe(n)′(τ + ∆tq) and qk,λ(τ + ∆tq) +19: +Fe(n) +incav(τ + 1 +2∆tq) = 1 +2Fe(n) +incav(τ) + 1 +2Fe(n) +incav(τ + ∆tq) +20: +if counter > 1 then +21: +if |Pe(n)(τ + ∆tq) − Pe(n) +test | < thres then +22: +Exit the while loop +23: +end if +24: +end if +25: +Pe(n) +test = Pe(n)(τ + ∆tq) +26: +counter += 1 +27: +end while +28: end for +S4 + +Algorithm S2 BO-RT-NEO-Ehrenfest dynamics. +1: ∆tNq = ∆tN/n, ∆tq = ∆tNq/m +2: for t = 0, ∆tN, 2∆tN, · · · do +3: +if scf_e then +4: +Converge Pe′(t) to ground state given Pn′(t) +5: +end if +6: +Compute forces F(t) using Pe′(t) and Pn′(t) +7: +P(t + 1 +2 ∆tN) = P(t − 1 +2 ∆tN) + F(t)∆tN +8: +R(t + ∆tN) = R(t) + P(t + 1 +2 ∆tN)∆tN/M +9: +for j = 1, 2, · · · , n do +10: +t′ = t + (j − 1)∆tNq +11: +R(t′ + 1 +2 ∆tNq) = R(t) + ((j − 1)∆tNq + 1 +2 ∆tNq)P(t + 1 +2 tN)/M +12: +Update basis center to R(t′ + 1 +2 ∆tNq) and recompute Se(n) +13: +for i = 1, 2, · · · , m do +14: +τ = t + (i − 1)∆tq +15: +Pe(n)′(τ) = [Se(n)]−1/2Pe(n)(τ)[Se(n)]−1/2 +16: +Build Fe(n)′(τ) using Pe(n)′(τ) and R(t′ + 1 +2 ∆tNq) +17: +Fe(n)(τ + 1 +2 ∆tq) = 2Fe(n)(τ) − Fe(n)(τ − 1 +2 ∆tq) +18: +counter = 1 +19: +while True do +20: +Pn(τ + ∆tq) = e−i∆tq[Fn(τ+ 1 +2 ∆tq)]Pn(τ)ei∆tq[Fn(τ+ 1 +2 ∆tq)] +21: +if scf_e AND counter == 1 then +22: +Converge Pe′(τ + ∆tq) to ground state given Pn′(τ + ∆tq) +23: +else +24: +Pe(τ + ∆tq) = e−i∆tqFe(τ+ 1 +2 ∆tq)Pe(τ)ei∆tqFe(τ+ 1 +2 ∆tq) +25: +end if +26: +Build Fe(n)′(τ + ∆tq) using Pe(n)′(τ + ∆tq) and R(t′ + 1 +2 ∆tNq) +27: +Fe(n)(τ + 1 +2 ∆tq) = 1 +2 Fe(n)(τ) + 1 +2 Fe(n)(τ + ∆tq) +28: +if counter > 1 then +29: +if |Pe(n)(τ + ∆tq) − Pe(n) +test | < thres then +30: +Exit the while loop +31: +end if +32: +end if +33: +Pe(n) +test = Pe(n)(τ + ∆tq) +34: +counter += 1 +35: +end while +36: +end for +37: +end for +38: end for +In the above two algorithms, when the electronic BO approximation is invoked, the +parameter scf_e is set to true, and otherwise this parameter is set to false. +The parameter thres in the above two algorithms controls the maximal error in the +density matrices during each time step. In our simulation, we have set a very loose threshold +S5 + +10−4 a.u. +In RT-NEO-Ehrenfest dynamics, the parameters m and n are used to reduce the number +of energy gradient evaluations,S2 which is the most time-consuming step. For our calculations +in the manuscript, we have set m = 1 and n = 10. +REFERENCES +[S1]L. Zhao, Z. Tao, F. Pavošević, A. Wildman, S. Hammes-Schiffer, and X. Li, “Real-Time +Time-Dependent Nuclear-Electronic Orbital Approach: Dynamics beyond the Born- +Oppenheimer Approximation,” J. Phys. Chem. Lett. 11, 4052–4058 (2020). +[S2]L. Zhao, A. Wildman, Z. Tao, P. Schneider, S. Hammes-Schiffer, and X. Li, “Nu- +clear–electronic orbital Ehrenfest dynamics,” J. Chem. Phys. 153, 224111 (2020). +S6 + diff --git a/V9E2T4oBgHgl3EQfuAiM/content/tmp_files/load_file.txt b/V9E2T4oBgHgl3EQfuAiM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2aabf6549a8e423d80cb6fd923c481b2af235869 --- /dev/null +++ b/V9E2T4oBgHgl3EQfuAiM/content/tmp_files/load_file.txt @@ -0,0 +1,1476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf,len=1475 +page_content='Electronic Born–Oppenheimer Approximation in Nuclear-Electronic Orbital Dynamics Tao E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Lia) and Sharon Hammes-Schifferb) Department of Chemistry, Yale University, New Haven, Connecticut, 06520, USA Within the nuclear-electronic orbital (NEO) framework, the real-time time-dependent density functional theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this approach, the electrons and quantum nuclei are propagated in time on the same footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A relatively small time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of long-time nuclear quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Herein, the electronic Born–Oppenheimer (BO) approximation within the NEO framework is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this approach, the electronic density is quenched to the ground state at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of- magnitude larger time step, thus greatly reducing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Moreover, invoking the electronic BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical RT-NEO-TDDFT simulations of polaritons under vibrational strong coupling even for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the real-time nuclear quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Thus, the BO RT-NEO approach provides the foundation for a wide range of chemical and biological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' INTRODUCTION Simulating nuclear quantum dynamics is important for understanding a wide range of chemical and biolog- ical processes, such as vibrationally excited chemistry and proton transfer reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A variety of theoreti- cal methods for simulating nuclear quantum dynamics have been developed, including the linearized semiclas- sical initial value representation (LSC-IVR) method,1,2 ring polymer molecular dynamics,3 the multiconfigura- tional time-dependent Hartree (MCTDH) method,4 and the exact factorization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 The nuclear-electronic orbital (NEO) method6,7 is another approach for simu- lating nuclear quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='8–14 Within the NEO framework, both electrons and selected nuclei, usually the protons, are described by first-principles methods such as density functional theory (DFT), while the re- maining heavy nuclei are treated classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As an exten- sion of conventional electronic structure theory, the NEO method can be combined with conventional nonadiabatic dynamics approaches, such as Ehrenfest dynamics15,16 and trajectory surface hopping,17 to simulate nonadi- abatic dynamics on vibronic surfaces rather than elec- tronic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A promising strategy for simulating nuclear quan- tum dynamics within the NEO framework is real-time NEO time-dependent density functional theory (RT- NEO-TDDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='11 In this approach, the quantum dy- namics of the electrons and quantum protons are prop- agated in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As a real-time version of a)Electronic mail: tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='li@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='edu b)Electronic mail: sharon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='hammes-schiffer@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='edu linear-response multicomponent TDDFT,18–21 RT-NEO- TDDFT can capture the linear-response electronic and protonic excited-state spectra by Fourier transforming the real-time dipole signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Moreover, this approach can also directly capture the coupled electron-proton nonadiabatic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' These calculations can be per- formed with fixed classical nuclei, or the classical nu- clei can be propagated on the mean-field potential en- ergy surface associated with the nonequilibrium elec- trons and quantum protons using the RT-NEO-Ehrenfest dynamics approach,12,13 which captures the full nona- diabatic dynamics of the electrons, quantum protons, and classical nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Another important extension of RT-NEO-TDDFT is semiclassical RT-NEO-TDDFT for polaritons,14 where the coupled dynamics between the NEO molecular subsystem and classical photon modes are propagated in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' When the light- matter coupling is large enough, this approach can de- scribe strong light-matter interactions under both elec- tronic and vibrational strong couplings, thus potentially providing a powerful scheme for simulating polariton chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='22–25 For these RT-NEO dynamics approaches, the electrons and quantum nuclei are propagated on the same footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As a result, a very small time step (≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='01 fs) is re- quired for the time propagation due to the fast electronic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='11–14 The requirement of a small time step may prohibit the study of long-time nuclear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' When nuclear quantum dynamics on the electronic ground state is considered, such a small time step can be avoided by invoking the electronic Born–Oppenheimer (BO) approx- imation between the electrons and both quantum and classical nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this case, the electrons are quenched to the ground state for each time step of the dynam- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='04076v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='chem-ph] 10 Jan 2023 2 ics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=', for each classical nuclear geometry and corre- sponding nonequilibrium quantum proton density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' This electronic BO approximation is different from the con- ventional BO approximation used in ab initio molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In the conventional BO approximation, the electronic ground state is determined solely by the ge- ometry of the classical nuclei, whereas in the NEO elec- tronic BO approximation, the electronic ground state is determined by the nonequilibrium proton density as well as the geometry of the classical nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For a given classical nuclear geometry, the nonequilibrium proton density in a BO-RT-NEO dynamics simulation can be a mixture of vibrational states, thereby also influenc- ing the electronic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The BO-RT-NEO dy- namics approach is also different from the constrained NEO (cNEO) dynamics method developed by Yang and coworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='26 The cNEO approach enables the inclusion of anharmonicity in vibrational spectra in a computation- ally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='9 In addition to this capability, the BO-RT-NEO approach also provides real-time dynamics associated with nonequilibrium proton densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this manuscript, we show that invoking the BO ap- proximation for RT-NEO dynamics allows an order-of- magnitude larger time step to be used during the time propagation compared to RT-NEO dynamics without the BO approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Thus, the BO approximation greatly reduces the computational cost, while producing nearly identical dynamics for the electronically adiabatic sys- tems studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Moreover, invoking the electronic BO ap- proximation also overcomes a serious drawback in semi- classical RT-NEO dynamics for polaritons, namely the previously observed unphysical asymmetirc Rabi split- ting under vibrational strong coupling even when the Rabi splitting is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='14 Herein we show that the BO- RT-NEO method produces symmetric Rabi splittings un- der these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Lastly, using intramolecular proton transfer in malonaldehyde27–31 as an example, we show that BO-RT-NEO-Ehrenfest dynamics can capture the proton delocalization associated with these types of pro- ton transfer processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' THEORY The equations of motion for RT-NEO dynamics, semi- classical RT-NEO dynamics for polaritons, and RT-NEO- Ehrenfest dynamics are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A more detailed review of these approaches is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RT-NEO dynamics with fixed classical nuclei Within the framework of the RT-NEO approach with fixed classical nuclei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='11 the dynamics of both electrons (assuming closed-shell) and quantum nuclei are propa- gated by the following von Neumann equations i ∂ ∂tPe(t) = [Fe(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pe(t)] (1a) i ∂ ∂tPn(t) = [Fn(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pn(t)] (1b) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' the density matrices are defined as Pe = CeCe† and Pn = CnCn†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' where Ce (or Cn) denotes the coeffi- cient vector of the electronic (or nuclear) wavefunction in the orthogonal atomic orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The transformation between the density matrices in the orthogonal (labeled without prime) and non-orthogonal (labeled with prime) atomic basis is governed by Pe = [Se]1/2Pe′[Se]1/2 (2) where Se is the electronic overlap matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Similarly, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (1a), the Kohn–Sham matrices Fe in the orthogonal atomic orbital basis are defined as Fe = [Se]−1/2Fe′[Se]−1/2 (3) Here, Fe′ denotes the Kohn–Sham matrix for the elec- trons in the non-orthogonal atomic orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The analogs to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (2) and (3) for the quantum nuclei are identical with the superscript e replaced by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' According to NEO-DFT,32,33 Fe′ and Fn′ are defined as Fe′(t) = He′ core + Jee′(Pe′(t)) + Ve′ xc(Pe′(t)) − Jen′(Pn′(t)) + Ven′ c (Pe′(t), Pn′(t)) (4a) Fn′(t) = Hn′ core + Jnn′(Pn′(t)) + Vn′ xc(Pn′(t)) − Jne′(Pe′(t)) + Vne′ c (Pn′(t), Pe′(t)) (4b) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (4), He′ core (or Hn′ core) denotes the core Hamilto- nian, which includes the kinetic energy and the Coulomb interaction between the electrons (or quantum nuclei) and the classical nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Jee′ (or Jnn′) denotes the Coulomb interactions among the electrons (or quantum nuclei);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Ve′ xc (or Vn′ xc) denotes the exchange-correlation potential for the electrons (or quantum nuclei);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Jne′ (or Jen′) denotes the Coulomb interaction between the elec- trons and quantum nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' and Ven′ c (or Ven′ c ) denotes the correlation potential between the electrons and quantum nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In the Hartree–Fock limit, Vne′ c = Ven′ c = 0, and Ve′ xc (or Vn′ xc) becomes the Hartree–Fock exchange term for electrons (or quantum nuclei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Similar to most RT- TDDFT implementations,34,35 the adiabatic approxima- tion is invoked, and the above functionals depend locally on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Semiclassical RT-NEO dynamics for polaritons Beyond RT-NEO with fixed classical nuclei, the semi- classical RT-NEO approach for polaritons14 can describe 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Equations of motion for different types of NEO dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RT-NEO11a Semiclassical RT-NEO for polaritons14 RT-NEO-Ehrenfest12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='13 electrons/non-BO i ∂ ∂tPe = [Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pe] i ∂ ∂tPe = � Fe + � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ εk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λˆµe λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pe� i ∂ ∂tPe = [Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pe] electrons/BO Pe′ = SCF[· · · ] Pe′ = SCF[ℜ(Pn′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' {RI}] Pe′ = SCF[· · · ] quantum nuclei i ∂ ∂tPn = [Fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pn] i ∂ ∂tPn = � Fn + � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ εk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λˆµn λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pn� i ∂ ∂tPn = [Fn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pn] classical nuclei fixed fixed MI ¨RI = −∇IE photons N/A ¨qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ = −ω2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ − εk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λµλ − γcpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ N/A a Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RT-NEO refers to RT-NEO dynamics with fixed classical nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' strong light-matter interactions between cavity photon modes and molecules36–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Within this approach, the cavity photons are propagated classically: ˙qk,λ = pk,λ (5a) ˙pk,λ = −ω2 k,λqk,λ − εk,λµλ − γcpk,λ (5b) Here, qk,λ, pk,λ, and ωk,λ denote the position, momen- tum, and frequency of the cavity photon mode charac- terized by the wave vector k = |k| and polarization unit vector ξλ, where k · ξλ = 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=', if the k direction is z, λ can be x or y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' εk,λ denotes the light-matter cou- pling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' µλ denotes the dipole moment of the molecule along the direction of ξλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' γc denotes the cavity loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In practice, when calculating µλ(t), we subtract the per- manent dipole contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=', µλ(t) = 2Tr [Pe(t)ˆµe λ] + Tr [Pn(t)ˆµn λ] − 2Tr [Pe(0)ˆµe λ] − Tr [Pn(0)ˆµn λ], so at time t = 0, qk,λ = pk,λ = 0 always represents the photonic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='14 Here, ˆµe λ (or ˆµn λ) denotes the dipole ma- trix of the electrons (or quantum nuclei) projected along the direction of ξλ in the orthogonal atomic orbital basis, and the prefactor 2 in the electronic dipole moment is in- cluded because of the restricted Kohn–Sham calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Due to the interaction with cavity photons, the dy- namics of the electrons and quantum nuclei become i ∂ ∂tPe(t) = � �Fe(t) + � k,λ εk,λqk,λˆµe λ, Pe(t) � � (6a) i ∂ ∂tPn(t) = � �Fn(t) + � k,λ εk,λqk,λˆµn λ, Pn(t) � � (6b) In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (5) and (6), although many cavity modes indexed by k, λ have been considered, in the simulation below, for simplicity, we will take into account only one cavity mode polarized along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Similar to RT-NEO dynamics with fixed classical nu- clei, here the classical nuclei are also assumed to be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (5) and (6) can be further combined with a mean- field propagation of the classical nuclei via Ehrenfest dy- namics, thus providing a full dynamics scheme for po- lariton chemistry applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because this extension is beyond the scope of this manuscript, we will report this development elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RT-NEO-Ehrenfest dynamics The RT-NEO-Ehrenfest dynamics12,13 combines the real-time dynamics of the electrons and quantum nuclei and the mean-field motion of the classical nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this approach, the electrons and quantum nuclei are propa- gated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (1), and the remaining nuclei are propagated classically by the following equations of motion: ˙RI = PI MI (7a) ˙PI = −∇IE[Pe′(t), Pn′(t), {RI}] (7b) Here, RI, PI, and MI denote the position, momentum, and mass of the I-th classical nucleus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' the total energy of the molecular system E[Pe′(t), Pn′(t), {RI}] is a func- tion of the nonequilibrium densities of the electrons and quantum nuclei (Pe′(t) and Pn′(t)) as well as the po- sitions of all classical nuclei RI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 12 provides the explicit form of the Ehrenfest gradients ∇IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' When using RT-NEO-Ehrenfest dynamics for describ- ing proton transfer, a reasonable choice for treating the proton basis function centers is to use a large proton basis set including several different fixed proton basis (FPB) function centers spanning the region sampled by the transferring proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='11 Another choice is the traveling proton basis (TPB) approach,12,13 in which the proton basis function centers are allowed to move semiclassically along the proton transfer trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because this TPB approach is a semiclassical approximation of the FPB approach, we will focus on the FPB approach in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' However, the RT-NEO-Ehrenfest dynam- ics simulations performed herein can also be performed with the TPB approach in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Moreover, as mentioned above, the RT-NEO-Ehrenfest approach can also be used in conjunction with semiclas- sical RT-NEO dynamics for polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Electronic BO Approximation When the electronic BO approximation is applied, the protonic dynamics is still propagated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (1b) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 4 (6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For the electrons, at each time step, the electronic density matrix is quenched to the ground state by solving the electronic self-consistent field (SCF) equation: Pe′(t) = SCF[ℜ(Pn′(t)), {RI(t)}] (8) Here, ℜ(Pn′(t)) denotes the real component of the pro- tonic density matrix in the non-orthogonal atomic orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because the converged Pe′ and electronic energy are real-valued, the imaginary component of Pn′ does not need to be included in the electronic Kohn-Sham ma- trix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (4a) for this SCF procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Hence, only ℜ(Pn′(t)) is used to solve the electronic SCF equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' More specifically, when solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (8), we iteratively find the converged electronic density satisfying the fol- lowing Hartee–Fock–Roothaan equation: Fe′Ce′ = SeCe′εe (9) where εe is the orbital energy matrix and Fe′ and Ce′ have been defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Here, Fe′ is a function of Pe′ (or Ce′), ℜ(Pn′), and {RI}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' SIMULATION DETAILS All the above approaches have been implemented in a developer version of Q-Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='43 The initial molecular geometries for the calculations below are given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The electrons and quantum nuclei were propagated by a modified midpoint unitary transform time-propagation scheme algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='34,44 Dur- ing time propagation, an additional predictor-corrector procedure45 was used to control the growth of numerical error in the electronic and nuclear quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The velocity Verlet algorithm was used to propagate the classical nuclei and cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The step-by-step al- gorithms of the above approaches are provided in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The RT-NEO method with fixed classical nuclei was applied to a single HCN molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The B3LYP functional46–48 was used for electron-electron exchange- correlation and the epc17-2 functional49,50 was used for electron-proton correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For the electronic basis, the cc-pVDZ electronic basis set51 was used for the heavy nuclei and the cc-pV5Z electronic basis set was used for the proton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' for the protonic basis, the PB4-F2 proton basis set52 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The initial densities for the elec- trons and quantum proton were obtained from the SCF ground state NEO-DFT solution32,33 with a tight energy convergence criterion of 10−12 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' During the real-time propagation, at time t = 0 a delta pulse was used to per- turbed the protonic Fock matrix as Fn′ + E · µn′, where E = (E0, E0, E0) and µn′ denotes the protonic dipole mo- ment matrix vector in the non-orthogonal atomic orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because we will compare the performance of calcu- lations with different time steps ∆tq, E0∆tq = 4 × 10−4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' is always assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' This restriction ensures that simulations with different time steps will produce dipole signals with the same amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As we will compare the performance with and without the electronic BO approx- imation, the delta pulse was not applied to the electronic subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The same HCN molecule was also used in the semiclas- sical RT-NEO calculations for polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A single cav- ity mode polarized along the x direction was resonantly coupled to the C H stretch mode of the molecule at ωc = 3685 cm−1 with light-matter coupling ε = 6 × 10−4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The initial conditions and computational methods for HCN were the same as those described for the RT- NEO calculations in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The initial condition for the cavity mode was set as pc(0) = 0 and qc(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=', and no delta pulse was applied to the molecular subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because the position of the cavity mode was displaced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=', in later times the excess energy in the cavity mode transferred to the C H stretch mode and generated real-time Rabi oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The cavity loss rate was assumed to be γc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The RT-NEO-Ehrenfest method was applied to in- tramolecular proton transfer in malonaldehyde with the transferring proton treated quantum mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The molecule was described at the B3LPY/epc17-2/cc- pVDZ/PB4-F2 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The initial electronic and quan- tum protonic densities were obtained from the NEO-DFT ground state solution with an energy convergence crite- rion of 10−9 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The initial velocities of the classical nuclei were set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because these classical nuclei were chosen to start out in a symmetric configuration, whereas the equilibrium geometry for the classical nu- clei is asymmetric, the classical nuclei experienced forces directed toward the asymmetric relaxed geometry, thus driving proton transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For the quantum proton, three fixed proton basis function centers were used, and each center contained a PB4-F2 proton basis set and a cc- pVDZ electronic basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' These three proton basis function centers were chosen to be near the donor oxy- gen atom (OD), the acceptor oxygen atom (OA), and the midpoint between the two centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The quantum proton position is defined as the expectation value of the pro- ton position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For RT-NEO-Ehrenfest dynamics without the electronic BO approximation, by default we set the time step for the electronic and protonic quantum dy- namics as ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs, and the nuclear gradients were evaluated every 10 time steps of the quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Under the electronic BO approximation, because a 10- fold larger time step was used for the protonic quantum dynamics, the nuclear gradients were evaluated at every time step of the quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RT-NEO vs BO-RT-NEO dynamics with fixed classical nuclei To compare the performance of RT-NEO and BO-RT- NEO with fixed classical nuclei, we applied both ap- proaches to a single HCN molecule oriented along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1a shows the RT-NEO-TDDFT dynam- ics of the x-component of the HCN nuclear dipole mo- ment, µn x(t) = Tr [Pn(t)ˆµn x], when the ground-state pro- ton density is perturbed by a delta pulse at time t = 0 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As shown in this figure, the RT-NEO-TDDFT approach yields the same dipole oscillations for any time step ∆tq ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs, whereas a larger time step ∆tq ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='097 fs leads to divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1b plots the corresponding power spectrum of the dipole signal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1a calculated by the following Fourier transform: Pn(ω) = � i=x,y,z ��F � µn i (t)e−γt��� (10) Here, for a better visualization of the spectrum, a small, artificial damping term e−γt (with γ = 10−5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=') pro- vides a small linewidth of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='8 cm−1 for the peaks in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The Padé approximation of the Fourier transform34,53 is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (10) for better fre- quency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1a, consistent spec- tra are obtained only when ∆tq ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs: the peak at 1787 cm−1 is the C H bending mode, and the peak at 3685 cm−1 is the C H stretch mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that the quantitative accuracy of these frequencies can be im- proved by increasing the size of the electronic and pro- tonic basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='54 When ∆tq ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs, the RT-NEO- TDDFT frequencies agree well with the linear-response NEO-TDDFT results, as denoted by the vertical gray dashed lines in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that the real-time and linear-response results also predict the same absorp- tion intensities as the linear-response results, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' S1 of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1c and d show the analogous results obtained with the BO-RT-NEO-TDDFT approach instead of the RT-NEO-TDDFT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Here, all the time steps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='001 ≤ ∆tq ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs yield the same real-time dynam- ics and spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1a, we find that the BO-RT-NEO-TDDFT approach produces reliable results with a 20-fold larger time step than that required for the RT-NEO-TDDFT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because the electronic density needs to be quenched to the ground state by solving the electronic SCF equation (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (8)) for each time step of the electronic BO dynamics, each time step for BO-RT-NEO-TDDFT is more expensive than a time step for RT-NEO-TDDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In practice, we find that for the calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1, the computational cost of BO-RT-NEO-TDDFT per time step is less than three times that of RT-NEO-TDDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Hence, overall, in- voking the electronic BO approximation can accelerate the calculation of the proton vibrational spectrum by a factor of > 20/3 ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Semiclassical RT-NEO vs semiclassical BO-RT-NEO dynamics for polaritons Beyond the conventional RT-NEO-TDDFT dynamics with fixed classical nuclei, when the molecular system is coupled to a classical cavity photon mode, the semiclassi- cal RT-NEO-TDDFT approach can provide a unified de- scription of vibrational and electronic strong couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2a shows µn x(t) for the x-oriented HCN molecule under vibrational strong coupling when the molecule is resonantly coupled to an x-polarized cavity mode with the cavity frequency the same as the C H stretch mode (ωc = 3685 cm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' After an initial perturbation of the cavity mode, the coherent energy transfer between the cavity mode and the molecule leads to vibrational Rabi oscillations in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' However, this behavior is only reliably captured when the simulation time step is ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 fs (solid black line), whereas using a larger time step ∆tq ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs (from red to magenta lines) leads to divergent time-domain dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In the frequency domain, the vibrational polariton spectrum can be calculated from the power spectrum of µn x(t) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Although the time-domain dy- namics is converged only when ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 fs (solid black), in the frequency domain the results with ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 fs (solid black) and ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs (dashed red) are virtually identical, presumably because the ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs real-time dipole dynamics is stable for a rela- tively long time (up to t ∼ 300 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For the converged spectrum (black or red line), the two polariton peaks are asymmetric with respect to the resonance frequency (ωc = 3685 cm−1, vertical thick gray line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because the Rabi splitting between the two polaritons is very small (within 100 cm−1) compared to the resonance frequency ωc, the polariton peaks are expected to be symmetric with respect to the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='55–57 Hence, this observed asymmetry is unphysical, reflecting a limitation of the semiclassical RT-NEO approach when describing vibrational strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2c and d show the dipole dynamics and spec- tra obtained with the electronic BO approximation un- der the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2a, we find that invoking the electronic BO approxi- mation captures stable real-time Rabi oscillations even with a 40-fold larger time step (∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs, magenta line) than that required for the semiclassical RT-NEO approach (∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 fs, gray line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Moreover, in the frequency domain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2d), the two polariton peaks are symmetric with respect to the resonance frequency (ωc = 3685 cm−1, vertical thick gray line), representing a significant advantage of the BO-RT-NEO-TDDFT ap- proach for simulating molecular polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The comparison between these two approaches allows us to understand why semiclassical RT-NEO-TDDFT 6 0 20 40 60 80 100 −1 0 1 µn x [×10−5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (a) RT-NEO fixed classical nuclei 1500 2000 2500 3000 3500 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 Pn(ω) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' units] (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='001 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='097 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs 0 20 40 60 80 100 time [fs] −1 0 1 µn x [×10−5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (c) BO-RT-NEO fixed classical nuclei 1500 2000 2500 3000 3500 4000 frequency [cm−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 Pn(ω) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' units] (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (a) RT-NEO-TDDFT dynamics of the x-component of the HCN nuclear dipole moment, µn x(t), in vacuum with fixed classical nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The quantum proton is perturbed by a delta pulse at t = 0 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Simulation results with different time steps are compared: ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='001 fs (solid black), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs (dashed red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='097 fs (dash-dotted blue), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs (dash-dotted magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (b) Corresponding nuclear dipole power spectrum, Pn(ω), as well as the linear-response NEO-TDDFT vibrationally excited state transition frequencies (vertical gray dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (c,d) The same plots as (a,b) except that the BO-RT-NEO-TDDFT approach is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that although the RT-NEO approach requires a relatively small ∆tq (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs), the BO-RT-NEO approach can still recover accurate nuclear dipole signals using a 20-fold larger ∆tq (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' without the electronic BO approximation predicts asym- metric polaritons even when the Rabi splitting is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2c, the nuclear dipole signals without the electronic BO approximation have smaller oscillation amplitudes, suggesting that the cavity energy is also transferred to the electronic degrees of freedom in the molecular system, as the cavity mode is coupled to both the electronic and nuclear Kohn–Sham matrices (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Due to this energy transfer, the higher-energy electronic excitations influence the vibra- tional polariton spectrum, leading to the asymmetry in the vibrational Rabi splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' This argument can also be understood by a simple three-state model: H = � � � ωc gv ge gv ωv 0 ge 0 ωe � � � (11) where gv (ge) denotes the coupling between the cavity mode and the vibrational (electronic) transition of fre- quency ωv (ωe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In the absence of the electronic transi- tion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=', let ge = 0), at resonance condition ωc = ωv = ω0, the two polariton frequencies are ω± = ω0 ± gv (12) which is symmetric with respect to ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Due to the exis- tence of the high-energy electronic transition, according to second-order perturbation theory, as previously shown by Shao and coworkers,58 the polariton frequencies are modified to ω′ ± = ω± − g2 e ωe − ω± (13) Because ωe ≫ ω±, the high-energy electronic transition would redshift the vibrational polaritons, as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2b compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The high-energy electronic transitions predicted in semiclassical RT-NEO-TDDFT probably arise from approximations underlying the light- matter Hamiltonian and are not physically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In contrast, if the electronic BO approximation is ap- plied, because the electrons are always quenched to the ground state, the influence of the unphysical high-energy electronic excitations is eliminated, thus preserving the symmetry of vibrational polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that the single- molecule vibrational strong coupling example described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2 is used to test our methods and should not be used to interpret polariton experiments in the collective regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' X7 0 200 400 600 800 1000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 µn x [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (a) semiclassical RT-NEO for polaritons 3600 3650 3700 3750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 Pn(ω) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' units] (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='024 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs 0 200 400 600 800 1000 time [fs] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 µn x [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (c) semiclassical BO-RT-NEO for polaritons 3600 3650 3700 3750 frequency [cm−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 Pn(ω) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' units] (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (a) Semiclassical RT-NEO-TDDFT dynamics of µn x(t) and (b) the corresponding power spectrum when a single HCN molecule is coupled to an x-polarized cavity mode with ωc = 3685 cm−1 (the vertical thick gray line in (b)) and coupling strength ε = 6 × 10−4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Simulation results with different time steps are compared: ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='005 fs (solid black), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs (dashed red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='024 fs (dash-dotted blue), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs (dash-dotted magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (c,d) The analogous plots as (a,b) except that the semiclassical BO-RT-NEO-TDDFT approach is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that invoking the BO approximation not only generates stable long-time simulation results when a larger ∆tq is used, but also removes the unphysical asymmetry in the Rabi splitting observed by the semiclassiacl RT-NEO approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' RT-NEO-Ehrenfest vs BO-RT-NEO-Ehrenfest dynamics Intramolecular proton transfer in malonaldehyde has been extensively studied both experimentally27,28 and theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='29–31 Within the NEO framework, the trans- ferring proton is treated quantum mechanically, and the remaining nuclei are treated classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For our sim- ulation, the classical nuclear geometry is chosen to be symmetric, obtained by averaging the equilibrium reac- tant and product geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The quantum proton is described by three proton basis function centers span- ning the region between the donor and acceptor oxygen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3a, the quantum proton density is chosen to be the NEO-DFT SCF so- lution localized near the donor oxygen, OD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The pro- ton remains localized at the NEO-DFT level because a multireference treatment is required to produce a delo- calized, bilobal proton density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='10 Note that this specific initial nuclear geometry and proton density serve as an illustration of the method and are not relevant to exper- imental studies of this molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3a, RT-NEO-Ehrenfest dynam- ics (black lines) predicts proton transfer from OD to OA within t = 100 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' This fast proton transfer re- action is induced by the nonequilibrium initial geome- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Specifically, the initial heavy nuclear geometry is symmetric, whereas the equilibrium geometry is asym- metric, with the proton bonded to one of the oxygen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Hence, the classical nuclei experience forces to- ward the asymmetric equilibrium geometry, accompanied by nonequilibrium quantum dynamics of the transferring proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' If the proton transfer time is defined as the time when the H OD distance (solid lines) is the same as the H OA distance (dashed lines), RT-NEO-Ehrenfest dy- namics (black lines) predicts that the proton transfer occurs at t = 48 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='59 The BO-RT-NEO-Ehrenfest dy- namics (red lines) is virtually identical to the RT-NEO- Ehrenfest dynamics, although the time step for BO-RT- NEO-Ehrenfest (∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='102 fs) is 10-folder larger than that for RT-NEO-Ehrenfest dynamics (∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For a better understanding of the proton transfer dy- namics, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 4 further depicts the nonequilibrium proton density at different times along the trajectory shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Interestingly, during and after proton transfer, the proton can become delocalized in the region between OD and OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' This delocalization is enabled by the use of three proton basis function centers spanning the re- gion that is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' A more accurate description of hydrogen tunneling is expected to require a multiref- erence NEO approach, such as multistate DFT (NEO- MSDFT)10 or a complete active space self-consistent-field (NEO-CASSCF) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='6 If the quantum proton is replaced by a classical proton and all the other conditions are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3a, y X (cavity polarization)8 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='6 H-O distance [Angstrom] (a) RT-NEO-Ehrenfest BO-RT-NEO-Ehrenfest 0 20 40 60 80 100 time [fs] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='6 H-O distance [Angstrom] (b) Ehrenfest classical proton BOMD classical proton FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (a) Electronic ground state proton transfer dynam- ics in malonaldehyde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The initial geometry and proton den- sity (in blue) are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Two approaches are compared: (i) RT-NEO-Ehrenfest dynamics (black, with ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs) and (ii) BO-RT-NEO-Ehrenfest dynamics (red, with ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='102 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The quantum proton is described by three fixed proton basis function centers spanning the region between OD and OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The solid (dashed) lines denote the distance between OD (OA) and the expectation value of the quantum proton position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' (b) The corresponding plot shown in (a) for the case when the quantum proton is replaced by a classical proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this case, both Ehrenfest dynamics (black) and BO molecular dynamics (red) predict no proton transfer, emphasizing the importance of a quantum treatment of the transferring proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In both plots, the black and red lines are virtually indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' both Ehrenfest dynamics (black lines) and BO molecu- lar dynamics (red lines) predict no proton transfer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The difference between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3b high- lights the importance of a quantum mechanical treatment of the transferring proton for proton transfer reactions, especially when hydrogen tunneling is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that the quantization of other modes for malonaldehyde has been shown to be important for a quantitatitvely ac- curate description of hydrogen tunneling30 and will be investigated within the NEO framework in future stud- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The total energy along the trajectories shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3 is plotted in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' CONCLUSION In this manuscript, we have explored the electronic BO approximation in three different flavors of NEO dynam- ics: (i) RT-NEO for proton dynamics with fixed classical nuclei, (ii) semiclassical RT-NEO for polariton dynamics with fixed classical nuclei, and (iii) RT-NEO-Ehrenfest dynamics for full molecular dynamics of molecular sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' When the BO approximation between the elec- trons and quantum protons is not invoked, the electronic and protonic dynamics are propagated on the same foot- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Because the electronic dynamics is much faster than the protonic dynamics, a small time step is needed for a converged result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' By invoking the electronic BO approx- imation, which involves quenching the electronic density to the SCF ground state at each time step, we can use an order-of-magnitude larger time step to perform the cal- culations, thus greatly reducing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' We emphasize that with this treatment, because the pro- ton density is still propagated in real time, the nonequi- librium quantum dynamics of the proton is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Moreover, we have also found that under vibrational strong coupling, the unphysical asymmetic Rabi split- ting observed in previous semiclassical RT-NEO simu- lations of polaritonic systems14 can be fixed by invoking the electronic BO approximation, demonstrating another significant advantage of this treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The application to intramolecular proton transfer in malonaldehyde highlights the importance of treating the transferring proton quantum mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In this sim- ulation, a quantum treatment leads to proton transfer, whereas a classical treatment predicts no proton trans- fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The ability to capture proton delocalization in this proton transfer system also demonstrates the capacity of the (BO)-RT-NEO-Ehrenfest approach for more exciting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' We emphasize that the BO-RT-NEO meth- ods are only applicable for electronically adiabatic sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Thus, this approximation should not be used when there are significant non-BO effects between the electrons and quantum nuclei, as in some proton-coupled electron transfer reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='60 Compared with the widely used ring polymer molecular dynamics approach3 for adiabatic nu- clear quantum dynamics, the BO-RT-NEO methods pro- vide a complementary perspective for describing vibra- tionally excited dynamics and nuclear quantum coher- ence, which will be topics of future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Overall, this work lays the groundwork for applying RT-NEO methods to a wide range of electronically adiabatic and nonadia- btic chemical and biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' ACKNOWLEDGMENTS This material is based upon work supported by the Air Force Office of Scientific Research under AFOSR Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' FA9550-18-1-0134 for the polariton simulations and by the National Science Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' CHE- 1954348 for the general NEO method developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' We 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Quantum proton density dynamics predicted by the BO-RT-NEO-Ehrenfest approach for the trajectory shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Virtually identical proton densities are predicted by the RT-NEO-Ehrenfest approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The dashed horizontal gray lines indicate the approximate equilibrium proton positions near OD and OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' During and after proton transfer, the delocalization of proton density between OD and OA is captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The quantum proton isosurface is plotted by setting the isovalue as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='001 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Note that the depiction of double bonds between classical nuclei at different times is due to the automatic rendering of the IQMol software and is not meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' thank Jonathan Fetherolf, Chris Malbon, Mathew Chow, Joseph Dickinson, and Eno Paenurk for useful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' VII.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 An(ω) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' units] (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='001 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='097 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='194 fs 0 20 40 60 80 100 time [fs] −1 0 1 µn x [×10−5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (c) BO-RT-NEO fixed classical nuclei 1500 2000 2500 3000 3500 4000 frequency [cm−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0 An(ω) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' units] (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The same plots as those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1 except that here the absorption spectrum is plotted: An = � i=x,y,z −ωIm � F � µn i (t)e−γt�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='S1 The linear-response peak heights (vertical gray solid lines) indicate the absorption intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Here, the peak ratios between the real-time signals and the linear-response signals (vertical gray lines) agree exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' S2 X0 20 40 60 80 100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0005 relative energy [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (a) RT-NEO-Ehrenfest (∆tq= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs) BO-RT-NEO-Ehrenfest (∆tq= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='102 fs) BO-RT-NEO-Ehrenfest (∆tq= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs) 0 20 40 60 80 100 time [fs] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='0005 relative energy [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='] (b) Ehrenfest classical proton (∆tq= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs) BOMD classical proton (∆tq= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='102 fs) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Energy conservation along the trajectories shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For both the NEO and classical simulations, the black lines indicate the trajectories without the BO approximation (with a time step ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='010 fs), and the red lines indicate the trajectories with the BO approximation (with a time step ∆tq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='102 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The additional yellow line in part (a) indicates the BO-RT- NEO-Ehrenfest trajectory with the same time step as the RT-NEO-Ehrenfest dynamics (black line), showing that the energy conservation behavior of BO-RT-NEO-Ehrenfest dynamics can be improved by reducing the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' S3 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' DETAILED ALGORITHMS FOR NEO DYNAMICS Algorithm S1 Semiclassical BO-RT-NEO dynamics for polaritons with fixed classical nu- clei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1: Calculate µλ(τ = 0) = Tr � Pn′(τ = 0)ˆµn′ λ � + 2Tr � Pe′(τ = 0)ˆµe′ λ � 2: for τ = ∆tq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2∆tq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' · · · do 3: Calculate µλ(τ) = Tr � Pn′(τ)ˆµn′ λ � + 2Tr � Pe′(τ)ˆµe′ λ � − µλ(τ = 0) 4: pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ + 1 2∆tq) = pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ − 1 2∆tq) − [ω2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ) + εk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λµk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ)]∆tq 5: qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ + ∆tq) = qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ) + pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ + 1 2∆tq)∆tq 6: pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ + 1 2∆tq) ∗= e−γc∆tq //cavity loss 7: Pe(n)′(τ) = [Se(n)]−1/2Pe(n)(τ)[Se(n)]−1/2 8: Build Fe(n)′ incav(τ) using Pe(n)′(τ) and qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ) 9: Fe(n) incav(τ + 1 2∆tq) = 2Fe(n) incav(τ) − Fe(n) incav(τ − 1 2∆tq) 10: counter = 1 11: while True do 12: Pn(τ + ∆tq) = e−i∆tqFn incav(τ+ 1 2 ∆tq)Pn(τ)ei∆tqFn incav(τ+ 1 2 ∆tq) 13: if scf_e AND counter == 1 then 14: Converge Pe′(τ + ∆tq) to ground state given Pn′(τ + ∆tq) 15: else 16: Pe(τ + ∆tq) = e−i∆tqFe incav(τ+ 1 2 ∆tq)Pe(τ)ei∆tqFe incav(τ+ 1 2 ∆tq) 17: end if 18: Build Fe(n)′ incav(τ + ∆tq) using Pe(n)′(τ + ∆tq) and qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='λ(τ + ∆tq) 19: Fe(n) incav(τ + 1 2∆tq) = 1 2Fe(n) incav(τ) + 1 2Fe(n) incav(τ + ∆tq) 20: if counter > 1 then 21: if |Pe(n)(τ + ∆tq) − Pe(n) test | < thres then 22: Exit the while loop 23: end if 24: end if 25: Pe(n) test = Pe(n)(τ + ∆tq) 26: counter += 1 27: end while 28: end for S4 Algorithm S2 BO-RT-NEO-Ehrenfest dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 1: ∆tNq = ∆tN/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' ∆tq = ∆tNq/m 2: for t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' ∆tN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2∆tN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' · · · do 3: if scf_e then 4: Converge Pe′(t) to ground state given Pn′(t) 5: end if 6: Compute forces F(t) using Pe′(t) and Pn′(t) 7: P(t + 1 2 ∆tN) = P(t − 1 2 ∆tN) + F(t)∆tN 8: R(t + ∆tN) = R(t) + P(t + 1 2 ∆tN)∆tN/M 9: for j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' n do 10: t′ = t + (j − 1)∆tNq 11: R(t′ + 1 2 ∆tNq) = R(t) + ((j − 1)∆tNq + 1 2 ∆tNq)P(t + 1 2 tN)/M 12: Update basis center to R(t′ + 1 2 ∆tNq) and recompute Se(n) 13: for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' m do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='τ = t + (i − 1)∆tq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Pe(n)′(τ) = [Se(n)]−1/2Pe(n)(τ)[Se(n)]−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Build Fe(n)′(τ) using Pe(n)′(τ) and R(t′ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tNq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Fe(n)(τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq) = 2Fe(n)(τ) − Fe(n)(τ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='counter = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='while True do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Pn(τ + ∆tq) = e−i∆tq[Fn(τ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq)]Pn(τ)ei∆tq[Fn(τ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='if scf_e AND counter == 1 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='22: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Converge Pe′(τ + ∆tq) to ground state given Pn′(τ + ∆tq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='23: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='24: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Pe(τ + ∆tq) = e−i∆tqFe(τ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq)Pe(τ)ei∆tqFe(τ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='25: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='26: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Build Fe(n)′(τ + ∆tq) using Pe(n)′(τ + ∆tq) and R(t′ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tNq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='27: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Fe(n)(τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 ∆tq) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 Fe(n)(τ) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='2 Fe(n)(τ + ∆tq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='28: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='if counter > 1 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='29: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='if |Pe(n)(τ + ∆tq) − Pe(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='test | < thres then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='30: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Exit the while loop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='31: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='32: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='33: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='Pe(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='test = Pe(n)(τ + ∆tq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='34: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='counter += 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='35: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='end while ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='36: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='37: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='38: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='In the above two algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' when the electronic BO approximation is invoked,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' the parameter scf_e is set to true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' and otherwise this parameter is set to false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' The parameter thres in the above two algorithms controls the maximal error in the density matrices during each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In our simulation, we have set a very loose threshold S5 10−4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' In RT-NEO-Ehrenfest dynamics, the parameters m and n are used to reduce the number of energy gradient evaluations,S2 which is the most time-consuming step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' For our calculations in the manuscript, we have set m = 1 and n = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' REFERENCES [S1]L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Tao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Pavošević, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Wildman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Hammes-Schiffer, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Li, “Real-Time Time-Dependent Nuclear-Electronic Orbital Approach: Dynamics beyond the Born- Oppenheimer Approximation,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 11, 4052–4058 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' [S2]L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Zhao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Wildman, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Tao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Schneider, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Hammes-Schiffer, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Li, “Nu- clear–electronic orbital Ehrenfest dynamics,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' 153, 224111 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} +page_content=' S6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfuAiM/content/2301.04076v1.pdf'} diff --git a/WNE5T4oBgHgl3EQfBw7L/content/tmp_files/2301.05390v1.pdf.txt b/WNE5T4oBgHgl3EQfBw7L/content/tmp_files/2301.05390v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ef6a08ed6649cb6578baef067304a32f835c821 --- /dev/null +++ b/WNE5T4oBgHgl3EQfBw7L/content/tmp_files/2301.05390v1.pdf.txt @@ -0,0 +1,1813 @@ +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC +CURVES +DETCHAT SAMART +Abstract. In this article, we study the logarithmic Mahler measure of the one-parameter +family +Qα = y2 + (x2 − αx)y + x, +denoted by m(Qα). The zero loci of Qα generically define elliptic curves Eα which are 3- +isogenous to the family of Hessian elliptic curves. We are particularly interested in the case +α ∈ (−1, 3), which has not been considered in the literature due to certain subtleties. For α +in this interval, we establish a hypergeometric formula for the (modified) Mahler measure +of Qα, denoted by ˜n(α). This formula coincides, up to a constant factor, with the known +formula for m(Qα) with |α| sufficiently large. In addition, we verify numerically that if α3 is +an integer, then ˜n(α) is a rational multiple of L′(Eα, 0). A proof of this identity for α = 2, +which is corresponding to an elliptic curve of conductor 19, is given. +1. Introduction +For any Laurent polynomial P ∈ C[x±1 +1 , . . . , x±1 +n ]\{0}, the (logarithmic) Mahler measure +of P, denoted by m(P), is the average of log |P| over the n-torus. In other words, +m(P) = +1 +(2πi)n +� +· · · +� +|x1|=···=|xn|=1 +log |P(x1, . . . , xn)|dx1 +x1 +· · · dxn +xn +. +Consider the following two families of bivariate polynomials +Pα(x, y) = x3 + y3 + 1 − αxy, +Qα(x, y) = y2 + (x2 − αx)y + x, +with the parameter α ∈ C. For α ̸= 3, the zero loci of Pα define a family of elliptic curves +known as the Hessian curves. There is a 3-isogeny between Pα(x, y) = 0 and the curve +Eα : Qα(x, y) = 0, +which is isomorphic to the curve in the Deuring form, defined by the zero locus of +Rα(x, y) = y2 + αxy + y − x3. +Observe that +(x2y)3Pα +� y +x2, 1 +xy +� += Qα(x3, y3), +from which we have m(Pα) = m(Qα) (see [20, Cor. 8]). Similarly, the change of variables +(x, y) �→ (−y, xy) transforms the family Rα into Qα without changing the Mahler measure. +For some technical reasons, we will focus on m(Qα) only. Following notation in previous +papers [13, 17, 18], we let +n(α) := m(Qα). +Date: January 16, 2023. +1 +arXiv:2301.05390v1 [math.NT] 13 Jan 2023 + +2 +DETCHAT SAMART +The Mahler measure of Qα (and its allies) was first studied by Boyd in his seminal paper +[4]. He verified numerically that for several α ∈ Z with α /∈ (−1, 3), +(1.1) +n(α) +?= rαL′(Eα, 0), +where rα ∈ Q and A +?= B means A and B are equal to at least 50 decimal places. Later, +Rodriguez Villegas [23] made an observation that (1.1) seems to hold for all sufficiently large +|α| which is a cube root of an integer. The values of α for which (1.1) has been proven +rigorously are given in Table 1. +α +Conductor of Eα +rα +Reference(s) +−6 +27 +3 +[23] +−3 +54 +1 +[7] +−2 +35 +1 +[7] +−1 +14 +2 +[16],[7] +3√ +32 +20 +8 +3 +[18] +3√ +54 +36 +3 +2 +[17] +5 +14 +7 +[16] +Table 1. Proven formulas for (1.1) +In addition to the results in this list, there are some known identities which relate n(α), +where α is a cube root of an algebraic integer, to a linear combination of L-values. For +example, the author proved in [19] that the following identity is true: +(1.2) +n +� +3� +6 − 6 +3√ +2 + 18 +3√ +4 +� += 1 +2 (L′(F108, 0) + L′(F36, 0) − 3L′(F27, 0)) , +where FN is an elliptic curve over Q of conductor N. In compliance with Boyd’s results, it +is worth noting that +3� +6 − 6 +3√ +2 + 18 +3√ +4 ≈ 3.0005 > 3. +We refer the interested reader to the aforementioned paper for more conjectural identities of +this type. +Recall that a polynomial P(x1, x2, . . . , xn) is said to be reciprocal if there exist integers +d1, d2, . . . , dn such that +xd1 +1 xd2 +2 · · · xdn +n P(1/x1, 1/x2, . . . , 1/xn) = P(x1, x2, . . . , xn), +and nonreciprocal otherwise. For a family of two-variable polynomials +˜Pα(x, y) = A(x)y2 + (B(x) + αx)y + C(x), +let Zα be the zero locus of ˜Pα(x, y) and let K be the set of α ∈ C for which ˜Pα vanishes on +the 2-torus. Boyd conjectured from his experiments that, for all integer α in the unbounded +component G∞ of C\K, if ˜Pα is tempered (see [23] for the definition), then m( ˜Pα) is related to +an L-value of elliptic curve (if Zα has genus one) or Dirichlet character (if Zα has genus zero). +If ˜Pα(x, y) is reciprocal, then it can be shown that K ⊆ R, implying G∞ = C. Hence by +continuity one could expect that identities like (1.1) hold for all α ∈ Z, with some exceptions +in the genus zero cases. Examples of polynomials satisfying these properties include the + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +3 +families x + 1/x + y + 1/y + α and (1 + x)(1 + y)(x + y) − αxy, whose Mahler measures have +been extensively studied over the past few decades (e.g. see [4, 12, 13, 14, 15, 17, 18, 23]). +The family Qα, on the other hand, is nonreciprocal, so the set K of α ∈ C for which Qα +vanishes on the 2-torus has nonempty interior. In fact, as described in [4, §2B] and [23, §14], +K is the region inside a hypocycloid whose vertices are the cube roots of 27 in the complex +plane and K ∩ R = (−1, 3). This is illustrated in Figure 1 below. It is known (see, for +Figure 1. +example, [17, Thm. 3.1]) that, for most complex numbers α, n(α) is expressible in terms of +a generalized hypergeometric function: if |α| is sufficiently large, then +(1.3) +n(α) = Re +� +log α − 2 +α3 4F3 +� 4 +3, +5 +3, 1, 1 +2, 2, 2 +���� +27 +α3 +�� +. +Since both sides of (1.3) are real parts of holomorphic functions that agree at every point +in an open subset of the region C\K, the formula (1.3) is valid for all α ∈ C\K; i.e., for +all α on the border and outside of the hypocycloid in Figure 1. Because of this anomalous +property of the family Qα (and other nonreciprocal families in general), to our knowledge, +there are no known results about n(α) for α ∈ K, with an exception for the case α = 0 due +to Smyth [21], namely +n(0) = m(x3 + y3 + 1) = m(x + y + 1) = L′(χ−3, −1), +where χ−N = +� N +· +� +. The aim of this paper is to give a thorough investigation of these omitted +values of n(α). In particular, we are interested in establishing formulas analogous to (1.1) +and (1.3) for α ∈ (−1, 3). +Let us first factorize Qα as +Qα(x, y) = y2 + (x2 − αx)y + x = (y − y+(x))(y − y−(x)), +where +y±(x) = −(x2 − αx) +� +1 +2 ± +� +1 +4 − +1 +x(x − α)2 +� +, +and denote +J(α) = 1 +π +� π +cos−1( α−1 +2 ) +log |y+(eiθ)|dθ. + +24 +DETCHAT SAMART +(Here and throughout we use the principal branch for the complex square root.) The signif- +icance of the function J(α), which can be seen as a part of m(Qα), will be made clear later. +For α ∈ (−1, 1) ∪ (1, 3), y±(x) are functions on T1 := {x ∈ C | |x| = 1}. If α = 1, y±(x) +have only one removable singularity on T1, namely x = 1, so we can extend its domain to +T1 by setting +y±(1) = lim +x→1 y±(x) = ∓i. +The first main result of this paper is the following hypergeometric formula, which extends +(1.3). +Theorem 1. Let ˜n(α) = n(α) − 3J(α). The following identity is true: +˜n(α) = +� +� +� +� +� +� +� +Re +� +log α − +2 +α3 4F3 +� +4 +3 , 5 +3 , 1, 1 +2, 2, 2 +���� +27 +α3 +�� +if α ∈ (−1, 0), +−2 Re +� +log α − +2 +α3 4F3 +� +4 +3 , 5 +3 , 1, 1 +2, 2, 2 +���� +27 +α3 +�� +if α ∈ (0, 3). +We also study ˜n(α) from the arithmetic point of view. We discovered from our numerical +computation that for α ∈ (−1, 3) which is a cube root of an integer ˜n(α) (conjecturally) +satisfies an identity analogous to (1.1). Numerical data for this identity are given in Table 2. +This identity can be proven rigorously in some cases using Brunault-Mellit-Zudilin’s formula +(see Theorem 8 below). As a concrete example, we prove the following result. +Theorem 2. Let ˜n(α) = n(α) − 3J(α) and let Eα be the elliptic curve defined by the zero +locus of Qα. Then the following evaluation is true: +˜n(2) = −3L′(E2, 0). +(1.4) +Note that E2 has conductor 19. What makes this curve special is that it admits a modular +unit parametrization. The celebrated modularity theorem asserts that every elliptic curve +over Q can be parametrized by modular functions. However, a recent result of Brunault [6] +reveals that there are only a finite number of them which can be parametrized by modular +units (i.e. modular functions whose zeros and poles are supported at the cusps). In order +to apply Brunault-Mellit-Zudilin’s formula, one needs to show that the integration path +corresponding to ˜n(α) becomes a closed path for the regulator integral defined on the curve +Qα(x, y) = 0. This path can then be translated into a path joining cusps on the modular +curve X0(19). The calculation for this part will be worked out in Section 3. +2. The hypergeometric formula +The goal of this section is to prove Theorem 1. To achieve this goal, we need some auxiliary +results as follows. +Lemma 3. Let α ∈ C and x ∈ C\{α}. If |x| = 1, then |y−(x)| ≤ 1 ≤ |y+(x)|. +Proof. Assume that |x| = 1 and write +� +1 +4 − +1 +x(x−α)2 = a + bi, where a, b ∈ R. Since the +square root is defined using the principal branch, we have a ≥ 0. Hence +|y−(x)| = |x2 − αx| +���� +1 +2 − a − bi +���� ≤ |x2 − αx| +���� +1 +2 + a + bi +���� = |y+(x)|. +Since |y+(x)||y−(x)| = |x| = 1, it follows that |y−(x)| ≤ 1 ≤ |y+(x)|, as desired. +□ + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +5 +By Lemma 3 and Jensen’s formula, we have +(2.1) +n(α) = 1 +2π +� π +−π +log |y+(eiθ)|dθ += 1 +π +� π +0 +log |y+(eiθ)|dθ += 1 +π Re +� π +0 +log +� +(x − α) +� +1 +2 + +� +1 +4 − +1 +x(x − α)2 +�� ���� +x=eiθ +dθ, +where the second equality follows from y+(e−iθ) = y+(eiθ). Next, we shall locate the toric +points, the points of intersection of the affine curve Qα = 0 and the 2-torus, explicitly. +Proposition 4. Let T2 = {(x, y) ∈ C2 | |x| = |y| = 1} and for each α ∈ C let Cα = {(x, y) ∈ +C2 | Qα(x, y) = 0}. Then for α ∈ (−1, 3), we have +Cα ∩ T2 = +�� +eit, y±(eit) +� +| t = 0, ± cos−1 +�α − 1 +2 +�� +. +Proof. Assume first that α ̸= 1. Suppose |x| = 1, so x = eit for some t ∈ (−π, π]. Since +y±(x) = −(x2 − αx) +� +1 +2 ± +� +1 +4 − +1 +x(x−α)2 +� +and |y+(x)||y−(x)| = |x| = 1, we have that the +condition |y+(x)| = 1 = |y−(x)| is equivalent to the equality +(2.2) +����� +1 +2 + +� +1 +4 − +1 +x(x − α)2 +����� = +����� +1 +2 − +� +1 +4 − +1 +x(x − α)2 +����� . +It is easily seen that (2.2) holds if and only if +� +1 +4 − +1 +x(x−α)2 is purely imaginary; equivalently, +x(x − α)2 ∈ (0, 4). Simple calculation yields +Re(x(x − α)2) = (cos t)((cos t − α)2 − sin2 t) − 2(cos t − α) sin2 t, +(2.3) +Im(x(x − α)2) = (sin t)(2 cos t − (α − 1))(2 cos t − (α + 1)), +(2.4) +|x(x − α)2| = |x − α|2 = α2 − 2α cos t + 1. +(2.5) +We have from (2.4) that x(x − α)2 ∈ R if and only if sin t = 0 or cos t = (α ± 1)/2. +If sin t = 0, then either cos t = 1 or cos t = −1. If cos t = −1, then x(x−α)2 = −(1−α)2 < 0. +If cos t = (α+1)/2, then α ∈ (−1, 1) and sin2 t = 1−((α + 1)/2)2 , from which we can deduce +using (2.3) that +x(x − α)2 = Re(x(x − α)2) = α − 1 < 0. +Also, it can be shown using (2.3) and (2.5) that the remaining cases, cos t = 1 and cos t = α−1 +2 , +imply 0 < x(x − α)2 < 4. As a consequence, the curve Cα = 0 intersects T2 exactly at +(eit, y±(eit)), where t = 0, ± cos−1 � α−1 +2 +� +. The same result also holds for α = 1 by continuity. +□ +Lemma 5. For λ ∈ [1, 2), let pλ(x) = x(λ2 − x) +� +x2 + +� 4 +λ − λ2� +x + 4 +λ2 +� +and γ = λ3−λ−2 +2λ ++ +λ+1 +2λ +� +(2 − λ)(λ3 + λ − 2)i. Then we have +(2.6) +� γ +λ−1 +1 +� +−pλ(x) +dx = +� −1/λ +0 +1 +� +−pλ(x) +dx, + +6 +DETCHAT SAMART +where the left (complex) integral is path-independent in the upper-half unit disk and the right +integral is a real integral. +Proof. Note first that |γ| = 1 and the nonzero roots of pλ(x) are +x1(λ) = λ2, +x2(λ) = λ3 − 4 + +� +λ3(λ3 − 8) +2λ +, and x3(λ) = λ3 − 4 − +� +λ3(λ3 − 8) +2λ +, +which lie outside the unit circle, so the integration path for the left integral can be chosen +to be any path joining λ − 1 and γ in the upper-half unit disk. For 1 < λ < 2 and x ∈ R, +x2 + +�4 +λ − λ2 +� +x + 4 +λ2 = +� +x + +�2 +λ − λ2 +2 +��2 +− λ +�λ3 +4 − 2 +� +> 0, +so −pλ(x) > 0 for all x ∈ (−1/λ, 0) and the integral on the right-hand side is real. Define +the symmetric polynomial1 Fλ(x, y) by +Fλ(x, y) := λ2(λ − 1)x2y2 − λ(λ − 1)(λ3 − λ2 + λ − 2)(x2y + xy2) + λ2(x2 + y2) ++ (λ7 − 2λ6 + 2λ5 − 5λ4 + 6λ3 − 6λ2 + 6λ − 4)xy − 2λ2(λ − 1)(x + y) + λ2(λ − 1)2. +Then, for λ ∈ [1, 2), Fλ(x, y) transforms the interval (−1/λ, 0) to a continuous path in the +upper-half unit disk joining γ and λ−1. Moreover, by implicitly differentiating Fλ(x, y) = 0, +it can be checked using a computer algebra system that the following equation holds on this +curve: +�dy +dx +�2 +− pλ(y) +pλ(x) = 0, +from which (2.6) follows immediately. +□ +Lemma 6. For α ∈ (−1, 3), if α = (λ3 − 2)/λ, then +d +dα (n(α) − 3J(α)) = − 1 +π +� λ2 +0 +1 +� +pλ(x) +dx, +where pλ(x) is defined as in Lemma 5. +Proof. Differentiating (2.1) with respect to α yields +d +dαn(α) = 1 +π Re +� π +0 +√x +� +x(x − α)2 − 4 +���� +x=eiθ +dθ. +Let c(α) = cos−1 � α−1 +2 +� +. Then, by Leibniz integral rule and Proposition 4, we have +d +dαJ(α) = 1 +π +� +− log +��y+ +� +eic(α)��� d +dαc(α) ++ Re +� π +c(α) +d +dα log +� +(x − α) +� +1 +2 + +� +1 +4 − +1 +x(x − α)2 +�� ���� +x=eiθ +dθ +� += 1 +π Re +� c(α) +0 +√x +� +x(x − α)2 − 4 +���� +x=eiθ +dθ. +1We obtain the polynomial Fλ(x, y) using numerical values of the integrals in (2.6). The PSLQ algorithm +plays an essential role in identifying its coefficients. + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +7 +It follows that +(2.7) +d +dα (n(α) − 3J(α)) = − 1 +π Re +�� +2 +� π +c(α) +− +� c(α) +0 +� +√x +� +x(x − α)2 − 4 +���� +x=eiθ +dθ +� +. +Let α = (λ3 − 2)/λ. Then α maps the interval (1, 2) bijectively onto (−1, 3) and +(2.8) +x(x − α)2 − 4 = (x − λ2) +� +x2 + +�4 +λ − λ2 +� +x + 4 +λ2 +� +. +An inspection of the signs of the square roots in the integrand reveals that +� π +c(α) +√x +� +x(x − α)2 − 4 +���� +x=eiθ +dθ = − +� −1 +γ +1 +� +pλ(x) +dx = +�� γ +0 +− +� −1 +0 +� +1 +� +pλ(x) +dx, +(2.9) +� c(α) +0 +√x +� +x(x − α)2 − 4 +���� +x=eiθ +dθ = +� γ +1 +1 +� +pλ(x) +dx = +�� γ +0 +− +� 1 +0 +� +1 +� +pλ(x) +dx, +(2.10) +where +γ = eic(α) = α − 1 +2 ++ +� +(3 − α)(α + 1) +2 +i = λ3 − λ − 2 +2λ ++ λ + 1 +2λ +� +(2 − λ)(λ3 + λ − 2)i. +Since pλ(x) < 0 for any x ∈ (−1, 0) and λ ∈ (1, 2), we have +(2.11) +Re +� −1 +0 +1 +� +pλ(x) +dx = 0. +Plugging (2.9),(2.10), and (2.11) into (2.7) gives +(2.12) +d +dα (n(α) − 3J(α)) = − 1 +π +�� 1 +0 +1 +� +pλ(x) +dx + Re +� γ +0 +1 +� +pλ(x) +dx +� +. +Note that the mapping +(2.13) +x �→ λ2 − x +λx + 1 +is the unique M¨obius transformation which interchanges the following values: +0 ↔ λ2, +1 ↔ λ − 1, +x2(λ) ↔ x3(λ), +where x2(λ) and x3(λ) are the roots of x2 + (4/λ − λ2)x + 4/λ2. Hence using (2.13) we have +� λ−1 +0 +1 +� +pλ(x) +dx = +� λ2 +1 +1 +� +pλ(x) +dx. +Finally, we have from Lemma 5 that +� γ +λ−1 +1 +� +pλ(x) +dx = +� −1/λ +0 +1 +� +pλ(x) +dx ∈ iR, +so (2.12) immediately gives the desired result. +□ + +8 +DETCHAT SAMART +Lemma 7. For α ∈ (−1, 0), we have +(2.14) +d +dα (n(α) − 3J(α)) = Re +� 1 +α +2F1 +� 1 +3, +2 +3 +1 +���� +27 +α3 +�� +. +For α ∈ (0, 3), we have +(2.15) +d +dα (n(α) − 3J(α)) = −2 Re +� 1 +α +2F1 +� 1 +3, +2 +3 +1 +���� +27 +α3 +�� +. +Proof. Let us first consider (2.15). We prove this identity by expressing both sides in terms +of the elliptic integral of the first kind +K(z) = +� 1 +0 +dx +� +(1 − x2)(1 − z2x2) +. +Again, let α = (λ3 − 2)/λ. Following a procedure in [11, Ch. 3], we let +u = −1 − +√ +λ3 + 1 +λ +, +v = −1 + +√ +λ3 + 1 +λ +, +x = ut − v +t − 1 . +This substitution transforms the integral in Lemma 6 (without the factor −1/π) into +λ +2 +√ +λ3 + 1 +� t2 +t1 +dt +� +(B1t2 + A1)(B2t2 + A2) +, +where +t1 = −λ3 + 2 − 2 +√ +λ3 + 1 +λ3 +, +t2 = −t1, +A1 = λ3 + 2 − 2 +√ +λ3 + 1 +4 +√ +λ3 + 1 +, +B1 = −λ3 − 2 − 2 +√ +λ3 + 1 +4 +√ +λ3 + 1 +, +A2 = −λ3 + 2 + 2 +√ +λ3 + 1 +4 +√ +λ3 + 1 +, +B2 = λ3 − 2 + 2 +√ +λ3 + 1 +4 +√ +λ3 + 1 +. +Observe that, for λ ∈ (1, 2), we have A1, A2, B2 > 0, B1 < 0, and +� +−A1/B1 = t2. Hence +the substitution t �→ +� +−A1/B1t yields +λ +2 +√ +λ3 + 1 +� t2 +t1 +dt +� +(B1t2 + A1)(B2t2 + A2) += +λ +2 +√ +λ3 + 1 +� +− +1 +A2B1 +� 1 +−1 +dt +� +(1 − t2) +� +1 − A1B2 +A2B1t2 +� += +4λ +��√ +λ3 + 1 + 1 +�3 � +3 − +√ +λ3 + 1 +�K +�� +A1B2 +A2B1 +� +. +Therefore, we obtain +(2.16) +d +dα (n(α) − 3J(α)) = − +4λ +π +��√ +λ3 + 1 + 1 +�3 � +3 − +√ +λ3 + 1 +�K +�� +A1B2 +A2B1 +� +. +On the other hand, we apply the hypergeometric transformation [18, p. 410] +(2.17) +Re 2F1 +� 1 +3, +2 +3 +1 +���� +27y +(y − 2)3 +� += y − 2 +y + 4 +2F1 +� 1 +3, +2 +3 +1 +���� +27y2 +(y + 4)3 +� +, + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +9 +which is valid for y ∈ (2, 8), to write the right-hand side of (2.15) as +− 2 +α Re +� +2F1 +� 1 +3, +2 +3 +1 +���� +27 +α3 +�� += +2λ +2 − λ3 Re +� +2F1 +� 1 +3, +2 +3 +1 +���� +27λ3 +(λ3 − 2)3 +�� += − +2λ +λ3 + 4 +2F1 +� 1 +3, +2 +3 +1 +���� +27λ6 +(λ3 + 4)3 +� +. +The substitution λ = +3� +4(p + p2) gives a bijection from the interval (( +√ +3 − 1)/2, 1) onto +( +3√ +2, 2), which is corresponding to the interval (0, 3) for α, with the inverse mapping p = +( +√ +λ3 + 1 − 1)/2. We apply this substitution together with a classical result of Ramanujan +[2, Thm 5.6] to deduce +− +2λ +λ3 + 4 +2F1 +� 1 +3, +2 +3 +1 +���� +27λ6 +(λ3 + 4)3 +� += − +3� +4(p + p2) +2(p2 + p + 1) +2F1 +� 1 +3, +2 +3 +1 +���� +27p2(1 + p)2 +4(1 + p + p2)3 +� += − +3� +4(p + p2) +2√1 + 2p +2F1 +� 1 +2, +1 +2 +1 +���� +p3(2 + p) +1 + 2p +� += − +λ +2 +4√ +λ3 + 1 +2F1 +� 1 +2, +1 +2 +1 +���� ρ(λ) +� +, +where +ρ(λ) = λ6 − 4λ3 − 8 + 8 +√ +λ3 + 1 +16 +√ +λ3 + 1 +. +Then by the identities [1, Eq. 3.2.3], [9, Eq. 15.8.1] +K(k) = π +2 +2F1 +� 1 +2, +1 +2 +1 +���� k2 +� +, +K(√r) = +1 +√1 − rK +�� +r +r − 1 +� +, +we arrive at +(2.18) +− +λ +2 +4√ +λ3 + 1 +2F1 +� 1 +2, +1 +2 +1 +���� ρ(λ) +� += − +4λ +π +��√ +λ3 + 1 + 1 +�3 � +3 − +√ +λ3 + 1 +�K +�� +ρ(λ) +ρ(λ) − 1 +� +. +It can be calculated directly that +ρ(λ) +ρ(λ) − 1 = λ6 − 4λ3 − 8 + 8 +√ +λ3 + 1 +λ6 − 4λ3 − 8 − 8 +√ +λ3 + 1 = A1B2 +A2B1 +, +so the right-hand side of (2.18) coincides with that of (2.16) and the proof is completed. +Equation 2.14 also follows from the arguments above, provided that (2.17) is replaced with +Re 2F1 +� 1 +3, +2 +3 +1 +���� +27y +(y − 2)3 +� += 4 − 2y +y + 4 +2F1 +� 1 +3, +2 +3 +1 +���� +27y2 +(y + 4)3 +� +, +which is valid for y ∈ (1, 2). +□ +Proof of Theorem 1. For α > 3, we can apply term-by-term differentiation to show that +d +dα Re +� +log α − 2 +α3 4F3 +� 4 +3, +5 +3, 1, 1 +2, 2, 2 +���� +27 +α3 +�� += Re +� 1 +α +2F1 +� 1 +3, +2 +3 +1 +���� +27 +α3 +�� +. + +10 +DETCHAT SAMART +By analytic continuation, the above equality also holds for α ∈ (−1, 0) ∪ (0, 3). Therefore, +integrating both sides of (2.14) and (2.15) yields +n(α) − 3J(α) = +� +� +� +� +� +� +� +Re +� +log α − +2 +α3 4F3 +� +4 +3 , 5 +3 , 1, 1 +2, 2, 2 +���� +27 +α3 +�� ++ C1, +if − 1 < α < 0, +−2 Re +� +log α − +2 +α3 4F3 +� +4 +3 , 5 +3 , 1, 1 +2, 2, 2 +���� +27 +α3 +�� ++ C2, +if 0 < α < 3, +for some constants C1 and C2. Since α = −1 and α = 3 are on the boundary of the set K +defined in Section 1, an argument underneath (1.3) implies that +(2.19) +n(−1) = Re +� +log(−1) + 24F3 +� 4 +3, +5 +3, 1, 1 +2, 2, 2 +���� − 27 +�� +, +n(3) = Re +� +log 3 − 2 +27 +4F3 +� 4 +3, +5 +3, 1, 1 +2, 2, 2 +���� 1 +�� +. +Hence, by continuity of n(α) and (2.19), we have +C1 = +lim +α→−1+(−3J(α)) = 0, +C2 = 3 lim +α→3−(n(3) − J(α)) = 0, +and the desired result follows. +□ +3. Relation to elliptic regulators and L-values +In this section, we prove Theorem 2, which resembles Boyd’s conjectures (1.1). The key +idea of the proof is to rewrite ˜n(α) as a regulator integral over a path joining two cusps and +apply Brunault-Mellit-Zudilin formula [25], which is stated below. As usual, we define the +real differential form η(f, g) for meromorphic functions f and g on a smooth curve C as +η(f, g) = log |f|d arg(g) − log |g|d arg(f), +where d arg(g) = Im(dg/g). +Theorem 8 (Brunault-Mellit-Zudilin). Let N be a positive integer and define +ga(τ) = qNB2(a/N)/2 +� +n≥1 +n≡a mod N +(1 − qn) +� +n≥1 +n≡−a mod N +(1 − qn), +q := e2πiτ, +where B2(x) = {x}2 − {x} + 1/6. Then for any a, b, c ∈ Z such that N ∤ ac and N ∤ bc, +� i∞ +c/N +η(ga, gb) = 1 +4πL(f(τ) − f(i∞), 2), +where f(τ) = fa,b;c(τ) is a weight 2 modular form given by +fa,b;c = ea,bceb,−ac − ea,−bceb,ac +and +ea,b(τ) = 1 +2 +�1 + ζa +N +1 − ζa +N ++ 1 + ζb +N +1 − ζb +N +� ++ +� +m,n≥1 +� +ζam+bn +N +− ζ−(am+bn) +N +� +qmn, +ζN := e +2πi +N . + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +11 +Let us first outline a general framework for computing ˜n(α) in terms of a regulator integral. +Recall from Deninger’s result [8, Prop. 3.3] that if Qα(x, y) is irreducible, then +n(α) = − 1 +2π +� +γα +η(x, y), +where γα is the Deninger path on the curve Eα : Qα(x, y) = 0; i.e., +γα = {(x, y) ∈ C2 | |x| = 1, |y| > 1, Qα(x, y) = 0}. +If Qα does not vanish on the torus, then γα becomes a closed path, so the Bloch-Beilinson +conjectures give a prediction that (1.1) holds for all sufficiently large |α| with suitable arith- +metic properties; in this case, we need that α be a cube root of an integer. On the other +hand, if α ∈ (−1, 3), then the functions y±(x) defined in Section 1 are discontinuous at the +toric points as given in Proposition 4, so γα is not closed in this case. We will show, however, +that the path on Eα corresponding to ˜n(α) is indeed closed, so that ˜n(α) is (conjecturally) +related to L-values. The numerical data supporting this hypothesis are given in Table 2. +Lemma 9. Let α ∈ (−1, 3) and let ˜n(α) = n(α) − 3J(α). Then +˜n(α) = − 1 +2π +� +˜γα +η(x, y) +for some ˜γα ∈ H1(Eα, Z)−. In other words, the integration path associated to the modified +Mahler measure ˜n(α) can be realized as a closed path which is anti-invariant under complex +conjugation. +Proof. We label the six toric points obtained from Proposition 4 as follows: +P ± +1 = (1, y±(1)) = (1, Y±) , +P ± +2 = (e±ic(α), y+(e±ic(α))) = (Y±, 1) , +P ± +3 = (e±ic(α), y−(e±ic(α))) = (Y±, Y±) , +where c(α) = cos−1 � α−1 +2 +� +and +Y± = α − 1 +2 +± +� +(3 − α)(α + 1) +2 +i. +Observe that ˜n(α) can be rewritten as ˜n(α) = I(α) − 2J(α), where +I(α) = 1 +2π +� c(α) +−c(α) +log |y+(eiθ)|dθ, +J(α) = 1 +2π +� 2π−c(α) +c(α) +log |y+(eiθ)|dθ. +Let S = {P ± +1 , P ± +2 , P ± +3 }. Then we may identify the paths corresponding to I(α) and J(α) as +elements in the relative homology H1(Eα, S, Z), say γI and γJ, respectively. In other words, +we write +I(α) = − 1 +2π +� +γI +η(x, y), +J(α) = − 1 +2π +� +γJ +η(x, y), + +12 +DETCHAT SAMART +and boundaries of these paths can be seen as 0-cycles on S. Computing the limits of y+(eiθ) +as θ approach 0, c(α), and −c(α) from both sides, we find that +lim +θ→−c(α)+ y+(eiθ) = +lim +θ→c(α)− y+(eiθ) = 1, +lim +θ→0+ y+(eiθ) = Y−, +lim +θ→0− y+(eiθ) = Y+. +Therefore, the path γI is discontinuous at θ = 0 and +(3.1) +∂γI = [[P + +1 ] − [P − +2 ]] + [[P + +2 ] − [P − +1 ]]. +This is illustrated in Figure 2 for α = 2, where the dashed curves in the upper-half plane +and the lower-half plane, both oriented counterclockwise, correspond to θ ∈ (−c(α), 0) and +θ ∈ (0, c(α)), respectively. Next, observe that +Figure 2. y+(eiθ), θ ∈ [0, 2π) +Figure 3. y−(eiθ), θ ∈ [0, 2π) +lim +θ→c(α)+ y−(eiθ) = 1 = +lim +θ→−c(α)− y−(eiθ), +and γJ can be identified as the path {(eiθ, y−(eiθ)) | c(α) < θ < 2π − c(α)} (with reversed +orientation), implying +(3.2) +∂γJ = [[P + +2 ] − [P − +2 ]]. +(For α = 2, the y-coordinate of this path is the bold curve inside the unit circle, as illustrated +in Figure 3, oriented clockwise.) Let +x±(y) = (αy − 1) +2y +� +1 ± +� +1 − +4y3 +(αy − 1)2 +� +, +which is obtained by solving Qα(x, y) = 0 in the variable x. Then y+(x−(eiθ)) = eiθ for any +θ ∈ R and +lim +θ→0± x−(eiθ) = e∓ic(α), +lim +θ→c(α)− x−(eiθ) = 1 = +lim +θ→−c(α)+ x−(eiθ). + +2- +1 +Y+ +-1 +0 +Y +-1 +21 +0 +-1MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +13 +Define +γ1 = {(x−(eiθ), y+(x−(eiθ))) | −c(α) < θ < 0}, +γ2 = {(x−(eiθ), y+(x−(eiθ))) | 0 < θ < c(α)}. +Then one can deduce using the limits of x− at the endpoints above that the closure of +γ1 ∪ γJ ∪ γ2, which we call γ′ +J, is a continuous path joining P − +1 and P + +1 . Moreover, since the +y-coordinates of γ1 and γ2 lie on the unit circle, we have +J(α) = − 1 +2π +� +γ′ +J +η(x, y), +where +(3.3) +∂γ′ +J = [[P + +1 ] − [P − +1 ]]. +Finally, we arrive at +˜n(α) = I(α) − 2J(α) = − 1 +2π +�� +γI +η(x, y) − +� +γJ +η(x, y) − +� +γ′ +J +η(x, y) +� += − 1 +2π +� +˜γα +η(x, y), +where, by (3.1),(3.2), and (3.3), ˜γα has trivial boundary, from which we can conclude that +˜γα ∈ H1(Eα, Z). It is clear from the construction of the paths γI, γJ, and γ′ +J that they are +anti-invariant under the action of complex conjugation. Therefore, we have ˜γα ∈ H1(Eα, Z)−, +as desired. +□ +We shall use Theorem 8 and Lemma 9 to prove Theorem 2. We essentially follow an +approach of Brunault [7] in identifying the path ˜γα as the push-forward of a path joining +cusps on X0(19) with the aid of Magma and Pari/GP. +Proof of Theorem 2. The elliptic curve E2 : y2 +(x2 −2x)y +x = 0 has Cremona label 19a3, +so it admits a modular parametrization ϕ : X0(19) → E2 by the modularity theorem. Let +f2 be the weight 2 newform of level 19 associated to the curve E2 and let ω = 2πif2(τ)dτ, +the pull-back of the holomorphic differential form on E2. Using Magma and Pari/GP codes +in [7, §6.1], we find that +� −4/19 +4/19 +ω = −Ω− ≈ −4.12709i, +where Ω− is the imaginary period of E2 obtained by subtracting twice the complex period +from the real period of E2. Hence it follows that ˜γα = ϕ∗ +� 4 +19, − 4 +19 +� +, where ˜γα is the path +associated to ˜n(α). Let +x(τ) = −g1g7g8 +g2g3g5 +, +y(τ) = g1g7g8 +g4g6g9 +, +where ga := ga(τ) is as given in Theorem 8 with N = 19. By a result of Yang [24, Cor. 3], +both x(τ) and y(τ) are modular functions on Γ1(19). Multiplying each term by a modular +form in M2(Γ1(19)), one can apply Sturm’s theorem [22, Cor. 9.19], with the Sturm bound +B(M2(Γ1(19))) = 60, to show that y(τ)2 + (x(τ)2 − 2x(τ))y(τ) + x(τ) vanishes identically; + +14 +DETCHAT SAMART +i.e., (x(τ), y(τ)) parametrizes the curve E2. Finally, by Lemma 9 and Theorem 8, we find +that +˜n(α) = − 1 +2π +� +˜γα +η(x, y) = 1 +2π +� 4/19 +−4/19 +η(x(τ), y(τ)) = − 1 +4π2L(57f2, 2) = −3L′(f2, 0), +where the last equality follows from the functional equation for L(f2, s). +□ +In addition to (1.4), we discovered that, for all α ∈ (−1, 3) which are a cube root of an +integer, the following identity holds numerically: +(3.4) +˜n(α) +?= rαL′(Eα, 0), +where rα ∈ Q. The data of rα and Eα are given in Table 2. +α3 +Cremona label of Eα +rα +α3 +Cremona label of Eα +rα +1 +26a3 +−1 +14 +2548d1 +1/36 +2 +20a1 +−5/3 +15 +1350i1 +1/18 +3 +54a1 +−2/3 +16 +44a1 +−4/3 +4 +92a1 +−1/3 +17 +2890e1 +−1/27 +5 +550d1 +−1/9 +18 +324b1 +−1/6 +6 +756f1 +−1/18 +19 +722a1 +1/9 +7 +490a1 +1/9 +20 +700i1 +−1/9 +8 +19a3 +−3 +21 +2464k1 +−1/27 +9 +162c1 +−1/3 +22 +2420d1 +1/26 +10 +1700c1 +1/36 +23 +1058b1 +−1/12 +11 +242b1 +−1/3 +24 +27a1 +−3 +12 +540d1 +1/9 +25 +50a1 +−5/3 +13 +2366d1 +−1/45 +26 +676c1 +−1/6 +Table 2. Data for (3.4) +It might be possible to prove some formulas in this list by relating ˜n(α) to known results +in Table 1. In particular, the conjectural formulas for the curves of conductor 20, 27, and 54 +are equivalent to the following identities: +˜n( +3√ +2) +?= −5 +8n( +3√ +32), +˜n( +3√ +24) +?= −n(−6), +˜n( +3√ +3) +?= −3 +2n(−3). +As a side note, the authors of [18] (incorrectly) proved +(3.5) +n( +3√ +2) = 5 +6L′(E 3√ +2, 0) + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +15 +(see the corollary under [18, Thm. 5]). In their arguments, they made use of the following +functional identity for Mahler measures [13, Thm. 2.4]: for sufficiently small |p| ̸= 0, +(3.6) +3g +�1 +p +� += n +�1 + 4p +3√p +� ++ 4n +� +1 − 2p +3� +p2 +� +, +where g(α) = m((x+1)(y+1)(x+y)−αxy). When any of the arguments of n in (3.6) enters +the region inside the hypocycloid in Figure 1 (e.g. p = −1/2 in this case), this functional +identity could be invalid due to discontinuity. Therefore, it is logically forbidden to deduce +(3.5) from (3.6). That said, the flawed identity (3.5) became a part of our motivation to +initiate this project. +4. Final remarks +The family Qα is among the several nonreciprocal families of two-variable polynomials +studied by Boyd. Our results give an evidence of how Mahler measure behaves when the +zero locus of a bivariate polynomial intersects the 2-torus nontrivially. +This could shed +some light on the discrepancies between Mahler measure and (elliptic) regulator, which is +conjecturally related to L-values under favorable conditions. Another family which possesses +similar properties (i.e. nonreciprocality and temperedness) to Qα is +Sα = y2 + (x2 + αx + 1)y + x3, +which is labeled (2-33) in [4]. Let K be as defined in Section 1. Then for the family Sα we +have K ∩R = [−4, 2]. For α in this range, the Mahler measure of Sα again splits naturally at +the points of intersection between the curve Sα = 0 and the 2-torus. If k = 0, these points +are ±i, and Boyd verified numerically that +(4.1) +1 +π +� π/2 +0 +log |y−(eiθ)|dθ − 1 +π +� π +π/2 +log |y−(eiθ)|dθ +?= −L′(E, 0), +where y−(x) = − (x2+1) +2 +� +1 − +� +1 − +4x3 +(x2+1)2 +� +and E is the conductor 11 elliptic curve defined +by S0 = 0. He also remarked +“This is in accord with our contention that in case P vanishes on the torus, it is the +integral of ω around a branch cut rather than m(P), which should be rationally related to +L′(E, 0).”. +One might try to prove this identity using the investigation carried out in Section 3 and a +result of Brunault [5] concerning Mahler measure of a conductor 11 elliptic curve. We also +discovered conjectural identities analogous to (4.1) for elliptic curves of conductor 17 and +53, which are corresponding to k = 1 and k = −1, respectively. As opposed to the family +Qα, we are unable to find a general formula, both analytically and arithmetically, for Mahler +measure (or its modification) of Sα, so the situation seems less apparent for this family. +We would also like to point out another related result in the literature which we find +incomplete. +In [10, Thm 3.1], Guillera and Rogers assert that for any |q| < 1 if α = +3 +� +1 + 27η12(3τ) +η12(τ) +� 1 +3 , then +(4.2) +n(α) = 9 +2π +∞ +� +n=−∞ +D +� +e2πi/3qn� +, + +16 +DETCHAT SAMART +where q = e2πiτ, η(τ) is the Dedekind eta function, and D(z) is the Bloch-Wigner dilogarithm. +The summation in the formula above can be seen as a value of the elliptic dilogarithm. +Consider the curve E2, which appears in Theorem 2 and is isomorphic to C/Z + Zτ, where +τ = 1/2 + 0.50586 . . . i. Then we have q = e2πiτ = −0.04165 . . .. However, the identity (4.2) +seems invalid in this case (and all other cases for −1 < α < 3). The right-hand side is +numerically equal to 3 +2L′(E2, 0), which is a conjecture of Bloch and Grayson [3], while n(2) +is not a rational multiple of L′(E2, 0). A correct formula for α ∈ (−1, 3) should be +˜n(α) = − 9 +π +∞ +� +n=−∞ +D +� +e2πi/3qn� +, +which can be proven using Lemma 9 and [7, Prop. 19]. +Finally, we propose some problems for the interested readers. +(i) The function ˜n(α) looks somewhat unnatural at first glance. Is it possible to write +it as the (full) Mahler measure of some polynomial? +(ii) Do there exist algebraic integers β for which +3√β ∈ (−1, 3) and ˜n( 3√β) is a linear +combination of L′(E, 0) (i.e. identities analogous to (1.2))? As suggested by a result +of Guillera and Rogers above, one might start by evaluating the function u(τ) = +3 +� +1 + 27η12(3τ) +η12(τ) +� 1 +3 at some suitable CM points and numerically compare ˜n(u(τ)) with +related elliptic L-values using the PSLQ algorithm. +Funding +This work was supported by the National Research Council of Thailand (NRCT) under +the Research Grant for Mid-Career Scholar [N41A640153 to D.S.]. +Acknowledgements +The author is indebted to Wadim Zudilin for helpful discussions and his suggestion about +integral and hypergeometric identities in the proofs of Lemma 6 and Lemma 7. The author +would also like to thank Fran¸cois Brunault for his guidance on an approach to proving +Lemma 9 and his explanation about Deninger’s results. +References +1. George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics +and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958 +2. Bruce C. 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Math. Soc. 10 (1997), no. 2, 259–281. MR 1415320 + +MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES +17 +9. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.23 of 2019-06-15, +F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. +Miller and B. V. Saunders, eds. +10. Jes´us Guillera and Mathew Rogers, Mahler measure and the WZ algorithm, Proc. Amer. Math. Soc. 143 +(2015), no. 7, 2873–2886. MR 3336612 +11. Leon M. Hall, Missouri S&T Math 483, Lecture Notes: Special Functions. +12. Matilde Lal´ın, Detchat Samart, and Wadim Zudilin, Further explorations of Boyd’s conjectures and a +conductor 21 elliptic curve, J. Lond. Math. Soc. (2) 93 (2016), no. 2, 341–360. MR 3483117 +13. Matilde N. Lalin and Mathew D. Rogers, Functional equations for Mahler measures of genus-one curves, +Algebra Number Theory 1 (2007), no. 1, 87–117. MR 2336636 +14. 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Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Appli- +cations, vol. 77, Cambridge University Press, Cambridge, 2000, With an appendix by Umberto Zannier. +MR 1770638 +21. C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc. 23 (1981), no. 1, +49–63. MR 615132 +22. William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, +American Mathematical Society, Providence, RI, 2007, With an appendix by Paul E. Gunnells. +MR 2289048 +23. F. Rodriguez Villegas, Modular Mahler measures. I, Topics in number theory (University Park, PA, +1997), Math. Appl., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 17–48. MR 1691309 +24. Yifan Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. +36 (2004), no. 5, 671–682. MR 2070444 +25. Wadim Zudilin, Regulator of modular units and Mahler measures, Math. Proc. Cambridge Philos. Soc. +156 (2014), no. 2, 313–326. MR 3177872 +Department of Mathematics, Faculty of Science, Burapha University, Chonburi, Thai- +land 20131 +Email address: petesamart@gmail.com + diff --git a/WNE5T4oBgHgl3EQfBw7L/content/tmp_files/load_file.txt b/WNE5T4oBgHgl3EQfBw7L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ce70b8aed209becfd452a4223ae397e46b2ffb5 --- /dev/null +++ b/WNE5T4oBgHgl3EQfBw7L/content/tmp_files/load_file.txt @@ -0,0 +1,568 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf,len=567 +page_content='MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES DETCHAT SAMART Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In this article, we study the logarithmic Mahler measure of the one-parameter family Qα = y2 + (x2 − αx)y + x, denoted by m(Qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The zero loci of Qα generically define elliptic curves Eα which are 3- isogenous to the family of Hessian elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We are particularly interested in the case α ∈ (−1, 3), which has not been considered in the literature due to certain subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α in this interval, we establish a hypergeometric formula for the (modified) Mahler measure of Qα, denoted by ˜n(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This formula coincides, up to a constant factor, with the known formula for m(Qα) with |α| sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In addition, we verify numerically that if α3 is an integer, then ˜n(α) is a rational multiple of L′(Eα, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' A proof of this identity for α = 2, which is corresponding to an elliptic curve of conductor 19, is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Introduction For any Laurent polynomial P ∈ C[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' , x±1 n ]\\{0}, the (logarithmic) Mahler measure of P, denoted by m(P), is the average of log |P| over the n-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In other words, m(P) = 1 (2πi)n � · · � |x1|=···=|xn|=1 log |P(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' , xn)|dx1 x1 · · dxn xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Consider the following two families of bivariate polynomials Pα(x, y) = x3 + y3 + 1 − αxy, Qα(x, y) = y2 + (x2 − αx)y + x, with the parameter α ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α ̸= 3, the zero loci of Pα define a family of elliptic curves known as the Hessian curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' There is a 3-isogeny between Pα(x, y) = 0 and the curve Eα : Qα(x, y) = 0, which is isomorphic to the curve in the Deuring form, defined by the zero locus of Rα(x, y) = y2 + αxy + y − x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Observe that (x2y)3Pα � y x2, 1 xy � = Qα(x3, y3), from which we have m(Pα) = m(Qα) (see [20, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Similarly, the change of variables (x, y) �→ (−y, xy) transforms the family Rα into Qα without changing the Mahler measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For some technical reasons, we will focus on m(Qα) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Following notation in previous papers [13, 17, 18], we let n(α) := m(Qα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='05390v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='NT] 13 Jan 2023 2 DETCHAT SAMART The Mahler measure of Qα (and its allies) was first studied by Boyd in his seminal paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' He verified numerically that for several α ∈ Z with α /∈ (−1, 3), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) n(α) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= rαL′(Eα, 0), where rα ∈ Q and A ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= B means A and B are equal to at least 50 decimal places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Later, Rodriguez Villegas [23] made an observation that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) seems to hold for all sufficiently large |α| which is a cube root of an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The values of α for which (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) has been proven rigorously are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' α Conductor of Eα rα Reference(s) −6 27 3 [23] −3 54 1 [7] −2 35 1 [7] −1 14 2 [16],[7] 3√ 32 20 8 3 [18] 3√ 54 36 3 2 [17] 5 14 7 [16] Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proven formulas for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) In addition to the results in this list, there are some known identities which relate n(α), where α is a cube root of an algebraic integer, to a linear combination of L-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For example, the author proved in [19] that the following identity is true: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2) n � 3� 6 − 6 3√ 2 + 18 3√ 4 � = 1 2 (L′(F108, 0) + L′(F36, 0) − 3L′(F27, 0)) , where FN is an elliptic curve over Q of conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In compliance with Boyd’s results, it is worth noting that 3� 6 − 6 3√ 2 + 18 3√ 4 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='0005 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We refer the interested reader to the aforementioned paper for more conjectural identities of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Recall that a polynomial P(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' , xn) is said to be reciprocal if there exist integers d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' , dn such that xd1 1 xd2 2 · · · xdn n P(1/x1, 1/x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' , 1/xn) = P(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' , xn), and nonreciprocal otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For a family of two-variable polynomials ˜Pα(x, y) = A(x)y2 + (B(x) + αx)y + C(x), let Zα be the zero locus of ˜Pα(x, y) and let K be the set of α ∈ C for which ˜Pα vanishes on the 2-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Boyd conjectured from his experiments that, for all integer α in the unbounded component G∞ of C\\K, if ˜Pα is tempered (see [23] for the definition), then m( ˜Pα) is related to an L-value of elliptic curve (if Zα has genus one) or Dirichlet character (if Zα has genus zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If ˜Pα(x, y) is reciprocal, then it can be shown that K ⊆ R, implying G∞ = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Hence by continuity one could expect that identities like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) hold for all α ∈ Z, with some exceptions in the genus zero cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Examples of polynomials satisfying these properties include the MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 3 families x + 1/x + y + 1/y + α and (1 + x)(1 + y)(x + y) − αxy, whose Mahler measures have been extensively studied over the past few decades (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' see [4, 12, 13, 14, 15, 17, 18, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The family Qα, on the other hand, is nonreciprocal, so the set K of α ∈ C for which Qα vanishes on the 2-torus has nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In fact, as described in [4, §2B] and [23, §14], K is the region inside a hypocycloid whose vertices are the cube roots of 27 in the complex plane and K ∩ R = (−1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This is illustrated in Figure 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' It is known (see, for Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' example, [17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1]) that, for most complex numbers α, n(α) is expressible in terms of a generalized hypergeometric function: if |α| is sufficiently large, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) n(α) = Re � log α − 2 α3 4F3 � 4 3, 5 3, 1, 1 2, 2, 2 ���� 27 α3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Since both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) are real parts of holomorphic functions that agree at every point in an open subset of the region C\\K, the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) is valid for all α ∈ C\\K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=', for all α on the border and outside of the hypocycloid in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Because of this anomalous property of the family Qα (and other nonreciprocal families in general), to our knowledge, there are no known results about n(α) for α ∈ K, with an exception for the case α = 0 due to Smyth [21], namely n(0) = m(x3 + y3 + 1) = m(x + y + 1) = L′(χ−3, −1), where χ−N = � N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The aim of this paper is to give a thorough investigation of these omitted values of n(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In particular, we are interested in establishing formulas analogous to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) for α ∈ (−1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let us first factorize Qα as Qα(x, y) = y2 + (x2 − αx)y + x = (y − y+(x))(y − y−(x)), where y±(x) = −(x2 − αx) � 1 2 ± � 1 4 − 1 x(x − α)2 � , and denote J(α) = 1 π � π cos−1( α−1 2 ) log |y+(eiθ)|dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 24 DETCHAT SAMART (Here and throughout we use the principal branch for the complex square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=') The signif- icance of the function J(α), which can be seen as a part of m(Qα), will be made clear later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α ∈ (−1, 1) ∪ (1, 3), y±(x) are functions on T1 := {x ∈ C | |x| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If α = 1, y±(x) have only one removable singularity on T1, namely x = 1, so we can extend its domain to T1 by setting y±(1) = lim x→1 y±(x) = ∓i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The first main result of this paper is the following hypergeometric formula, which extends (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let ˜n(α) = n(α) − 3J(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The following identity is true: ˜n(α) = � � � � � � � Re � log α − 2 α3 4F3 � 4 3 , 5 3 , 1, 1 2, 2, 2 ���� 27 α3 �� if α ∈ (−1, 0), −2 Re � log α − 2 α3 4F3 � 4 3 , 5 3 , 1, 1 2, 2, 2 ���� 27 α3 �� if α ∈ (0, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We also study ˜n(α) from the arithmetic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We discovered from our numerical computation that for α ∈ (−1, 3) which is a cube root of an integer ˜n(α) (conjecturally) satisfies an identity analogous to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Numerical data for this identity are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This identity can be proven rigorously in some cases using Brunault-Mellit-Zudilin’s formula (see Theorem 8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' As a concrete example, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let ˜n(α) = n(α) − 3J(α) and let Eα be the elliptic curve defined by the zero locus of Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then the following evaluation is true: ˜n(2) = −3L′(E2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4) Note that E2 has conductor 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' What makes this curve special is that it admits a modular unit parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The celebrated modularity theorem asserts that every elliptic curve over Q can be parametrized by modular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' However, a recent result of Brunault [6] reveals that there are only a finite number of them which can be parametrized by modular units (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' modular functions whose zeros and poles are supported at the cusps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In order to apply Brunault-Mellit-Zudilin’s formula, one needs to show that the integration path corresponding to ˜n(α) becomes a closed path for the regulator integral defined on the curve Qα(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This path can then be translated into a path joining cusps on the modular curve X0(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The calculation for this part will be worked out in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The hypergeometric formula The goal of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' To achieve this goal, we need some auxiliary results as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let α ∈ C and x ∈ C\\{α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If |x| = 1, then |y−(x)| ≤ 1 ≤ |y+(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Assume that |x| = 1 and write � 1 4 − 1 x(x−α)2 = a + bi, where a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Since the square root is defined using the principal branch, we have a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Hence |y−(x)| = |x2 − αx| ���� 1 2 − a − bi ���� ≤ |x2 − αx| ���� 1 2 + a + bi ���� = |y+(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Since |y+(x)||y−(x)| = |x| = 1, it follows that |y−(x)| ≤ 1 ≤ |y+(x)|, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 5 By Lemma 3 and Jensen’s formula, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) n(α) = 1 2π � π −π log |y+(eiθ)|dθ = 1 π � π 0 log |y+(eiθ)|dθ = 1 π Re � π 0 log � (x − α) � 1 2 + � 1 4 − 1 x(x − α)2 �� ���� x=eiθ dθ, where the second equality follows from y+(e−iθ) = y+(eiθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Next, we shall locate the toric points, the points of intersection of the affine curve Qα = 0 and the 2-torus, explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let T2 = {(x, y) ∈ C2 | |x| = |y| = 1} and for each α ∈ C let Cα = {(x, y) ∈ C2 | Qα(x, y) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then for α ∈ (−1, 3), we have Cα ∩ T2 = �� eit, y±(eit) � | t = 0, ± cos−1 �α − 1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Assume first that α ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Suppose |x| = 1, so x = eit for some t ∈ (−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Since y±(x) = −(x2 − αx) � 1 2 ± � 1 4 − 1 x(x−α)2 � and |y+(x)||y−(x)| = |x| = 1, we have that the condition |y+(x)| = 1 = |y−(x)| is equivalent to the equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2) ����� 1 2 + � 1 4 − 1 x(x − α)2 ����� = ����� 1 2 − � 1 4 − 1 x(x − α)2 ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' It is easily seen that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2) holds if and only if � 1 4 − 1 x(x−α)2 is purely imaginary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' equivalently, x(x − α)2 ∈ (0, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Simple calculation yields Re(x(x − α)2) = (cos t)((cos t − α)2 − sin2 t) − 2(cos t − α) sin2 t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) Im(x(x − α)2) = (sin t)(2 cos t − (α − 1))(2 cos t − (α + 1)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4) |x(x − α)2| = |x − α|2 = α2 − 2α cos t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='5) We have from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4) that x(x − α)2 ∈ R if and only if sin t = 0 or cos t = (α ± 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If sin t = 0, then either cos t = 1 or cos t = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If cos t = −1, then x(x−α)2 = −(1−α)2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If cos t = (α+1)/2, then α ∈ (−1, 1) and sin2 t = 1−((α + 1)/2)2 , from which we can deduce using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) that x(x − α)2 = Re(x(x − α)2) = α − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Also, it can be shown using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='5) that the remaining cases, cos t = 1 and cos t = α−1 2 , imply 0 < x(x − α)2 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' As a consequence, the curve Cα = 0 intersects T2 exactly at (eit, y±(eit)), where t = 0, ± cos−1 � α−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The same result also holds for α = 1 by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For λ ∈ [1, 2), let pλ(x) = x(λ2 − x) � x2 + � 4 λ − λ2� x + 4 λ2 � and γ = λ3−λ−2 2λ + λ+1 2λ � (2 − λ)(λ3 + λ − 2)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6) � γ λ−1 1 � −pλ(x) dx = � −1/λ 0 1 � −pλ(x) dx, 6 DETCHAT SAMART where the left (complex) integral is path-independent in the upper-half unit disk and the right integral is a real integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Note first that |γ| = 1 and the nonzero roots of pλ(x) are x1(λ) = λ2, x2(λ) = λ3 − 4 + � λ3(λ3 − 8) 2λ , and x3(λ) = λ3 − 4 − � λ3(λ3 − 8) 2λ , which lie outside the unit circle, so the integration path for the left integral can be chosen to be any path joining λ − 1 and γ in the upper-half unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For 1 < λ < 2 and x ∈ R, x2 + �4 λ − λ2 � x + 4 λ2 = � x + �2 λ − λ2 2 ��2 − λ �λ3 4 − 2 � > 0, so −pλ(x) > 0 for all x ∈ (−1/λ, 0) and the integral on the right-hand side is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Define the symmetric polynomial1 Fλ(x, y) by Fλ(x, y) := λ2(λ − 1)x2y2 − λ(λ − 1)(λ3 − λ2 + λ − 2)(x2y + xy2) + λ2(x2 + y2) + (λ7 − 2λ6 + 2λ5 − 5λ4 + 6λ3 − 6λ2 + 6λ − 4)xy − 2λ2(λ − 1)(x + y) + λ2(λ − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then, for λ ∈ [1, 2), Fλ(x, y) transforms the interval (−1/λ, 0) to a continuous path in the upper-half unit disk joining γ and λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Moreover, by implicitly differentiating Fλ(x, y) = 0, it can be checked using a computer algebra system that the following equation holds on this curve: �dy dx �2 − pλ(y) pλ(x) = 0, from which (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α ∈ (−1, 3), if α = (λ3 − 2)/λ, then d dα (n(α) − 3J(α)) = − 1 π � λ2 0 1 � pλ(x) dx, where pλ(x) is defined as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Differentiating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) with respect to α yields d dαn(α) = 1 π Re � π 0 √x � x(x − α)2 − 4 ���� x=eiθ dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let c(α) = cos−1 � α−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then, by Leibniz integral rule and Proposition 4, we have d dαJ(α) = 1 π � − log ��y+ � eic(α)��� d dαc(α) + Re � π c(α) d dα log � (x − α) � 1 2 + � 1 4 − 1 x(x − α)2 �� ���� x=eiθ dθ � = 1 π Re � c(α) 0 √x � x(x − α)2 − 4 ���� x=eiθ dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 1We obtain the polynomial Fλ(x, y) using numerical values of the integrals in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The PSLQ algorithm plays an essential role in identifying its coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 7 It follows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='7) d dα (n(α) − 3J(α)) = − 1 π Re �� 2 � π c(α) − � c(α) 0 � √x � x(x − α)2 − 4 ���� x=eiθ dθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let α = (λ3 − 2)/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then α maps the interval (1, 2) bijectively onto (−1, 3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='8) x(x − α)2 − 4 = (x − λ2) � x2 + �4 λ − λ2 � x + 4 λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' An inspection of the signs of the square roots in the integrand reveals that � π c(α) √x � x(x − α)2 − 4 ���� x=eiθ dθ = − � −1 γ 1 � pλ(x) dx = �� γ 0 − � −1 0 � 1 � pλ(x) dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='9) � c(α) 0 √x � x(x − α)2 − 4 ���� x=eiθ dθ = � γ 1 1 � pλ(x) dx = �� γ 0 − � 1 0 � 1 � pλ(x) dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='10) where γ = eic(α) = α − 1 2 + � (3 − α)(α + 1) 2 i = λ3 − λ − 2 2λ + λ + 1 2λ � (2 − λ)(λ3 + λ − 2)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Since pλ(x) < 0 for any x ∈ (−1, 0) and λ ∈ (1, 2), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='11) Re � −1 0 1 � pλ(x) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='9),(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='10), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='11) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='7) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='12) d dα (n(α) − 3J(α)) = − 1 π �� 1 0 1 � pλ(x) dx + Re � γ 0 1 � pλ(x) dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Note that the mapping (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='13) x �→ λ2 − x λx + 1 is the unique M¨obius transformation which interchanges the following values: 0 ↔ λ2, 1 ↔ λ − 1, x2(λ) ↔ x3(λ), where x2(λ) and x3(λ) are the roots of x2 + (4/λ − λ2)x + 4/λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Hence using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='13) we have � λ−1 0 1 � pλ(x) dx = � λ2 1 1 � pλ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Finally, we have from Lemma 5 that � γ λ−1 1 � pλ(x) dx = � −1/λ 0 1 � pλ(x) dx ∈ iR, so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='12) immediately gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ 8 DETCHAT SAMART Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α ∈ (−1, 0), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='14) d dα (n(α) − 3J(α)) = Re � 1 α 2F1 � 1 3, 2 3 1 ���� 27 α3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α ∈ (0, 3), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='15) d dα (n(α) − 3J(α)) = −2 Re � 1 α 2F1 � 1 3, 2 3 1 ���� 27 α3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let us first consider (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We prove this identity by expressing both sides in terms of the elliptic integral of the first kind K(z) = � 1 0 dx � (1 − x2)(1 − z2x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Again, let α = (λ3 − 2)/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Following a procedure in [11, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 3], we let u = −1 − √ λ3 + 1 λ , v = −1 + √ λ3 + 1 λ , x = ut − v t − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This substitution transforms the integral in Lemma 6 (without the factor −1/π) into λ 2 √ λ3 + 1 � t2 t1 dt � (B1t2 + A1)(B2t2 + A2) , where t1 = −λ3 + 2 − 2 √ λ3 + 1 λ3 , t2 = −t1, A1 = λ3 + 2 − 2 √ λ3 + 1 4 √ λ3 + 1 , B1 = −λ3 − 2 − 2 √ λ3 + 1 4 √ λ3 + 1 , A2 = −λ3 + 2 + 2 √ λ3 + 1 4 √ λ3 + 1 , B2 = λ3 − 2 + 2 √ λ3 + 1 4 √ λ3 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Observe that, for λ ∈ (1, 2), we have A1, A2, B2 > 0, B1 < 0, and � −A1/B1 = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Hence the substitution t �→ � −A1/B1t yields λ 2 √ λ3 + 1 � t2 t1 dt � (B1t2 + A1)(B2t2 + A2) = λ 2 √ λ3 + 1 � − 1 A2B1 � 1 −1 dt � (1 − t2) � 1 − A1B2 A2B1t2 � = 4λ ��√ λ3 + 1 + 1 �3 � 3 − √ λ3 + 1 �K �� A1B2 A2B1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Therefore, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='16) d dα (n(α) − 3J(α)) = − 4λ π ��√ λ3 + 1 + 1 �3 � 3 − √ λ3 + 1 �K �� A1B2 A2B1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' On the other hand, we apply the hypergeometric transformation [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 410] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='17) Re 2F1 � 1 3, 2 3 1 ���� 27y (y − 2)3 � = y − 2 y + 4 2F1 � 1 3, 2 3 1 ���� 27y2 (y + 4)3 � , MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 9 which is valid for y ∈ (2, 8), to write the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='15) as − 2 α Re � 2F1 � 1 3, 2 3 1 ���� 27 α3 �� = 2λ 2 − λ3 Re � 2F1 � 1 3, 2 3 1 ���� 27λ3 (λ3 − 2)3 �� = − 2λ λ3 + 4 2F1 � 1 3, 2 3 1 ���� 27λ6 (λ3 + 4)3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The substitution λ = 3� 4(p + p2) gives a bijection from the interval (( √ 3 − 1)/2, 1) onto ( 3√ 2, 2), which is corresponding to the interval (0, 3) for α, with the inverse mapping p = ( √ λ3 + 1 − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We apply this substitution together with a classical result of Ramanujan [2, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6] to deduce − 2λ λ3 + 4 2F1 � 1 3, 2 3 1 ���� 27λ6 (λ3 + 4)3 � = − 3� 4(p + p2) 2(p2 + p + 1) 2F1 � 1 3, 2 3 1 ���� 27p2(1 + p)2 4(1 + p + p2)3 � = − 3� 4(p + p2) 2√1 + 2p 2F1 � 1 2, 1 2 1 ���� p3(2 + p) 1 + 2p � = − λ 2 4√ λ3 + 1 2F1 � 1 2, 1 2 1 ���� ρ(λ) � , where ρ(λ) = λ6 − 4λ3 − 8 + 8 √ λ3 + 1 16 √ λ3 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then by the identities [1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3], [9, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1] K(k) = π 2 2F1 � 1 2, 1 2 1 ���� k2 � , K(√r) = 1 √1 − rK �� r r − 1 � , we arrive at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='18) − λ 2 4√ λ3 + 1 2F1 � 1 2, 1 2 1 ���� ρ(λ) � = − 4λ π ��√ λ3 + 1 + 1 �3 � 3 − √ λ3 + 1 �K �� ρ(λ) ρ(λ) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' It can be calculated directly that ρ(λ) ρ(λ) − 1 = λ6 − 4λ3 − 8 + 8 √ λ3 + 1 λ6 − 4λ3 − 8 − 8 √ λ3 + 1 = A1B2 A2B1 , so the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='18) coincides with that of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='16) and the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='14 also follows from the arguments above, provided that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='17) is replaced with Re 2F1 � 1 3, 2 3 1 ���� 27y (y − 2)3 � = 4 − 2y y + 4 2F1 � 1 3, 2 3 1 ���� 27y2 (y + 4)3 � , which is valid for y ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α > 3, we can apply term-by-term differentiation to show that d dα Re � log α − 2 α3 4F3 � 4 3, 5 3, 1, 1 2, 2, 2 ���� 27 α3 �� = Re � 1 α 2F1 � 1 3, 2 3 1 ���� 27 α3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 10 DETCHAT SAMART By analytic continuation, the above equality also holds for α ∈ (−1, 0) ∪ (0, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Therefore, integrating both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='15) yields n(α) − 3J(α) = � � � � � � � Re � log α − 2 α3 4F3 � 4 3 , 5 3 , 1, 1 2, 2, 2 ���� 27 α3 �� + C1, if − 1 < α < 0, −2 Re � log α − 2 α3 4F3 � 4 3 , 5 3 , 1, 1 2, 2, 2 ���� 27 α3 �� + C2, if 0 < α < 3, for some constants C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Since α = −1 and α = 3 are on the boundary of the set K defined in Section 1, an argument underneath (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) implies that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='19) n(−1) = Re � log(−1) + 24F3 � 4 3, 5 3, 1, 1 2, 2, 2 ���� − 27 �� , n(3) = Re � log 3 − 2 27 4F3 � 4 3, 5 3, 1, 1 2, 2, 2 ���� 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Hence, by continuity of n(α) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='19), we have C1 = lim α→−1+(−3J(α)) = 0, C2 = 3 lim α→3−(n(3) − J(α)) = 0, and the desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Relation to elliptic regulators and L-values In this section, we prove Theorem 2, which resembles Boyd’s conjectures (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The key idea of the proof is to rewrite ˜n(α) as a regulator integral over a path joining two cusps and apply Brunault-Mellit-Zudilin formula [25], which is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' As usual, we define the real differential form η(f, g) for meromorphic functions f and g on a smooth curve C as η(f, g) = log |f|d arg(g) − log |g|d arg(f), where d arg(g) = Im(dg/g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Theorem 8 (Brunault-Mellit-Zudilin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let N be a positive integer and define ga(τ) = qNB2(a/N)/2 � n≥1 n≡a mod N (1 − qn) � n≥1 n≡−a mod N (1 − qn), q := e2πiτ, where B2(x) = {x}2 − {x} + 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then for any a, b, c ∈ Z such that N ∤ ac and N ∤ bc, � i∞ c/N η(ga, gb) = 1 4πL(f(τ) − f(i∞), 2), where f(τ) = fa,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='c(τ) is a weight 2 modular form given by fa,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='c = ea,bceb,−ac − ea,−bceb,ac and ea,b(τ) = 1 2 �1 + ζa N 1 − ζa N + 1 + ζb N 1 − ζb N � + � m,n≥1 � ζam+bn N − ζ−(am+bn) N � qmn, ζN := e 2πi N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 11 Let us first outline a general framework for computing ˜n(α) in terms of a regulator integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Recall from Deninger’s result [8, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3] that if Qα(x, y) is irreducible, then n(α) = − 1 2π � γα η(x, y), where γα is the Deninger path on the curve Eα : Qα(x, y) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=', γα = {(x, y) ∈ C2 | |x| = 1, |y| > 1, Qα(x, y) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If Qα does not vanish on the torus, then γα becomes a closed path, so the Bloch-Beilinson conjectures give a prediction that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) holds for all sufficiently large |α| with suitable arith- metic properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' in this case, we need that α be a cube root of an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' On the other hand, if α ∈ (−1, 3), then the functions y±(x) defined in Section 1 are discontinuous at the toric points as given in Proposition 4, so γα is not closed in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We will show, however, that the path on Eα corresponding to ˜n(α) is indeed closed, so that ˜n(α) is (conjecturally) related to L-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The numerical data supporting this hypothesis are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let α ∈ (−1, 3) and let ˜n(α) = n(α) − 3J(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then ˜n(α) = − 1 2π � ˜γα η(x, y) for some ˜γα ∈ H1(Eα, Z)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In other words, the integration path associated to the modified Mahler measure ˜n(α) can be realized as a closed path which is anti-invariant under complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We label the six toric points obtained from Proposition 4 as follows: P ± 1 = (1, y±(1)) = (1, Y±) , P ± 2 = (e±ic(α), y+(e±ic(α))) = (Y±, 1) , P ± 3 = (e±ic(α), y−(e±ic(α))) = (Y±, Y±) , where c(α) = cos−1 � α−1 2 � and Y± = α − 1 2 ± � (3 − α)(α + 1) 2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Observe that ˜n(α) can be rewritten as ˜n(α) = I(α) − 2J(α), where I(α) = 1 2π � c(α) −c(α) log |y+(eiθ)|dθ, J(α) = 1 2π � 2π−c(α) c(α) log |y+(eiθ)|dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let S = {P ± 1 , P ± 2 , P ± 3 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then we may identify the paths corresponding to I(α) and J(α) as elements in the relative homology H1(Eα, S, Z), say γI and γJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In other words, we write I(α) = − 1 2π � γI η(x, y), J(α) = − 1 2π � γJ η(x, y), 12 DETCHAT SAMART and boundaries of these paths can be seen as 0-cycles on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Computing the limits of y+(eiθ) as θ approach 0, c(α), and −c(α) from both sides, we find that lim θ→−c(α)+ y+(eiθ) = lim θ→c(α)− y+(eiθ) = 1, lim θ→0+ y+(eiθ) = Y−, lim θ→0− y+(eiθ) = Y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Therefore, the path γI is discontinuous at θ = 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) ∂γI = [[P + 1 ] − [P − 2 ]] + [[P + 2 ] − [P − 1 ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This is illustrated in Figure 2 for α = 2, where the dashed curves in the upper-half plane and the lower-half plane, both oriented counterclockwise, correspond to θ ∈ (−c(α), 0) and θ ∈ (0, c(α)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Next, observe that Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' y+(eiθ), θ ∈ [0, 2π) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' y−(eiθ), θ ∈ [0, 2π) lim θ→c(α)+ y−(eiθ) = 1 = lim θ→−c(α)− y−(eiθ), and γJ can be identified as the path {(eiθ, y−(eiθ)) | c(α) < θ < 2π − c(α)} (with reversed orientation), implying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2) ∂γJ = [[P + 2 ] − [P − 2 ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' (For α = 2, the y-coordinate of this path is the bold curve inside the unit circle, as illustrated in Figure 3, oriented clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=') Let x±(y) = (αy − 1) 2y � 1 ± � 1 − 4y3 (αy − 1)2 � , which is obtained by solving Qα(x, y) = 0 in the variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then y+(x−(eiθ)) = eiθ for any θ ∈ R and lim θ→0± x−(eiθ) = e∓ic(α), lim θ→c(α)− x−(eiθ) = 1 = lim θ→−c(α)+ x−(eiθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 2- 1 Y+ 1 0 Y 1 21 0 1MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 13 Define γ1 = {(x−(eiθ), y+(x−(eiθ))) | −c(α) < θ < 0}, γ2 = {(x−(eiθ), y+(x−(eiθ))) | 0 < θ < c(α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then one can deduce using the limits of x− at the endpoints above that the closure of γ1 ∪ γJ ∪ γ2, which we call γ′ J, is a continuous path joining P − 1 and P + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Moreover, since the y-coordinates of γ1 and γ2 lie on the unit circle, we have J(α) = − 1 2π � γ′ J η(x, y), where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3) ∂γ′ J = [[P + 1 ] − [P − 1 ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Finally, we arrive at ˜n(α) = I(α) − 2J(α) = − 1 2π �� γI η(x, y) − � γJ η(x, y) − � γ′ J η(x, y) � = − 1 2π � ˜γα η(x, y), where, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='3), ˜γα has trivial boundary, from which we can conclude that ˜γα ∈ H1(Eα, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' It is clear from the construction of the paths γI, γJ, and γ′ J that they are anti-invariant under the action of complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Therefore, we have ˜γα ∈ H1(Eα, Z)−, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ We shall use Theorem 8 and Lemma 9 to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We essentially follow an approach of Brunault [7] in identifying the path ˜γα as the push-forward of a path joining cusps on X0(19) with the aid of Magma and Pari/GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The elliptic curve E2 : y2 +(x2 −2x)y +x = 0 has Cremona label 19a3, so it admits a modular parametrization ϕ : X0(19) → E2 by the modularity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let f2 be the weight 2 newform of level 19 associated to the curve E2 and let ω = 2πif2(τ)dτ, the pull-back of the holomorphic differential form on E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Using Magma and Pari/GP codes in [7, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1], we find that � −4/19 4/19 ω = −Ω− ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='12709i, where Ω− is the imaginary period of E2 obtained by subtracting twice the complex period from the real period of E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Hence it follows that ˜γα = ϕ∗ � 4 19, − 4 19 � , where ˜γα is the path associated to ˜n(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let x(τ) = −g1g7g8 g2g3g5 , y(τ) = g1g7g8 g4g6g9 , where ga := ga(τ) is as given in Theorem 8 with N = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' By a result of Yang [24, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 3], both x(τ) and y(τ) are modular functions on Γ1(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Multiplying each term by a modular form in M2(Γ1(19)), one can apply Sturm’s theorem [22, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='19], with the Sturm bound B(M2(Γ1(19))) = 60, to show that y(τ)2 + (x(τ)2 − 2x(τ))y(τ) + x(τ) vanishes identically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 14 DETCHAT SAMART i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=', (x(τ), y(τ)) parametrizes the curve E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Finally, by Lemma 9 and Theorem 8, we find that ˜n(α) = − 1 2π � ˜γα η(x, y) = 1 2π � 4/19 −4/19 η(x(τ), y(τ)) = − 1 4π2L(57f2, 2) = −3L′(f2, 0), where the last equality follows from the functional equation for L(f2, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' □ In addition to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4), we discovered that, for all α ∈ (−1, 3) which are a cube root of an integer, the following identity holds numerically: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4) ˜n(α) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= rαL′(Eα, 0), where rα ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The data of rα and Eα are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' α3 Cremona label of Eα rα α3 Cremona label of Eα rα 1 26a3 −1 14 2548d1 1/36 2 20a1 −5/3 15 1350i1 1/18 3 54a1 −2/3 16 44a1 −4/3 4 92a1 −1/3 17 2890e1 −1/27 5 550d1 −1/9 18 324b1 −1/6 6 756f1 −1/18 19 722a1 1/9 7 490a1 1/9 20 700i1 −1/9 8 19a3 −3 21 2464k1 −1/27 9 162c1 −1/3 22 2420d1 1/26 10 1700c1 1/36 23 1058b1 −1/12 11 242b1 −1/3 24 27a1 −3 12 540d1 1/9 25 50a1 −5/3 13 2366d1 −1/45 26 676c1 −1/6 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Data for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4) It might be possible to prove some formulas in this list by relating ˜n(α) to known results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In particular, the conjectural formulas for the curves of conductor 20, 27, and 54 are equivalent to the following identities: ˜n( 3√ 2) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= −5 8n( 3√ 32), ˜n( 3√ 24) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= −n(−6), ˜n( 3√ 3) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= −3 2n(−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' As a side note, the authors of [18] (incorrectly) proved (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='5) n( 3√ 2) = 5 6L′(E 3√ 2, 0) MAHLER MEASURE OF A NONRECIPROCAL FAMILY OF ELLIPTIC CURVES 15 (see the corollary under [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In their arguments, they made use of the following functional identity for Mahler measures [13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='4]: for sufficiently small |p| ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6) 3g �1 p � = n �1 + 4p 3√p � + 4n � 1 − 2p 3� p2 � , where g(α) = m((x+1)(y+1)(x+y)−αxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' When any of the arguments of n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6) enters the region inside the hypocycloid in Figure 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' p = −1/2 in this case), this functional identity could be invalid due to discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Therefore, it is logically forbidden to deduce (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='5) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' That said, the flawed identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='5) became a part of our motivation to initiate this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Final remarks The family Qα is among the several nonreciprocal families of two-variable polynomials studied by Boyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Our results give an evidence of how Mahler measure behaves when the zero locus of a bivariate polynomial intersects the 2-torus nontrivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' This could shed some light on the discrepancies between Mahler measure and (elliptic) regulator, which is conjecturally related to L-values under favorable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Another family which possesses similar properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' nonreciprocality and temperedness) to Qα is Sα = y2 + (x2 + αx + 1)y + x3, which is labeled (2-33) in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Let K be as defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then for the family Sα we have K ∩R = [−4, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' For α in this range, the Mahler measure of Sα again splits naturally at the points of intersection between the curve Sα = 0 and the 2-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' If k = 0, these points are ±i, and Boyd verified numerically that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) 1 π � π/2 0 log |y−(eiθ)|dθ − 1 π � π π/2 log |y−(eiθ)|dθ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='= −L′(E, 0), where y−(x) = − (x2+1) 2 � 1 − � 1 − 4x3 (x2+1)2 � and E is the conductor 11 elliptic curve defined by S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' He also remarked “This is in accord with our contention that in case P vanishes on the torus, it is the integral of ω around a branch cut rather than m(P), which should be rationally related to L′(E, 0).”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' One might try to prove this identity using the investigation carried out in Section 3 and a result of Brunault [5] concerning Mahler measure of a conductor 11 elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We also discovered conjectural identities analogous to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1) for elliptic curves of conductor 17 and 53, which are corresponding to k = 1 and k = −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' As opposed to the family Qα, we are unable to find a general formula, both analytically and arithmetically, for Mahler measure (or its modification) of Sα, so the situation seems less apparent for this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' We would also like to point out another related result in the literature which we find incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' In [10, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='1], Guillera and Rogers assert that for any |q| < 1 if α = 3 � 1 + 27η12(3τ) η12(τ) � 1 3 , then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2) n(α) = 9 2π ∞ � n=−∞ D � e2πi/3qn� , 16 DETCHAT SAMART where q = e2πiτ, η(τ) is the Dedekind eta function, and D(z) is the Bloch-Wigner dilogarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The summation in the formula above can be seen as a value of the elliptic dilogarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Consider the curve E2, which appears in Theorem 2 and is isomorphic to C/Z + Zτ, where τ = 1/2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='50586 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Then we have q = e2πiτ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='04165 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='. However, the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2) seems invalid in this case (and all other cases for −1 < α < 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The right-hand side is numerically equal to 3 2L′(E2, 0), which is a conjecture of Bloch and Grayson [3], while n(2) is not a rational multiple of L′(E2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' A correct formula for α ∈ (−1, 3) should be ˜n(α) = − 9 π ∞ � n=−∞ D � e2πi/3qn� , which can be proven using Lemma 9 and [7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Finally, we propose some problems for the interested readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' (i) The function ˜n(α) looks somewhat unnatural at first glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Is it possible to write it as the (full) Mahler measure of some polynomial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' (ii) Do there exist algebraic integers β for which 3√β ∈ (−1, 3) and ˜n( 3√β) is a linear combination of L′(E, 0) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' identities analogous to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='2))?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' As suggested by a result of Guillera and Rogers above, one might start by evaluating the function u(τ) = 3 � 1 + 27η12(3τ) η12(τ) � 1 3 at some suitable CM points and numerically compare ˜n(u(τ)) with related elliptic L-values using the PSLQ algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Funding This work was supported by the National Research Council of Thailand (NRCT) under the Research Grant for Mid-Career Scholar [N41A640153 to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' Acknowledgements The author is indebted to Wadim Zudilin for helpful discussions and his suggestion about integral and hypergeometric identities in the proofs of Lemma 6 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' The author would also like to thank Fran¸cois Brunault for his guidance on an approach to proving Lemma 9 and his explanation about Deninger’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfBw7L/content/2301.05390v1.pdf'} +page_content=' References 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a/WNE5T4oBgHgl3EQfcQ-J/content/tmp_files/2301.05602v1.pdf.txt b/WNE5T4oBgHgl3EQfcQ-J/content/tmp_files/2301.05602v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..736995622e1505e1323065ed421eb2ab7114397f --- /dev/null +++ b/WNE5T4oBgHgl3EQfcQ-J/content/tmp_files/2301.05602v1.pdf.txt @@ -0,0 +1,1588 @@ +Hybrid Parametric Classes of Isotropic Covariance +Functions for Spatial Random Fields +Alfredo Alegr´ıa∗1, Fabi´an Ram´ırez1, and Emilio Porcu2 +1Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Chile +2Department of Mathematics, Khalifa University, The Arab Emirates +January 16, 2023 +Abstract +Covariance functions are the core of spatial statistics, stochastic processes, machine learning as +well as many other theoretical and applied disciplines. The properties of the covariance func- +tion at small and large distances determine the geometric attributes of the associated Gaussian +random field. Having covariance functions that allow to specify both local and global proper- +ties is certainly on demand. This paper provides a method to find new classes of covariance +functions having such properties. We term these models hybrid as they are obtained as scale +mixtures of piecewise covariance kernels against measures that are also defined as piecewise lin- +ear combination of parametric families of measures. In order to illustrate our methodology, we +provide new families of covariance functions that are proved to be richer with respect to other +well known families that have been proposed by earlier literature. More precisely, we derive a +hybrid Cauchy-Mat´ern model, which allows us to index both long memory and mean square +differentiability of the random field, and a hybrid Hole-Effect-Mat´ern model, which is capable +of attaining negative values (hole effect), while preserving the local attributes of the traditional +Mat´ern model. Our findings are illustrated through numerical studies with both simulated and +real data. +Keywords: Cauchy model; Gaussian scale mixtures; Hole effect; Long memory; Mat´ern model; +Mean square differentiability. +1 +Introduction +Covariance functions are central to many disciplines such as spatial statistics (Cressie, 1993; Chil`es +and Delfiner, 2012; Hristopulos, 2020), stochastic processes (Porcu et al., 2018a,b), machine learning +(Schaback and Wendland, 2006; James et al., 2013; Barp et al., 2022), numerical analysis (Pazouki +and Schaback, 2011; Cockayne et al., 2019) and stochastic mechanics (Ostoja-Starzewski, 2006, +with the references therein). Recent applications in climatology (Guinness and Hammerling, 2018; +Edwards et al., 2019), oceanography (Furrer et al., 2007; Di Lorenzo et al., 2014), environmental +sciences (Cressie and Kornak, 2003; Stein, 2007) and natural resources engineering (Chen et al., +2018; Emery and S´eguret, 2020) witness on the importance of covariance functions. +It is very customary to assume the covariance function to depend on the distance between any +pair of random variables located at two different points at the input space. Such an assumption is +∗Corresponding author. Email: alfredo.alegria@usm.cl +1 +arXiv:2301.05602v1 [math.ST] 13 Jan 2023 + +termed isotropy in spatial statistics and machine learning, and it is termed radial symmetry in other +areas of applied mathematics. The behaviour of the covariance function at short or long distances +(we call this local and global properties, respectively) is crucial to understand the properties of +random processes with a given covariance function. Specifically, the local properties are related to +the fractal dimension as well as the geometric properties (e.g., mean square differentiability) of the +associated random process, as well as to its sample paths. On the other hand, the global behaviour +of the covariance function allows to characterize persistency or antipersistency (i.e., the long term +behaviour) of the associated process. Another global behaviour of great interest is the so-called hole +effect, which means that the covariance function could take negative values in a certain interval. +Finding parametric families of isotropic covariance functions that allow to index both local and +global behaviour is a major challenge that has been tackled to a very limited extent. The Mat´ern +family has been the cornerstone in spatial statistics for over half a century now (Stein, 1999). Its +popularity is due to a parameter that controls the degree of mean square differentiability and fractal +dimension of the corresponding random field (Stein, 1999). Recently, Bevilacqua et al. (2022) have +shown that the Mat´ern class is a special case of a richer class of models that, additionally to indexing +local properties, allow to switch between compact or global supports. In turn, compactly supported +models lead to sparse covariance matrices (Furrer et al., 2006; Kaufman et al., 2008) and this implies +considerable computational gains in both estimation and prediction. Unfortunately, the Mat´ern +class does not allow to index global behaviour of the associated random process. The Generalized +Cauchy family (Gneiting and Schlather, 2004) allows to index the fractal dimension and the long +memory behaviour. Notably, it does not allow to index mean square differentiability, as the model +is either non differentiable or infinitely differentiable at the origin. The same properties are shared +by the Dagum model (Berg et al., 2008), which does not allow to index mean square differentiability +either. None of the aforementioned models allow to attain negative spatial dependencies. +Spectral approaches can be a promising avenue to find flexible families of covariance functions. +Laga and Kleiber (2017) proposed a modified version of the spectral density associated with the +Mat´ern family. The new class has two additional parameters that can be loosely interpreted as a +continuous version of a moving average process. More recently, Ma and Bhadra (2022) have proved +that a two-fold application of Gaussian scale mixtures can provide models with polynomial decays +while preserving the local properties of the candidate covariance function. Other non conventional +properties of covariance functions have been studied by Alegr´ıa (2020) and Alegr´ıa et al. (2021), +who proposed some modified scale mixtures representations to obtain classes of cross-covariance +functions with non-monotonic behaviours (the so-called cross-dimple effect) for vector-valued ran- +dom fields. In Schlather and Moreva (2017), models that allow for a smooth transition between +stationary and intrinsically stationary Gaussian random fields are derived. +All the previously mentioned parametric classes of covariance functions admit a scale mixture +representation of a Gaussian kernel against a continuous, positive and bounded measure. +Our +paper starts from the Schoenberg integral representation of isotropic covariance functions on Rd +(Schoenberg, 1938), for all natural numbers, d. We specifically assume the Schoenberg measures +to be parametric families of measures that are defined piecewise. Such a strategy is then shown +to provide hybrid classes that generalize classes proposed in earlier literature. We illustrate this +methodology by constructing a model that combines the global attributes of the Cauchy class +and the local properties of the Mat´ern class. We show that the proposed model admits a closed +form expression and examine its theoretical properties. +Additionally, we study a more flexible +formulation, where the Gaussian kernel involved in the scale mixture is replaced with a covariance +kernel that is also defined piecewise. Following this approach, we derive a hybrid model with local +2 + +behaviour of Mat´ern type, and global behaviour that allows for covariance functions with negative +values. We conduct numerical experiments with both simulated and real data in order to assess +the statistical performance of the proposed models. +The article is organized as follows. +Section 2 contains a concise review of random fields and +covariance functions coming from scale mixtures. Section 3 presents general methodologies to build +hybrid covariance models. Then, we derive the hybrid Cauchy-Mat´ern and the hybrid Hole-Effect- +Mat´ern classes. Section 4 guides the reader through some numerical studies. We finally provide +a critical discussion in Section 5, including a description of technical extensions of the present +work such as the multivariate case, where covariance functions are matrix-valued, and the case of +spherically indexed fields, where isotropy is defined in terms of the geodesic distance. +2 +Background +Let {Z(s) : s ∈ Rd} be a (centered) second-order stationary Gaussian random field on Rd. Such +a field is completely characterized by its covariance function (or kernel). +The isotropy of the +covariance function is defined through a mapping ϕ : [0, ∞) → R such that cov[Z(s), Z(s′)] = ϕ(h), +for every s, s′ ∈ Rd, where h = ∥s − s′∥. The covariance function must satisfy the positive (semi) +definiteness condition: for any k ∈ N, {a1, . . . , ak} ⊂ R and {s1, . . . , sk} ⊂ Rd, +k +� +i,j=1 +aiajϕ(∥si − sj∥) ≥ 0. +We use the notation ϕ(·; λ) for a parametric family of continuous covariance functions, where +λ ∈ Rp is a vector of parameters. Further, we make use of the celebrated Schoenberg’ theorem +(Schoenberg, 1938): the functions ϕ that are valid in any dimension d ∈ N are uniquely written as +Gaussian scale mixtures of positive and bounded measures, that is +ϕ(h; λ) = +� ∞ +0 +exp(−uh2)G(du; λ), +h ≥ 0, +where {G(d·; λ), λ ∈ Rp} is a parametric family of measures, that are termed Schoenberg measures +in Daley and Porcu (2014). Most of the covariance classes listed in the introduction admit such +a representation against a measure that is absolutely continuous with respect to the Lebesgue +measure, that is +ϕ(h; λ) = +� ∞ +0 +exp(−uh2)g(u; λ)du, +h ≥ 0, +(2.1) +for {g(·; λ), λ ∈ Rp} a parametric family of nonnegative functions. Throughout, we call g the +mixing function. +We now describe examples of some parametric classes of functions ϕ that are determined according +to (2.1). +Special attention is devoted to the Mat´ern, Cauchy and Generalized Cauchy models. +Other examples, including the stable and generalized hyperbolic models, can be found in Yaglom +(1987), Barndorff-Nielsen (1978), Schlather (2010) and Porcu et al. (2018b). +Example 2.1 (Mat´ern). This class of covariance functions is defined as (Mat´ern, 1986) +ϕM (h; λ) = 21−ν +Γ(ν)(h/α)νKν(h/α), +h ≥ 0, +(2.2) +3 + +where Γ is the gamma function and Kν is the modified Bessel function of the second kind (Abramowitz +and Stegun, 1972). Here, λ = [α, ν]⊤, with α and ν being positive parameters that control the scale +(the rate of decay of the covariance in terms of h) and shape of (2.2), respectively. More precisely, +ν regulates the degree of mean square differentiability of the random field (large values of ν are +associated with smoother sample paths) (Stein, 1999). When λ = [α, 1/2]⊤, (2.2) simplifies into +the exponential model, exp(−h/α). On the other hand, as ν → ∞, a reparameterization of (2.2) +tends to the Gaussian covariance function, defined as exp(−h2/α). +Example 2.2 (Cauchy). This class of covariance functions is given by (Chil`es and Delfiner, 2012) +ϕC (h; λ) = +� +1 + h2/α +�−ν/2 , +h ≥ 0, +(2.3) +with λ = [α, ν]⊤. +As in the Mat´ern model, α > 0 is a scale parameter. +However, unlike the +Mat´ern model which decays exponentially with distance (Stein, 1999), (2.3) has a polynomial +decay regulated by ν > 0. When ν ∈ (0, 2), such a polynomial decay is connected with the Hurst +parameter, a measure of long term memory, given by H = 1 − ν/2. +Example 2.3 (Generalized Cauchy). This class of covariance functions is defined as (Gneiting and +Schlather, 2004 and the references therein) +ϕGC (h; λ) = +� +1 + hδ/α +�−ν/δ +, +h ≥ 0, +(2.4) +with λ = [α, ν, δ]⊤, where δ ∈ (0, 2], α > 0 and ν > 0. +This generalized class preserves the +polynomial decay of (2.3), but it is more flexible in the sense that the fractal dimension can be +arbitrarily regulated through δ (see Gneiting and Schlather, 2004 for details). Maybe surprisingly, +this model does not allow to control the mean square differentiability of the respective random +field, as the model is either non differentiable or infinitely differentiable at the origin. +Additional classes of covariance functions can be obtained from the more general mixture +ϕ(h; λ, ϑ) = +� ∞ +0 +φ(h; u, ϑ)g(u; λ)du, +h ≥ 0, +(2.5) +where φ(·; u, ϑ) is an arbitrary covariance kernel, for every u > 0, and ϑ is a vector of parameters. +Since the class of positive definite functions is a convex cone that is closed under the topology of +pointwise convergence, if φ is valid (positive definite) in Rd for d ≤ d′, for some d′ ∈ N, then ϕ +is valid in Rd for d ≤ d′ as well. We refer the reader to Emery and Lantu´ejoul (2006) for several +explicit examples. +3 +Hybrid Classes of Covariance Functions +3.1 +General Construction +In this study, we propose new parametric classes of isotropic covariance functions, �ϕ(·; λ, ω, ξ), +determined according to +�ϕ(h; λ, ω, ξ) = ω1 +� ξ1 +0 +exp(−uh2)g1(u; λ1)du + ω2 +� ∞ +ξ2 +exp(−uh2)g2(u; λ2)du, +(3.1) +where g1 and g2 are nonnegative functions on [0, ξ1) and [ξ2, ∞), respectively, and λ = [λ⊤ +1 , λ⊤ +2 ]⊤, +ω = [ω1, ω2]⊤ and ξ = [ξ1, ξ2]⊤ are vectors of parameters, with ωi, ξi > 0, for i = 1, 2. In other +4 + +words, we replace the mixing function, g, in Equation (2.1) with a function �g that is defined +piecewise, i.e., +�g(u; λ, ω, ξ) = ω1 g1(u; λ1)1[0,ξ1)(u) + ω2 g2(u; λ2)1[ξ2,∞)(u), +u ≥ 0, +(3.2) +with 1A(·) standing for the indicator function of a set A. Note that �g may have discontinuities as +it is built by gluing two individual pieces. If the functions gi are continuous and bounded on their +domains, a direct application of the dominated convergence theorem implies that the proposed +covariance function (3.1) is continuous on [0, ∞). Throughout this manuscript, each function gi is +positively proportional to a continuous probability density function. Hence, the parametric family +proposed in Equation (3.1) belongs to the Schoenberg class as defined through Equation (2.1). +A more general construction considers different kernels in each segment of the mixture, i.e., +�ϕ(h; λ, ω, ξ, ϑ) = ω1 +� ξ1 +0 +φ1(h; u, ϑ1)g1(u; λ1)du + ω2 +� ∞ +ξ2 +φ2(h; u, ϑ2)g2(u; λ2)du, +(3.3) +where ϑ = [ϑ⊤ +1 , ϑ⊤ +2 ]⊤. If φi is a valid covariance function in Rd for d ≤ d′ +i, for some d′ +i ∈ N, i = 1, 2, +then (3.3) is a valid model in Rd if and only if d ≤ min(d′ +1, d′ +2). The continuity of (3.3) can be +justified by following the same arguments used for the continuity of (3.1). +Remark 3.1. Let us point out some additional remarks on this methodology. +1. When ξ1 = ξ2 = ξ, this parameter produces a continuous bridge between two apparently +disunited marginal models. More precisely, as it increases from 0 to ∞, we gradually go from +ω2 +� ∞ +0 φ2(h; u, ϑ2)g2(u; λ2)du to ω1 +� ∞ +0 φ1(h; u, ϑ1)g1(u; λ1)du. +2. When ξ1 > ξ2, instead, there is a superposition of the marginal structures in the interval +[ξ2, ξ1). +As ξ2 → 0 and ξ1 → ∞, we obtain the greatest possible superposition, which +corresponds to a linear combination of the marginal models, ω1 +� ∞ +0 φ1(h; u, ϑ1)g1(u; λ1)du + +ω2 +� ∞ +0 φ2(h; u, ϑ2)g2(u; λ2)du. +The apparent flexibility of the proposed mixtures is justified by classical theory on local and global +behaviour of covariance functions. In particular, a direct application of Tauberian theorems (Stein, +1999) proves that mean square differentiability of �ϕ will be determined by g2. On the other hand, +direct inspection in concert with Equation (4) in Gneiting and Schlather (2004) shows that the +long term behaviour of �ϕ is decided by g1. The next sections show that it is possible to provide +examples in algebraically closed form that allow to attain the desired flexibility. +3.2 +A Hybrid Cauchy-Mat´ern Class +We present a hybrid Cauchy-Mat´ern model, for which the acronym CM is used. This model is a +special case of (3.1). Let us first introduce the generalized incomplete gamma function (Chaudhry +and Zubair, 1994), +Γ(a; b; c) = +� ∞ +b +ta−1 exp(−t − ct−1) dt, +and the lower incomplete gamma function, γ(a, b) = Γ(a; 0; 0) − Γ(a; b; 0). +Proposition 3.1. Let λ = [λ⊤ +1 , λ⊤ +2 ]⊤, with λi = [αi, νi]⊤, ω = [ω1, ω2]⊤ and ξ = [ξ1, ξ2]⊤ be +vectors having positive elements. Let +�ϕCM(h; λ, ω, ξ) = ω1 �ϕ (1) +C (h; λ1, ξ1) + ω2 �ϕ (2) +M (h; λ2, ξ2), +h ≥ 0, +(3.4) +5 + +where +�ϕ (1) +C (h; λ1, ξ1) = γ(ν1/2, (h2 + α1)ξ1) +Γ(ν1/2) +ϕC(h; λ1) +(3.5) +and +�ϕ (2) +M (h; λ2, ξ2) = ϕM(h; λ2) − +1 +Γ(ν2)Γ +� +ν2; +1 +4ξ2α2 +2 +; h2 +4α2 +2 +� +, +(3.6) +with ϕM and ϕC being, respectively, the Mat´ern and the Cauchy models defined at (2.2) and (2.3). +Then, �ϕCM is positive definite in Rd for all d ∈ N. +Proof 3.1. We provide a proof of the constructive type, by showing that �ϕCM admits the rep- +resentation (3.1), with g1(u; λ1) = gC(u; λ1) and g2(u; λ2) = gM(u; λ2), with gC and gM that are +respectively defined as +gC(u; λ1) = +αν1/2 +1 +Γ(ν1/2)uν1/2−1 exp (−α1u) , +(3.7) +and +gM(u; λ2) = +1 +Γ(ν2) +� 1 +2α2 +�2ν2 +u−ν2−1 exp +� +− +1 +4uα2 +2 +� +, +(3.8) +where for both cases all the parameters are positive. To attain the analytical expression of �ϕ(1) +C , +we notice that +� ξ1 +0 +exp(−uh2)gC(u; λ1)du += +ϕC(h; λ1) +� ξ1 +0 +(h2 + α1)ν1/2 +Γ(ν1/2) +uν1/2−1 exp(−(h2 + α1)u)du += +ϕC(h; λ1)γ(ν1/2, (h2 + α1)ξ1) +Γ(ν1/2) +, +where the second equality is due to the fact that the integral on the right hand side of the first +line amounts to the cumulative distribution function of a gamma random variable with parameters +h2 + α1 and ν1/2. +To attain the expression of �ϕ(2) +M, we invoke Equation (10) in Alegr´ıa et al. (2021), so that +� ξ2 +0 +exp(−uh2)gM(u; λ2)du = +1 +Γ(ν2)Γ +� +ν2; +1 +4ξ2α2 +2 +; h2 +4α2 +2 +� +. +(3.9) +The function �ϕ(2) +M is thus attained by invoking formula 3.471.9 in Gradshteyn and Ryzhik (2007), +for which we have +� ∞ +0 exp(−uh2)gM(u; λ2)du = ϕM(h; λ2). +When ν2 = n + 1/2, for some n ∈ N, (3.6) can be expressed in terms of complementary error +functions and modified Bessel functions of first and second kinds. We refer the reader to Alegr´ıa +et al. (2021) for a more detailed study of these special cases. +The flexibility of the proposed structure is now illustrated through the following result, where we +use the notation f1(h) ∼ f2(h), h → ∞, to represent that, for some positive constant c0, the +asymptotic relationship limh→∞ f1(h)/f2(h) = c0 holds. +Proposition 3.2. Let Z be a Gaussian random field with covariance function of the form (3.4). +Then, Z is κ-times mean square differentiable if and only if ν2 > κ ≥ 0. Moreover, it is true that +�ϕCM(h; λ, ω, ξ) ∼ h−ν1, h → ∞. Hence, the Hurst parameter associated with Z is solely indexed +by the parameter ν1. +6 + +Proof 3.2. Arguments in Chapter 2 of Stein (1999) show that an isotropic random field with +covariance function ϕ is κ-times mean square differentiable if and only if ϕ(2κ)(0; λ) exists and +is finite. +Direct inspection in concert with dominated convergence on the invoked Schoenberg +representation (2.1) show that this happens if and only if the mixing function g satisfies +� ∞ +0 +uκg(u; λ)du < ∞. +(3.10) +We use the latter argument for the special case of the function �ϕCM, for which the tale of the +resulting mixing function is uniquely determined by the mixing function associated with ϕ(2) +M as in +Proposition 3.1. Direct inspection shows that (3.10) is true if and only if ν2 > κ. The first part of +the proposition is established. +For the second part, note that (3.5) behaves as h−ν1, as h → ∞, because the lower incomplete +gamma function involved in such an equation tends to Γ(ν1/2), and the Cauchy class with parameter +ν1 decays as h−ν1. The result follows by noting that (3.6) is dominated by the traditional Mat´ern +model, which decays exponentially. +To wrap up, the hybrid Cauchy-Mat´ern model allows to index both mean square differentiability and +long term behaviour of the associated Gaussian random field. We also note that these properties +are independently addressed by the two parameters ν1 and ν2, and hence those parameters are +statistically identifiable and allow to decouple local and global properties. +From a statistical viewpoint, a parsimonious choice may be considered by setting ω1 = ω2 = ω, +α1 = α2 = α and ξ1 = ξ2 = ξ. Thus, we obtain that Proposition 3.1 provides a five parameter +family where ω indexes the variance, α the scale, ν2 the mean square differentiability, and ν1 the +Hurst effect, whereas ξ is a parameter that balances the shapes of the marginal structures involved +in this model. Hence, (3.4) generalizes the Mat´ern model in that it allows for polynomial decay +while indexing continuously mean square differentiability. +Figure 3.1 shows the parsimonious hybrid Cauchy-Mat´ern model for different values of ξ. The +traditional Mat´ern and traditional Cauchy, as well as their average, which are also special cases of +the hybrid construction, are reported for comparison purposes. Note that the curves have a linear +or parabolic decay near the origin according to ν2 = 1/2 or ν2 = 3/2, respectively, and then the +decay is more gradual (polynomial rate) for large distances according to ν1, which is consistent +with the local and global patterns that are coexisting. We observe that ξ has a manifest impact +on the shape of the covariance function, as it produces some interesting forms (apparent changes +of concavity) that could be useful in practice. +3.3 +A Hybrid Hole-Effect-Mat´ern Class +We now present a hybrid class of covariance functions, with local attributes of Mat´ern type, at- +taining negative values at large distances. We use the acronym HM for this model, termed hybrid +Hole-Effect-Mat´ern. The proposed class comes from the mixture (3.3), where φ1 is chosen in such +a way that the resulting model can take negative values. +Proposition 3.3. Let λ = [λ⊤ +1 , λ⊤ +2 ]⊤, with λi = [αi, νi]⊤, ω = [ω1, ω2]⊤ and ξ = [ξ1, ξ2]⊤ be +vectors having positive elements, and ϑ = [τ, η]⊤ be a vector of additional parameters. Let +�ϕHM(h; λ, ω, ξ, ϑ) = ω1 �ϕ (1) +H (h; λ1, ξ1, ϑ) + ω2 �ϕ (2) +M (h; λ2, ξ2), +h ≥ 0, +(3.11) +7 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +ξ = 3 +ξ = 32 +ξ = 33 +ξ = 34 +ξ = 35 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +ξ = 3 +ξ = 32 +ξ = 33 +ξ = 34 +ξ = 35 +Figure 3.1: Parsimonious hybrid Cauchy-Mat´ern model for ω = 1/2, α = 1/8, ν1 = 3/4 and +different values of ξ. (Left) ν2 = 1/2 and (Right) ν2 = 3/2. The dashed lines represent the purely +Cauchy, purely Mat´ern, and their average. All the models have been appropriately rescaled in order +to obtain correlation functions. +where +�ϕ (1) +H (h; λ1, ξ1, ϑ) = +τ +Γ(ν1)Γ +� +ν1; +1 +4ξ1α2 +1 +; ηh2 +4α2 +1 +� +− +1 +Γ(ν1)Γ +� +ν1; +1 +4ξ1α2 +1 +; h2 +4α2 +1 +� +, +(3.12) +and �ϕ (2) +M as in (3.6). Then, �ϕHM is positive definite in Rd if and only if 1 < η < τ 2/d. +Proof 3.3. We consider the construction (3.3), with both g1 and g2 of the form (3.8), and φ2 of +Gaussian type. Thus, the derivation of �ϕ (2) +M follows the same arguments employed in the proof of +Proposition 3.1. +Before deriving (3.12), let us introduce the following lemma, which is a combination of Corollaries +4, 8 and 11 in Posa (2022). +Lemma 3.1. The mapping h �→ A exp(−ah2) − B exp(−bh2) is positive definite in Rd if and only +if +1 < a +b < +�A +B +�2/d +. +(3.13) +Although Posa (2022) focused on dimensions d ≤ 3, the same proof can be used in arbitrary +dimensions. To obtain the expression (3.12), we take the following covariance kernel in the first +segment of the scale mixture +φ1(h; u, ϑ) = τ exp(−uηh2) − exp(−uh2), +h ≥ 0. +(3.14) +Lemma 3.1 ensures that (3.14) is positive definite in Rd, provided that u > 0 and 1 < η < τ 2/d. +Thus, +�ϕ (1) +H (h; λ1, ξ1, ϑ) = τ +� ξ1 +0 +exp(−uηh2)gM(u; λ1)du − +� ξ1 +0 +exp(−uh2)gM(u; λ1)du. +(3.15) +8 + +Finally, we invoke the identity (3.9), and we apply it to each integral involved in the right hand +side of Equation (3.15). +The covariance function (3.14) always takes negative values (Posa, 2022), so it is a natural building +block to achieve hybrid models with hole effect. +The parameters in ϑ are responsible for the +sharpness of the hole effect. More precisely, as η approaches τ 2/d, the hole effect is more pronounced +because the positive term in the right hand size of (3.14) has less dominance. Moreover, when d = 1, +we have the least restrictive condition on η, and the resulting hole effect is more marked. It is well +known that the possibility of significant negative correlations vanishes as the dimension increases +(see, e.g., page 45 in Stein, 1999). +The next proposition characterizes the local attributes of (3.11) and provides a lower bound for +this model. +Proposition 3.4. Let Z be a Gaussian random field with covariance function of the form (3.11). +Then, Z is κ-times mean square differentiable if and only if ν2 > κ ≥ 0. Moreover, we have the +lower bound +�ϕHM(h; λ, ω, ξ, ϑ) ≥ ω1(τη)−1/(η−1) +�1 − η +η +� � +1 − γ(ν1; α1/ξ1) +Γ(ν1) +� +, +h ≥ 0. +(3.16) +Proof 3.4. The fact that ν2 controls the mean square differentiability is a direct consequence of +the arguments used in the proof of Proposition 3.2. On the other hand, to find a lower bound, we +note that +�ϕHM(h; λ, ω, ξ, ϑ) +≥ +ω1 inf +h≥0 �ϕ (1) +H (h; λ1, ξ1, ϑ) + ω2 inf +h≥0 �ϕ (2) +M (h; λ2, ξ2) += +ω1 +� ξ1 +0 +inf +h≥0 φ1(h; u, ϑ)g1(u; λ1)du. +A straightforward calculation shows that φ1 attains its minimum value at h∗ = +� +log(τη) +u(η−1). Thus, +φ1(h; u, ϑ) ≥ φ1(h∗; u, ϑ) = τ exp +� +−η log(τη) +η − 1 +� +− exp +� +−log(τη) +η − 1 +� += (τη)−1/(η−1) +�1 − η +η +� +. +Since g1 is given by (3.8), we invoke the formula of the cumulative distribution function of an +inverse gamma random variable to establish that +� ξ1 +0 +g1(u; λ1)du = 1 − γ(ν1; α1/ξ1) +Γ(ν1) +. +The proof is completed. +Note that as ξ1 → ∞ (i.e., as the hole effect predominates), the lower bound in Equation (3.16) +decreases to (τη)−1/(η−1)(1−η)/η. On the contrary, as ξ1 → 0, such a bound increases to zero, i.e., +the hole effect becomes negligible, which is not surprising, because in such a case the Mat´ern class +is predominant. A similar conclusion can be obtained in the limit case η → 1. +A parsimonious variant of this model consists of taking ω1 = ω2 = ω (variance parameter), α1 = +α2 = α (scale parameter) and ν1 = ν2 = ν (smoothness parameter), whereas ϑ regulates the +hole effect (as discussed above) and ξ1 = ξ2 = ξ has a similar interpretation as in the hybrid +Cauchy-Mat´ern model. +9 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +ξ = 3 +ξ = 32 +ξ = 33 +ξ = 34 +ξ = 35 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance +Correlation +ξ = 3 +ξ = 32 +ξ = 33 +ξ = 34 +ξ = 35 +Figure 3.2: Parsimonious hybrid Hole-Effect-Mat´ern model in dimension one, for ω = 1/2, α = 1/8, +τ = 2, η = 7/2 and different values of ξ. (Left) ν = 1/2 and (Right) ν = 3/2. The dashed lines +represent the limit cases reported in Remark 3.1. All the models have been appropriately rescaled +in order to obtain correlation functions. +Figure 3.2 shows the parsimonious hybrid Hole-Effect-Mat´ern model for different values of ξ. The +limit cases described in Remark 3.1 are also reported, in a similar fashion to Figure 3.1. It can be +seen that negative values coexist with different levels of smoothness at the origin, as expected. +4 +Numerical Experiments +4.1 +Simulated Data +We conduct simulation studies to assess the performance of maximum likelihood inference when +a hybrid covariance structure is present. We focus on the parsimonious hybrid Cauchy-Mat´ern +dependence structure, as it will be applied to real data in the next section. We consider ω = 1, +α = 1/8, ν1 = 3/4 and the following scenarios for [ν2, ξ]: (a) [1/2, 40], (b) [1/2, 120], (c) [3/2, 40] and +(d) [3/2, 120]. For each scenario, we simulate 200 independent realizations of a Gaussian random +field on 100 uniformly sampled points in the square [0, 3]2 and estimate the parameters through +maximum likelihood. We then repeat the experiment with 256 spatial locations. We only estimate +ω , α and ξ, whereas ν1 and ν2 are fixed, which is a common practice in geostatistics. Instead +of directly estimating ξ, we consider the following alternative parameterization: �ξ = √ξα, which +seems to be a natural choice according to Equations (3.5) and (3.6). +Figure 4.1 displays the results. The estimates are approximately unbiased and the variance de- +creases as the sample size increases from 100 to 256, which is an expected behaviour. The vari- +ability of the estimates substantially decreases in scenarios (c) and (d), i.e., when the random field +is smoother, which is a typical attribute of likelihood-based estimates in this context (Bevilacqua +and Gaetan, 2015). On the contrary, such a variability deteriorates as ξ increases from 40 to 120. +Figure 4.2 shows the log-likelihood in terms of ξ and α, with fixed ω, for a single realization of the +random field, under scenario (b). Although the surface has a clear maximum value, the objective +function is apparently more flat in the direction of ξ. This could explain the increased variability +10 + +-1 +0 +1 +2 +3 +n = 100 +n = 256 +Sample Size +Estimates +Scenario +(a) +(b) +(c) +(d) +ω +0.00 +0.25 +0.50 +0.75 +n = 100 +n = 256 +Sample Size +Estimates +Scenario +(a) +(b) +(c) +(d) +α +0.0 +2.5 +5.0 +n = 100 +n = 256 +Sample Size +Estimates +Scenario +(a) +(b) +(c) +(d) +ξ~ +Figure 4.1: Centered boxplots of the maximum likelihood estimates for the parsimonious hybrid +Cauchy-Mat´ern model in scenarios (a)-(d). +in scenarios (b) and (d), with respect to (a) and (c). Despite the previous remarks, in general, the +estimates appear to be reasonable in each scenario and no identifiability issues are observed. +We now explore the predictive performance of the proposed class through a cross validation analysis. +We simulate 200 independent realizations on 100 uniformly sampled locations in [0, 3]2 according +to the scenarios (a)-(d) described above. We assess the accuracy through a leave-one-out predic- +tion strategy in terms of the mean squared error (MSE), mean absolute error (MAE), log-score +(LSCORE) and continuous ranked probability score (CRPS) (see Zhang and Wang, 2010). Small +values of these indicators suggest superior predictions. We evaluate the performance of the hybrid +Cauchy-Mat´ern model, using the Generalized Cauchy class as benchmark. Thus, for each realiza- +tion, we estimate the parameters with both models and proceed to make the predictions through a +simple kriging approach. The Generalized Cauchy model (2.4) has been augmented with a multi- +plicative parameter ω, namely h �→ ω(1+hδ/α)−ν/δ, so it is parameterized by ω and α, and ν = 3/4 +and δ = 1, 2 are fixed. +Table 4.1 shows that, in each scenario, the proposed hybrid model outperforms its competitor. All +the cross-validation scores substantially decrease in scenarios (c) and (d). From this brief study, we +observe that when the true underlying covariance has a hybrid structure, an incorrect specification +11 + +alpha +Log Likelihood +xi +Log Likelihood +Figure 4.2: Log-likelihood function, with respect to α and ξ, for scenario (b). Left and right panels +correspond to the same plot from different viewpoints. +of the spatial association has a negative impact on the posterior predictions. Since the behaviour +of an isotropic covariance function near the origin has a strong impact on the quality of predictions +(Stein, 1999), our simulation experiment suggests that in some circumstances the local shape of +the proposed model cannot be replicated by other appealing existing structures. +4.2 +A Real Data Illustration +The estimation of recoverable resources is a task of fundamental importance in modern mining +processes. +A sound evaluation of such resources is crucial from an economic viewpoint and is +critical for assessing the long-term availability of mineral resources and its impact on society. We +consider a data set from a lateritic nickel deposit mined by open pit in Colombia, which contains +measurements of the grades of nickel, iron, chrome, alumina, magnesia and silica. +This study focuses on nickel concentrations that are placed at an elevation of about 120 meters, +where 199 irregularly spaced observations are available. We apply a log-transformation to reduce +the skewness, and then the sample mean is subtracted. The resulting values are approximately +Gaussian. The left panel of Figure 4.4 shows the transformed data set. We fit two covariance +models: the former is the parsimonious hybrid Cauchy-Mat´ern, parameterized by ω, α and �ξ, with +fixed ν1 = 1/4 and ν2 = 1/2, and the latter is the Generalized Cauchy, parameterized as in Section +4.1, with fixed ν = 1/4 and δ = 0.95. +The values of the fixed parameters have been selected +after some experimental trials, taking into account the local behavior of the sample covariance (see +Figure 4.3). +Table 4.2 reports the likelihood estimates, with the corresponding standard errors, and the Akaike +information criterion (AIC). We observe that the hybrid Cauchy-Mat´ern model outperforms its +competitor in terms of AIC. Figure 4.3 shows that the fitted covariance models seem to be reason- +ably close to the sample covariance. The fitted models differ substantially near the origin (distances +less than 3 meters), since the hybrid model decays faster. On the contrary, for larger distances the +hybrid model decays slower, although the difference between the curves becomes slight for distances +greater than 15 meters. +12 + +Table 4.1: Cross-validation scores for the parsimonious hybrid Cauchy-Mat´ern and Generalized +Cauchy (with δ = 1, 2) models in scenarios (a)-(d). +Scenario +Model +MSE +MAE +LSCORE +CRPS +(a) +Hybrid Cauchy-Mat´ern +0.706 +0.668 +1.231 +1.773 +Generalized Cauchy (δ = 1) +0.714 +0.672 +1.238 +1.787 +Generalized Cauchy (δ = 2) +0.718 +0.674 +1.241 +1.798 +(b) +Hybrid Cauchy-Mat´ern +0.480 +0.549 +1.034 +1.462 +Generalized Cauchy (δ = 1) +0.489 +0.555 +1.046 +1.480 +Generalized Cauchy (δ = 2) +0.497 +0.559 +1.055 +1.506 +(c) +Hybrid Cauchy-Mat´ern +0.172 +0.316 +0.398 +0.840 +Generalized Cauchy (δ = 1) +0.176 +0.319 +0.446 +0.851 +Generalized Cauchy (δ = 2) +0.176 +0.319 +0.414 +0.862 +(d) +Hybrid Cauchy-Mat´ern +0.075 +0.210 +0.024 +0.561 +Generalized Cauchy (δ = 1) +0.077 +0.213 +0.052 +0.575 +Generalized Cauchy (δ = 2) +0.082 +0.219 +0.103 +0.612 +Table 4.2: Parameter estimates and Akaike Information Criterion (AIC) of fitted covariance models. +Standard errors are reported in parentheses. +Model +ω +α +�ξ +AIC +Hybrid Cauchy-Mat´ern +1.055 +12.31 +0.063 +−34.03 +(0.2516) +(2.801) +(0.042) +Generalized Cauchy +0.164 +1.959 +− +−31.93 +(0.040) +(0.849) +− +13 + +0 +5 +10 +15 +20 +25 +0.04 +0.08 +0.12 +0.16 +Distance (m) +Covariance of log Ni +0 +5 +10 +15 +20 +25 +0.04 +0.08 +0.12 +0.16 +Distance (m) +Covariance of log Ni +0 +5 +10 +15 +20 +25 +0.04 +0.08 +0.12 +0.16 +Distance (m) +Covariance of log Ni +Hybrid Cauchy-Matern +Generalized Cauchy +Figure 4.3: Sample (circles) and modeled (solid lines) covariances of log-nickel concentrations. +Table 4.3: Scores for the leave-one-out cross-validation study of log-nickel concentrations. +Model +MSE +MAE +LSCORE +CRPS +Hybrid Cauchy-Mat´ern +0.0428 +0.1431 +−0.1840 +0.4113 +Generalized Cauchy +0.0443 +0.1462 +−0.1677 +0.4159 +In order to compare the models in terms of predictive performance, we conduct a cross-validation +study, in a similar fashion to the experiments performed with simulated data. Table 4.3 shows +evidence, based on a leave-one-out cross-validation scheme, that the hybrid model has a better +performance for this specific data set. In percentage terms, the MSE shows an improvement of +approximately 3.4%. The largest difference occurs when we compare the LSCORE’s (about 9% +improvement). +We conclude this section with an illustration of a downscaled map of log-nickel concentrations +(see Figure 4.4), using the hybrid Cauchy-Mat´ern model. The interpolated spatial map, which +is obtained through simple kriging, is exhibited on a spatial grid of approximately 1 meter (7500 +locations). This kriged surface could be useful in small-scale mining processes, as it is a crucial +step for industrial exploration and to quantify mineral reserves. +5 +Conclusions and Perspectives +We introduced a simple formalism to build sophisticated parametric families of covariance functions. +We focused on a combination between the Mat´ern and Cauchy models, where local (mean square +differentiability) and global (long memory) properties coexist in a single family. +We have also +illustrated the use of our methodology by constructing a model that behaves as the Mat´ern class +at short distances and attains negative values at large distances. Simulation studies show that +a parsimonious hybrid Cauchy-Mat´ern model has statistically identifiable parameters. Also, this +model provides improvements in terms of predictive performance in comparison to existing models, +14 + +Easting (m) +Northing (m) +−1.0 +−0.5 +0.0 +0.5 +1.0 +Easting (m) +Northing (m) +1650 +1700 +1750 +1670 +1680 +1690 +1700 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Easting (m) +Northing (m) +1650 +1700 +1750 +1670 +1680 +1690 +1700 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Figure 4.4: Log-nickel concentrations (left), with the kriged surface (middle) and the corresponding +variance (right). +when a hybrid inherent dependence structure is present. We reach similar conclusions when we +apply this methodology to a mining dataset. While similar numerical studies could be performed +for the hybrid Hole-Effect-Mat´ern model, we avoid them for the sake of simplicity and brevity. +Additional interesting extensions of this work can be tackled in future investigations. We now +provide two concrete research lines that could emerge from this work. +Multivariate Hybrid Covariance Models +In many practical situations, two or more variables are simultaneously recorded. Thus, our findings +can be generalized to the case of multivariate fields {Z(s) = (Z1(s), . . . , Zp(s))⊤, s ∈ Rd}, having an +isotropic matrix-valued covariance function Φ : [0, ∞) → Rp×p, that is, cov[Zi(s), Zj(s′)] = Φij(h), +h ≥ 0, where h = ∥s − s′∥ and i, j = 1, . . . , p. We propose the hybrid model +�Φ(h; λ, ω, ξ) = ω1 +� ξ1 +0 +exp(−uh2)G1(u; λ1)du + ω2 +� ∞ +ξ2 +exp(−uh2)G2(u; λ2)du, +that generalizes (3.1), where the vectors of parameters λi must be chosen in such a way that the p×p +matrices Gi(u; λi) are positive semi-definite for every fixed u ≥ 0. Hence, a straight application +of Proposition 4 in Porcu and Zastavnyi (2011) would ensure �Φ to be positive semi-definite. A +multivariate version of the hybrid Cauchy-Mat´ern covariance function is a natural candidate. The +works of Gneiting et al. (2010) and Moreva and Schlather (2022) are relevant to tackle this challenge. +A multivariate version of the formulation (3.3) could be deduced similarly. +Hybrid Covariance Models on Spheres +For random fields that are indexed by the d-dimensional unit sphere, Sd, which is a useful framework +when analyzing global data (S2 is used as an approximation of the Earth), the isotropy assumption +is given by cov[Z(s), Z(s′)] = ψ(θ), s, s′ ∈ Sd, where ψ : [0, π] → R is a continuous mapping and +θ = arccos(s⊤s′) ∈ [0, π] is the geodesic distance. Schoenberg’s characterization (Schoenberg, 1942) +establishes that a parametric isotropic covariance function ψ(; λ) is valid in any dimension d, if +15 + +and only if, it can be written as ψ(θ; λ) = �∞ +ℓ=0 βℓ(λ)(cos θ)ℓ, θ ∈ [0, π], for some nonnegative +and summable parametric sequence {βℓ(λ)}∞ +ℓ=0. Thus, the hybrid models can be adapted to the +spherical context by considering a modified sequence of the form +�βℓ(λ, ω, ξ) = ω1 β(1) +ℓ (λ1)1[0,⌊ξ1⌋)(ℓ) + ω2 β(2) +ℓ (λ2)1[⌊ξ2⌋,∞)(ℓ), +ℓ = 0, 1, . . . , +where ⌊ξi⌋ ≥ 0, for i = 1, 2, with ⌊·⌋ standing for the floor function, and β(i) +ℓ +being a nonnegative +and summable sequence. +The local properties of spherically indexed random fields, and their +connections with the covariance function, have been studied in past literature (Bingham, 1973; +Guinness and Fuentes, 2016). However, global properties such as long memory are less intuitive +in this scenario as the spatial domain is a compact set. Covariance functions with hole effect, for +low-dimensional spheres, could be obtained by adapting formulation (3.3). +Acknowledgements +Alfredo Alegr´ıa was partially supported by the National Agency for Research and Development of +Chile, through grant ANID/FONDECYT/INICIACI´ON/No. 11190686. 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Environ- +metrics, 21(3-4):290–304. +19 + diff --git a/WNE5T4oBgHgl3EQfcQ-J/content/tmp_files/load_file.txt b/WNE5T4oBgHgl3EQfcQ-J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c34f47e618368652bb6275eba65a26fb31ace924 --- /dev/null +++ b/WNE5T4oBgHgl3EQfcQ-J/content/tmp_files/load_file.txt @@ -0,0 +1,1345 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf,len=1344 +page_content='Hybrid Parametric Classes of Isotropic Covariance Functions for Spatial Random Fields Alfredo Alegr´ıa∗1, Fabi´an Ram´ırez1, and Emilio Porcu2 1Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Chile 2Department of Mathematics, Khalifa University, The Arab Emirates January 16, 2023 Abstract Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The properties of the covariance func- tion at small and large distances determine the geometric attributes of the associated Gaussian random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Having covariance functions that allow to specify both local and global proper- ties is certainly on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This paper provides a method to find new classes of covariance functions having such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We term these models hybrid as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise lin- ear combination of parametric families of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' In order to illustrate our methodology, we provide new families of covariance functions that are proved to be richer with respect to other well known families that have been proposed by earlier literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' More precisely, we derive a hybrid Cauchy-Mat´ern model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid Hole-Effect-Mat´ern model, which is capable of attaining negative values (hole effect), while preserving the local attributes of the traditional Mat´ern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Our findings are illustrated through numerical studies with both simulated and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Keywords: Cauchy model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Gaussian scale mixtures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hole effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Long memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Mat´ern model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Mean square differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 1 Introduction Covariance functions are central to many disciplines such as spatial statistics (Cressie, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Chil`es and Delfiner, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hristopulos, 2020), stochastic processes (Porcu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2018a,b), machine learning (Schaback and Wendland, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Barp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2022), numerical analysis (Pazouki and Schaback, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Cockayne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2019) and stochastic mechanics (Ostoja-Starzewski, 2006, with the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Recent applications in climatology (Guinness and Hammerling, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2019), oceanography (Furrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Di Lorenzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2014), environmental sciences (Cressie and Kornak, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Stein, 2007) and natural resources engineering (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Emery and S´eguret, 2020) witness on the importance of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' It is very customary to assume the covariance function to depend on the distance between any pair of random variables located at two different points at the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Such an assumption is ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Email: alfredo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='alegria@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='cl 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='05602v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='ST] 13 Jan 2023 termed isotropy in spatial statistics and machine learning, and it is termed radial symmetry in other areas of applied mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The behaviour of the covariance function at short or long distances (we call this local and global properties, respectively) is crucial to understand the properties of random processes with a given covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Specifically, the local properties are related to the fractal dimension as well as the geometric properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', mean square differentiability) of the associated random process, as well as to its sample paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the other hand, the global behaviour of the covariance function allows to characterize persistency or antipersistency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', the long term behaviour) of the associated process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Another global behaviour of great interest is the so-called hole effect, which means that the covariance function could take negative values in a certain interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Finding parametric families of isotropic covariance functions that allow to index both local and global behaviour is a major challenge that has been tackled to a very limited extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The Mat´ern family has been the cornerstone in spatial statistics for over half a century now (Stein, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Its popularity is due to a parameter that controls the degree of mean square differentiability and fractal dimension of the corresponding random field (Stein, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Recently, Bevilacqua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (2022) have shown that the Mat´ern class is a special case of a richer class of models that, additionally to indexing local properties, allow to switch between compact or global supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' In turn, compactly supported models lead to sparse covariance matrices (Furrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Kaufman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2008) and this implies considerable computational gains in both estimation and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Unfortunately, the Mat´ern class does not allow to index global behaviour of the associated random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The Generalized Cauchy family (Gneiting and Schlather, 2004) allows to index the fractal dimension and the long memory behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Notably, it does not allow to index mean square differentiability, as the model is either non differentiable or infinitely differentiable at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The same properties are shared by the Dagum model (Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', 2008), which does not allow to index mean square differentiability either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' None of the aforementioned models allow to attain negative spatial dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Spectral approaches can be a promising avenue to find flexible families of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Laga and Kleiber (2017) proposed a modified version of the spectral density associated with the Mat´ern family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The new class has two additional parameters that can be loosely interpreted as a continuous version of a moving average process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' More recently, Ma and Bhadra (2022) have proved that a two-fold application of Gaussian scale mixtures can provide models with polynomial decays while preserving the local properties of the candidate covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Other non conventional properties of covariance functions have been studied by Alegr´ıa (2020) and Alegr´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (2021), who proposed some modified scale mixtures representations to obtain classes of cross-covariance functions with non-monotonic behaviours (the so-called cross-dimple effect) for vector-valued ran- dom fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' In Schlather and Moreva (2017), models that allow for a smooth transition between stationary and intrinsically stationary Gaussian random fields are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' All the previously mentioned parametric classes of covariance functions admit a scale mixture representation of a Gaussian kernel against a continuous, positive and bounded measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Our paper starts from the Schoenberg integral representation of isotropic covariance functions on Rd (Schoenberg, 1938), for all natural numbers, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We specifically assume the Schoenberg measures to be parametric families of measures that are defined piecewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Such a strategy is then shown to provide hybrid classes that generalize classes proposed in earlier literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We illustrate this methodology by constructing a model that combines the global attributes of the Cauchy class and the local properties of the Mat´ern class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We show that the proposed model admits a closed form expression and examine its theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Additionally, we study a more flexible formulation, where the Gaussian kernel involved in the scale mixture is replaced with a covariance kernel that is also defined piecewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Following this approach, we derive a hybrid model with local 2 behaviour of Mat´ern type, and global behaviour that allows for covariance functions with negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We conduct numerical experiments with both simulated and real data in order to assess the statistical performance of the proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Section 2 contains a concise review of random fields and covariance functions coming from scale mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Section 3 presents general methodologies to build hybrid covariance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Then, we derive the hybrid Cauchy-Mat´ern and the hybrid Hole-Effect- Mat´ern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Section 4 guides the reader through some numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We finally provide a critical discussion in Section 5, including a description of technical extensions of the present work such as the multivariate case, where covariance functions are matrix-valued, and the case of spherically indexed fields, where isotropy is defined in terms of the geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 2 Background Let {Z(s) : s ∈ Rd} be a (centered) second-order stationary Gaussian random field on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Such a field is completely characterized by its covariance function (or kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The isotropy of the covariance function is defined through a mapping ϕ : [0, ∞) → R such that cov[Z(s), Z(s′)] = ϕ(h), for every s, s′ ∈ Rd, where h = ∥s − s′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The covariance function must satisfy the positive (semi) definiteness condition: for any k ∈ N, {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' , ak} ⊂ R and {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' , sk} ⊂ Rd, k � i,j=1 aiajϕ(∥si − sj∥) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We use the notation ϕ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) for a parametric family of continuous covariance functions, where λ ∈ Rp is a vector of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Further, we make use of the celebrated Schoenberg’ theorem (Schoenberg, 1938): the functions ϕ that are valid in any dimension d ∈ N are uniquely written as Gaussian scale mixtures of positive and bounded measures, that is ϕ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) = � ∞ 0 exp(−uh2)G(du;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ), h ≥ 0, where {G(d·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ), λ ∈ Rp} is a parametric family of measures, that are termed Schoenberg measures in Daley and Porcu (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Most of the covariance classes listed in the introduction admit such a representation against a measure that is absolutely continuous with respect to the Lebesgue measure, that is ϕ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) = � ∞ 0 exp(−uh2)g(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ)du, h ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1) for {g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ), λ ∈ Rp} a parametric family of nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Throughout, we call g the mixing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We now describe examples of some parametric classes of functions ϕ that are determined according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Special attention is devoted to the Mat´ern, Cauchy and Generalized Cauchy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Other examples, including the stable and generalized hyperbolic models, can be found in Yaglom (1987), Barndorff-Nielsen (1978), Schlather (2010) and Porcu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 (Mat´ern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This class of covariance functions is defined as (Mat´ern, 1986) ϕM (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) = 21−ν Γ(ν)(h/α)νKν(h/α), h ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2) 3 where Γ is the gamma function and Kν is the modified Bessel function of the second kind (Abramowitz and Stegun, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Here, λ = [α, ν]⊤, with α and ν being positive parameters that control the scale (the rate of decay of the covariance in terms of h) and shape of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' More precisely, ν regulates the degree of mean square differentiability of the random field (large values of ν are associated with smoother sample paths) (Stein, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' When λ = [α, 1/2]⊤, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2) simplifies into the exponential model, exp(−h/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the other hand, as ν → ∞, a reparameterization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2) tends to the Gaussian covariance function, defined as exp(−h2/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 (Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This class of covariance functions is given by (Chil`es and Delfiner, 2012) ϕC (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) = � 1 + h2/α �−ν/2 , h ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3) with λ = [α, ν]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' As in the Mat´ern model, α > 0 is a scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' However, unlike the Mat´ern model which decays exponentially with distance (Stein, 1999), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3) has a polynomial decay regulated by ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' When ν ∈ (0, 2), such a polynomial decay is connected with the Hurst parameter, a measure of long term memory, given by H = 1 − ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 (Generalized Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This class of covariance functions is defined as (Gneiting and Schlather, 2004 and the references therein) ϕGC (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) = � 1 + hδ/α �−ν/δ , h ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4) with λ = [α, ν, δ]⊤, where δ ∈ (0, 2], α > 0 and ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This generalized class preserves the polynomial decay of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3), but it is more flexible in the sense that the fractal dimension can be arbitrarily regulated through δ (see Gneiting and Schlather, 2004 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Maybe surprisingly, this model does not allow to control the mean square differentiability of the respective random field, as the model is either non differentiable or infinitely differentiable at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Additional classes of covariance functions can be obtained from the more general mixture ϕ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ϑ) = � ∞ 0 φ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ)g(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ)du, h ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5) where φ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ) is an arbitrary covariance kernel, for every u > 0, and ϑ is a vector of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Since the class of positive definite functions is a convex cone that is closed under the topology of pointwise convergence, if φ is valid (positive definite) in Rd for d ≤ d′, for some d′ ∈ N, then ϕ is valid in Rd for d ≤ d′ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We refer the reader to Emery and Lantu´ejoul (2006) for several explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 3 Hybrid Classes of Covariance Functions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 General Construction In this study, we propose new parametric classes of isotropic covariance functions, �ϕ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ), determined according to �ϕ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ) = ω1 � ξ1 0 exp(−uh2)g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du + ω2 � ∞ ξ2 exp(−uh2)g2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1) where g1 and g2 are nonnegative functions on [0, ξ1) and [ξ2, ∞), respectively, and λ = [λ⊤ 1 , λ⊤ 2 ]⊤, ω = [ω1, ω2]⊤ and ξ = [ξ1, ξ2]⊤ are vectors of parameters, with ωi, ξi > 0, for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' In other 4 words, we replace the mixing function, g, in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1) with a function �g that is defined piecewise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', �g(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ) = ω1 g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)1[0,ξ1)(u) + ω2 g2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)1[ξ2,∞)(u), u ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2) with 1A(·) standing for the indicator function of a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Note that �g may have discontinuities as it is built by gluing two individual pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' If the functions gi are continuous and bounded on their domains, a direct application of the dominated convergence theorem implies that the proposed covariance function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1) is continuous on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Throughout this manuscript, each function gi is positively proportional to a continuous probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hence, the parametric family proposed in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1) belongs to the Schoenberg class as defined through Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A more general construction considers different kernels in each segment of the mixture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', �ϕ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ, ϑ) = ω1 � ξ1 0 φ1(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ1)g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du + ω2 � ∞ ξ2 φ2(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ2)g2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3) where ϑ = [ϑ⊤ 1 , ϑ⊤ 2 ]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' If φi is a valid covariance function in Rd for d ≤ d′ i, for some d′ i ∈ N, i = 1, 2, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3) is a valid model in Rd if and only if d ≤ min(d′ 1, d′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The continuity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3) can be justified by following the same arguments used for the continuity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let us point out some additional remarks on this methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' When ξ1 = ξ2 = ξ, this parameter produces a continuous bridge between two apparently disunited marginal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' More precisely, as it increases from 0 to ∞, we gradually go from ω2 � ∞ 0 φ2(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ2)g2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du to ω1 � ∞ 0 φ1(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ1)g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' When ξ1 > ξ2, instead, there is a superposition of the marginal structures in the interval [ξ2, ξ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' As ξ2 → 0 and ξ1 → ∞, we obtain the greatest possible superposition, which corresponds to a linear combination of the marginal models, ω1 � ∞ 0 φ1(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ1)g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du + ω2 � ∞ 0 φ2(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ2)g2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The apparent flexibility of the proposed mixtures is justified by classical theory on local and global behaviour of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' In particular, a direct application of Tauberian theorems (Stein, 1999) proves that mean square differentiability of �ϕ will be determined by g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the other hand, direct inspection in concert with Equation (4) in Gneiting and Schlather (2004) shows that the long term behaviour of �ϕ is decided by g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The next sections show that it is possible to provide examples in algebraically closed form that allow to attain the desired flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 A Hybrid Cauchy-Mat´ern Class We present a hybrid Cauchy-Mat´ern model, for which the acronym CM is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This model is a special case of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let us first introduce the generalized incomplete gamma function (Chaudhry and Zubair, 1994), Γ(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' c) = � ∞ b ta−1 exp(−t − ct−1) dt, and the lower incomplete gamma function, γ(a, b) = Γ(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 0) − Γ(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let λ = [λ⊤ 1 , λ⊤ 2 ]⊤, with λi = [αi, νi]⊤, ω = [ω1, ω2]⊤ and ξ = [ξ1, ξ2]⊤ be vectors having positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let �ϕCM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ) = ω1 �ϕ (1) C (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1, ξ1) + ω2 �ϕ (2) M (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2, ξ2), h ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4) 5 where �ϕ (1) C (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1, ξ1) = γ(ν1/2, (h2 + α1)ξ1) Γ(ν1/2) ϕC(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5) and �ϕ (2) M (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2, ξ2) = ϕM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2) − 1 Γ(ν2)Γ � ν2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 1 4ξ2α2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' h2 4α2 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6) with ϕM and ϕC being, respectively, the Mat´ern and the Cauchy models defined at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Then, �ϕCM is positive definite in Rd for all d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Proof 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We provide a proof of the constructive type, by showing that �ϕCM admits the rep- resentation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1), with g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1) = gC(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1) and g2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2) = gM(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2), with gC and gM that are respectively defined as gC(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1) = αν1/2 1 Γ(ν1/2)uν1/2−1 exp (−α1u) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='7) and gM(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2) = 1 Γ(ν2) � 1 2α2 �2ν2 u−ν2−1 exp � − 1 4uα2 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8) where for both cases all the parameters are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' To attain the analytical expression of �ϕ(1) C , we notice that � ξ1 0 exp(−uh2)gC(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du = ϕC(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1) � ξ1 0 (h2 + α1)ν1/2 Γ(ν1/2) uν1/2−1 exp(−(h2 + α1)u)du = ϕC(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)γ(ν1/2, (h2 + α1)ξ1) Γ(ν1/2) , where the second equality is due to the fact that the integral on the right hand side of the first line amounts to the cumulative distribution function of a gamma random variable with parameters h2 + α1 and ν1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' To attain the expression of �ϕ(2) M, we invoke Equation (10) in Alegr´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (2021), so that � ξ2 0 exp(−uh2)gM(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du = 1 Γ(ν2)Γ � ν2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 1 4ξ2α2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' h2 4α2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='9) The function �ϕ(2) M is thus attained by invoking formula 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='9 in Gradshteyn and Ryzhik (2007), for which we have � ∞ 0 exp(−uh2)gM(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du = ϕM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' When ν2 = n + 1/2, for some n ∈ N, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6) can be expressed in terms of complementary error functions and modified Bessel functions of first and second kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We refer the reader to Alegr´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (2021) for a more detailed study of these special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The flexibility of the proposed structure is now illustrated through the following result, where we use the notation f1(h) ∼ f2(h), h → ∞, to represent that, for some positive constant c0, the asymptotic relationship limh→∞ f1(h)/f2(h) = c0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let Z be a Gaussian random field with covariance function of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Then, Z is κ-times mean square differentiable if and only if ν2 > κ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Moreover, it is true that �ϕCM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ) ∼ h−ν1, h → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hence, the Hurst parameter associated with Z is solely indexed by the parameter ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 6 Proof 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Arguments in Chapter 2 of Stein (1999) show that an isotropic random field with covariance function ϕ is κ-times mean square differentiable if and only if ϕ(2κ)(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) exists and is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Direct inspection in concert with dominated convergence on the invoked Schoenberg representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1) show that this happens if and only if the mixing function g satisfies � ∞ 0 uκg(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ)du < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='10) We use the latter argument for the special case of the function �ϕCM, for which the tale of the resulting mixing function is uniquely determined by the mixing function associated with ϕ(2) M as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Direct inspection shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='10) is true if and only if ν2 > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The first part of the proposition is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' For the second part, note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5) behaves as h−ν1, as h → ∞, because the lower incomplete gamma function involved in such an equation tends to Γ(ν1/2), and the Cauchy class with parameter ν1 decays as h−ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The result follows by noting that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6) is dominated by the traditional Mat´ern model, which decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' To wrap up, the hybrid Cauchy-Mat´ern model allows to index both mean square differentiability and long term behaviour of the associated Gaussian random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We also note that these properties are independently addressed by the two parameters ν1 and ν2, and hence those parameters are statistically identifiable and allow to decouple local and global properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' From a statistical viewpoint, a parsimonious choice may be considered by setting ω1 = ω2 = ω, α1 = α2 = α and ξ1 = ξ2 = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, we obtain that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 provides a five parameter family where ω indexes the variance, α the scale, ν2 the mean square differentiability, and ν1 the Hurst effect, whereas ξ is a parameter that balances the shapes of the marginal structures involved in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4) generalizes the Mat´ern model in that it allows for polynomial decay while indexing continuously mean square differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 shows the parsimonious hybrid Cauchy-Mat´ern model for different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The traditional Mat´ern and traditional Cauchy, as well as their average, which are also special cases of the hybrid construction, are reported for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Note that the curves have a linear or parabolic decay near the origin according to ν2 = 1/2 or ν2 = 3/2, respectively, and then the decay is more gradual (polynomial rate) for large distances according to ν1, which is consistent with the local and global patterns that are coexisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We observe that ξ has a manifest impact on the shape of the covariance function, as it produces some interesting forms (apparent changes of concavity) that could be useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 A Hybrid Hole-Effect-Mat´ern Class We now present a hybrid class of covariance functions, with local attributes of Mat´ern type, at- taining negative values at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We use the acronym HM for this model, termed hybrid Hole-Effect-Mat´ern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The proposed class comes from the mixture (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3), where φ1 is chosen in such a way that the resulting model can take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let λ = [λ⊤ 1 , λ⊤ 2 ]⊤, with λi = [αi, νi]⊤, ω = [ω1, ω2]⊤ and ξ = [ξ1, ξ2]⊤ be vectors having positive elements, and ϑ = [τ, η]⊤ be a vector of additional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let �ϕHM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ, ϑ) = ω1 �ϕ (1) H (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1, ξ1, ϑ) + ω2 �ϕ (2) M (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2, ξ2), h ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='11) 7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation ξ = 3 ξ = 32 ξ = 33 ξ = 34 ξ = 35 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1: Parsimonious hybrid Cauchy-Mat´ern model for ω = 1/2, α = 1/8, ν1 = 3/4 and different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (Left) ν2 = 1/2 and (Right) ν2 = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The dashed lines represent the purely Cauchy, purely Mat´ern, and their average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' All the models have been appropriately rescaled in order to obtain correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' where �ϕ (1) H (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1, ξ1, ϑ) = τ Γ(ν1)Γ � ν1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 1 4ξ1α2 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' ηh2 4α2 1 � − 1 Γ(ν1)Γ � ν1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 1 4ξ1α2 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' h2 4α2 1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='12) and �ϕ (2) M as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Then, �ϕHM is positive definite in Rd if and only if 1 < η < τ 2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Proof 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We consider the construction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3), with both g1 and g2 of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8), and φ2 of Gaussian type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, the derivation of �ϕ (2) M follows the same arguments employed in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Before deriving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='12), let us introduce the following lemma, which is a combination of Corollaries 4, 8 and 11 in Posa (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The mapping h �→ A exp(−ah2) − B exp(−bh2) is positive definite in Rd if and only if 1 < a b < �A B �2/d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='13) Although Posa (2022) focused on dimensions d ≤ 3, the same proof can be used in arbitrary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' To obtain the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='12), we take the following covariance kernel in the first segment of the scale mixture φ1(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ) = τ exp(−uηh2) − exp(−uh2), h ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='14) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 ensures that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='14) is positive definite in Rd, provided that u > 0 and 1 < η < τ 2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, �ϕ (1) H (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1, ξ1, ϑ) = τ � ξ1 0 exp(−uηh2)gM(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du − � ξ1 0 exp(−uh2)gM(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='15) 8 Finally, we invoke the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='9), and we apply it to each integral involved in the right hand side of Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The covariance function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='14) always takes negative values (Posa, 2022), so it is a natural building block to achieve hybrid models with hole effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The parameters in ϑ are responsible for the sharpness of the hole effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' More precisely, as η approaches τ 2/d, the hole effect is more pronounced because the positive term in the right hand size of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='14) has less dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Moreover, when d = 1, we have the least restrictive condition on η, and the resulting hole effect is more marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' It is well known that the possibility of significant negative correlations vanishes as the dimension increases (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', page 45 in Stein, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The next proposition characterizes the local attributes of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='11) and provides a lower bound for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Let Z be a Gaussian random field with covariance function of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Then, Z is κ-times mean square differentiable if and only if ν2 > κ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Moreover, we have the lower bound �ϕHM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ, ϑ) ≥ ω1(τη)−1/(η−1) �1 − η η � � 1 − γ(ν1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' α1/ξ1) Γ(ν1) � , h ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='16) Proof 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The fact that ν2 controls the mean square differentiability is a direct consequence of the arguments used in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the other hand, to find a lower bound, we note that �ϕHM(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ, ϑ) ≥ ω1 inf h≥0 �ϕ (1) H (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1, ξ1, ϑ) + ω2 inf h≥0 �ϕ (2) M (h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2, ξ2) = ω1 � ξ1 0 inf h≥0 φ1(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ)g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A straightforward calculation shows that φ1 attains its minimum value at h∗ = � log(τη) u(η−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, φ1(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ) ≥ φ1(h∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' u, ϑ) = τ exp � −η log(τη) η − 1 � − exp � −log(τη) η − 1 � = (τη)−1/(η−1) �1 − η η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Since g1 is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8), we invoke the formula of the cumulative distribution function of an inverse gamma random variable to establish that � ξ1 0 g1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du = 1 − γ(ν1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' α1/ξ1) Γ(ν1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Note that as ξ1 → ∞ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', as the hole effect predominates), the lower bound in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='16) decreases to (τη)−1/(η−1)(1−η)/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the contrary, as ξ1 → 0, such a bound increases to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', the hole effect becomes negligible, which is not surprising, because in such a case the Mat´ern class is predominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A similar conclusion can be obtained in the limit case η → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A parsimonious variant of this model consists of taking ω1 = ω2 = ω (variance parameter), α1 = α2 = α (scale parameter) and ν1 = ν2 = ν (smoothness parameter), whereas ϑ regulates the hole effect (as discussed above) and ξ1 = ξ2 = ξ has a similar interpretation as in the hybrid Cauchy-Mat´ern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation ξ = 3 ξ = 32 ξ = 33 ξ = 34 ξ = 35 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Distance Correlation ξ = 3 ξ = 32 ξ = 33 ξ = 34 ξ = 35 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2: Parsimonious hybrid Hole-Effect-Mat´ern model in dimension one, for ω = 1/2, α = 1/8, τ = 2, η = 7/2 and different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (Left) ν = 1/2 and (Right) ν = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The dashed lines represent the limit cases reported in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' All the models have been appropriately rescaled in order to obtain correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 shows the parsimonious hybrid Hole-Effect-Mat´ern model for different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The limit cases described in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 are also reported, in a similar fashion to Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' It can be seen that negative values coexist with different levels of smoothness at the origin, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 4 Numerical Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 Simulated Data We conduct simulation studies to assess the performance of maximum likelihood inference when a hybrid covariance structure is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We focus on the parsimonious hybrid Cauchy-Mat´ern dependence structure, as it will be applied to real data in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We consider ω = 1, α = 1/8, ν1 = 3/4 and the following scenarios for [ν2, ξ]: (a) [1/2, 40], (b) [1/2, 120], (c) [3/2, 40] and (d) [3/2, 120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' For each scenario, we simulate 200 independent realizations of a Gaussian random field on 100 uniformly sampled points in the square [0, 3]2 and estimate the parameters through maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We then repeat the experiment with 256 spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We only estimate ω , α and ξ, whereas ν1 and ν2 are fixed, which is a common practice in geostatistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Instead of directly estimating ξ, we consider the following alternative parameterization: �ξ = √ξα, which seems to be a natural choice according to Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 displays the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The estimates are approximately unbiased and the variance de- creases as the sample size increases from 100 to 256, which is an expected behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The vari- ability of the estimates substantially decreases in scenarios (c) and (d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=', when the random field is smoother, which is a typical attribute of likelihood-based estimates in this context (Bevilacqua and Gaetan, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the contrary, such a variability deteriorates as ξ increases from 40 to 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 shows the log-likelihood in terms of ξ and α, with fixed ω, for a single realization of the random field, under scenario (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Although the surface has a clear maximum value, the objective function is apparently more flat in the direction of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This could explain the increased variability 10 1 0 1 2 3 n = 100 n = 256 Sample Size Estimates Scenario (a) (b) (c) (d) ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='75 n = 100 n = 256 Sample Size Estimates Scenario (a) (b) (c) (d) α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 n = 100 n = 256 Sample Size Estimates Scenario (a) (b) (c) (d) ξ~ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1: Centered boxplots of the maximum likelihood estimates for the parsimonious hybrid Cauchy-Mat´ern model in scenarios (a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' in scenarios (b) and (d), with respect to (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Despite the previous remarks, in general, the estimates appear to be reasonable in each scenario and no identifiability issues are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We now explore the predictive performance of the proposed class through a cross validation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We simulate 200 independent realizations on 100 uniformly sampled locations in [0, 3]2 according to the scenarios (a)-(d) described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We assess the accuracy through a leave-one-out predic- tion strategy in terms of the mean squared error (MSE), mean absolute error (MAE), log-score (LSCORE) and continuous ranked probability score (CRPS) (see Zhang and Wang, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Small values of these indicators suggest superior predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We evaluate the performance of the hybrid Cauchy-Mat´ern model, using the Generalized Cauchy class as benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, for each realiza- tion, we estimate the parameters with both models and proceed to make the predictions through a simple kriging approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The Generalized Cauchy model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4) has been augmented with a multi- plicative parameter ω, namely h �→ ω(1+hδ/α)−ν/δ, so it is parameterized by ω and α, and ν = 3/4 and δ = 1, 2 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1 shows that, in each scenario, the proposed hybrid model outperforms its competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' All the cross-validation scores substantially decrease in scenarios (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' From this brief study, we observe that when the true underlying covariance has a hybrid structure, an incorrect specification 11 alpha Log Likelihood xi Log Likelihood Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2: Log-likelihood function, with respect to α and ξ, for scenario (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Left and right panels correspond to the same plot from different viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' of the spatial association has a negative impact on the posterior predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Since the behaviour of an isotropic covariance function near the origin has a strong impact on the quality of predictions (Stein, 1999), our simulation experiment suggests that in some circumstances the local shape of the proposed model cannot be replicated by other appealing existing structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 A Real Data Illustration The estimation of recoverable resources is a task of fundamental importance in modern mining processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A sound evaluation of such resources is crucial from an economic viewpoint and is critical for assessing the long-term availability of mineral resources and its impact on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We consider a data set from a lateritic nickel deposit mined by open pit in Colombia, which contains measurements of the grades of nickel, iron, chrome, alumina, magnesia and silica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This study focuses on nickel concentrations that are placed at an elevation of about 120 meters, where 199 irregularly spaced observations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We apply a log-transformation to reduce the skewness, and then the sample mean is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The resulting values are approximately Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The left panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4 shows the transformed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We fit two covariance models: the former is the parsimonious hybrid Cauchy-Mat´ern, parameterized by ω, α and �ξ, with fixed ν1 = 1/4 and ν2 = 1/2, and the latter is the Generalized Cauchy, parameterized as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1, with fixed ν = 1/4 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The values of the fixed parameters have been selected after some experimental trials, taking into account the local behavior of the sample covariance (see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2 reports the likelihood estimates, with the corresponding standard errors, and the Akaike information criterion (AIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We observe that the hybrid Cauchy-Mat´ern model outperforms its competitor in terms of AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 shows that the fitted covariance models seem to be reason- ably close to the sample covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The fitted models differ substantially near the origin (distances less than 3 meters), since the hybrid model decays faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' On the contrary, for larger distances the hybrid model decays slower, although the difference between the curves becomes slight for distances greater than 15 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 12 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1: Cross-validation scores for the parsimonious hybrid Cauchy-Mat´ern and Generalized Cauchy (with δ = 1, 2) models in scenarios (a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Scenario Model MSE MAE LSCORE CRPS (a) Hybrid Cauchy-Mat´ern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='668 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='231 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='773 Generalized Cauchy (δ = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='672 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='238 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='787 Generalized Cauchy (δ = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='674 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='241 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='798 (b) Hybrid Cauchy-Mat´ern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='549 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='034 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='462 Generalized Cauchy (δ = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='555 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='046 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='480 Generalized Cauchy (δ = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='559 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='055 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='506 (c) Hybrid Cauchy-Mat´ern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='840 Generalized Cauchy (δ = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='851 Generalized Cauchy (δ = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='862 (d) Hybrid Cauchy-Mat´ern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='561 Generalized Cauchy (δ = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='575 Generalized Cauchy (δ = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='612 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2: Parameter estimates and Akaike Information Criterion (AIC) of fitted covariance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Standard errors are reported in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Model ω α �ξ AIC Hybrid Cauchy-Mat´ern 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='055 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='063 −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='2516) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='801) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='042) Generalized Cauchy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='164 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='959 − −31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='040) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='849) − 13 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='16 Distance (m) Covariance of log Ni 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='16 Distance (m) Covariance of log Ni 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='16 Distance (m) Covariance of log Ni Hybrid Cauchy-Matern Generalized Cauchy Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3: Sample (circles) and modeled (solid lines) covariances of log-nickel concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3: Scores for the leave-one-out cross-validation study of log-nickel concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Model MSE MAE LSCORE CRPS Hybrid Cauchy-Mat´ern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1431 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4113 Generalized Cauchy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1462 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4159 In order to compare the models in terms of predictive performance, we conduct a cross-validation study, in a similar fashion to the experiments performed with simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3 shows evidence, based on a leave-one-out cross-validation scheme, that the hybrid model has a better performance for this specific data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' In percentage terms, the MSE shows an improvement of approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The largest difference occurs when we compare the LSCORE’s (about 9% improvement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We conclude this section with an illustration of a downscaled map of log-nickel concentrations (see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4), using the hybrid Cauchy-Mat´ern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The interpolated spatial map, which is obtained through simple kriging, is exhibited on a spatial grid of approximately 1 meter (7500 locations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' This kriged surface could be useful in small-scale mining processes, as it is a crucial step for industrial exploration and to quantify mineral reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 5 Conclusions and Perspectives We introduced a simple formalism to build sophisticated parametric families of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We focused on a combination between the Mat´ern and Cauchy models, where local (mean square differentiability) and global (long memory) properties coexist in a single family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We have also illustrated the use of our methodology by constructing a model that behaves as the Mat´ern class at short distances and attains negative values at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Simulation studies show that a parsimonious hybrid Cauchy-Mat´ern model has statistically identifiable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Also, this model provides improvements in terms of predictive performance in comparison to existing models, 14 Easting (m) Northing (m) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Easting (m) Northing (m) 1650 1700 1750 1670 1680 1690 1700 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='0 Easting (m) Northing (m) 1650 1700 1750 1670 1680 1690 1700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='4: Log-nickel concentrations (left), with the kriged surface (middle) and the corresponding variance (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' when a hybrid inherent dependence structure is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We reach similar conclusions when we apply this methodology to a mining dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' While similar numerical studies could be performed for the hybrid Hole-Effect-Mat´ern model, we avoid them for the sake of simplicity and brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Additional interesting extensions of this work can be tackled in future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We now provide two concrete research lines that could emerge from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Multivariate Hybrid Covariance Models In many practical situations, two or more variables are simultaneously recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, our findings can be generalized to the case of multivariate fields {Z(s) = (Z1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' , Zp(s))⊤, s ∈ Rd}, having an isotropic matrix-valued covariance function Φ : [0, ∞) → Rp×p, that is, cov[Zi(s), Zj(s′)] = Φij(h), h ≥ 0, where h = ∥s − s′∥ and i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' We propose the hybrid model �Φ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ, ω, ξ) = ω1 � ξ1 0 exp(−uh2)G1(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ1)du + ω2 � ∞ ξ2 exp(−uh2)G2(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ2)du, that generalizes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='1), where the vectors of parameters λi must be chosen in such a way that the p×p matrices Gi(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λi) are positive semi-definite for every fixed u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hence, a straight application of Proposition 4 in Porcu and Zastavnyi (2011) would ensure �Φ to be positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A multivariate version of the hybrid Cauchy-Mat´ern covariance function is a natural candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The works of Gneiting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (2010) and Moreva and Schlather (2022) are relevant to tackle this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A multivariate version of the formulation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3) could be deduced similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Hybrid Covariance Models on Spheres For random fields that are indexed by the d-dimensional unit sphere, Sd, which is a useful framework when analyzing global data (S2 is used as an approximation of the Earth), the isotropy assumption is given by cov[Z(s), Z(s′)] = ψ(θ), s, s′ ∈ Sd, where ψ : [0, π] → R is a continuous mapping and θ = arccos(s⊤s′) ∈ [0, π] is the geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Schoenberg’s characterization (Schoenberg, 1942) establishes that a parametric isotropic covariance function ψ(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) is valid in any dimension d, if 15 and only if, it can be written as ψ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' λ) = �∞ ℓ=0 βℓ(λ)(cos θ)ℓ, θ ∈ [0, π], for some nonnegative and summable parametric sequence {βℓ(λ)}∞ ℓ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Thus, the hybrid models can be adapted to the spherical context by considering a modified sequence of the form �βℓ(λ, ω, ξ) = ω1 β(1) ℓ (λ1)1[0,⌊ξ1⌋)(ℓ) + ω2 β(2) ℓ (λ2)1[⌊ξ2⌋,∞)(ℓ), ℓ = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' , where ⌊ξi⌋ ≥ 0, for i = 1, 2, with ⌊·⌋ standing for the floor function, and β(i) ℓ being a nonnegative and summable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' The local properties of spherically indexed random fields, and their connections with the covariance function, have been studied in past literature (Bingham, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Guinness and Fuentes, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' However, global properties such as long memory are less intuitive in this scenario as the spatial domain is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Covariance functions with hole effect, for low-dimensional spheres, could be obtained by adapting formulation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Acknowledgements Alfredo Alegr´ıa was partially supported by the National Agency for Research and Development of Chile, through grant ANID/FONDECYT/INICIACI´ON/No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' 11190686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Fabi´an Ramirez was par- tially supported by the Direcci´on de Postgrados y Programas (DPP) of the Universidad T´ecnica Federico Santa Mar´ıa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' Emilio Porcu is supported by the Khalifa University of Science and Tech- nology under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' FSU-2021-016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' References Abramowitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' and Stegun, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} +page_content=' (1972).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE5T4oBgHgl3EQfcQ-J/content/2301.05602v1.pdf'} diff --git a/XdFRT4oBgHgl3EQf-jgZ/content/tmp_files/2301.13691v1.pdf.txt b/XdFRT4oBgHgl3EQf-jgZ/content/tmp_files/2301.13691v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4af57652b1a16258be24ec4712b14f294b3c93f2 --- /dev/null +++ b/XdFRT4oBgHgl3EQf-jgZ/content/tmp_files/2301.13691v1.pdf.txt @@ -0,0 +1,3769 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +1 +Time Series Forecasting via Semi-Asymmetric +Convolutional Architecture with Global Atrous +Sliding Window +Yuanpeng He +Abstract—The proposed method in this paper is designed to address the problem of time series forecasting. Although some +exquisitely designed models achieve excellent prediction performances, how to extract more useful information and make accurate +predictions is still an open issue. Most of modern models only focus on a short range of information, which are fatal for problems such +as time series forecasting which needs to capture long-term information characteristics. As a result, the main concern of this work is to +further mine relationship between local and global information contained in time series to produce more precise predictions. In this +paper, to satisfactorily realize the purpose, we make three main contributions that are experimentally verified to have performance +advantages. Firstly, original time series is transformed into difference sequence which serves as input to the proposed model. And +secondly, we introduce the global atrous sliding window into the forecasting model which references the concept of fuzzy time series to +associate relevant global information with temporal data within a time period and utilizes central-bidirectional atrous algorithm to +capture underlying-related features to ensure validity and consistency of captured data. Thirdly, a variation of widely-used asymmetric +convolution which is called semi-asymmetric convolution is devised to more flexibly extract relationships in adjacent elements and +corresponding associated global features with adjustable ranges of convolution on vertical and horizontal directions. The proposed +model in this paper achieves state-of-the-art on most of time series datasets provided compared with competitive modern models. +Index Terms—Local and global information, Difference sequence, Global atrous sliding window, Semi-asymmetric convolution +! +1 +INTRODUCTION +T +IME series is a sequence taken at successive equally +spaced points in time which is also known as dynamic +series. Precise prediction of time series has close connections +to human society, for instance, it may help people format +schedules and company make adjustments on investment +strategy. Moreover, foreseeing future behaviour based on +analysis of known historical data is of great importance in +lots of fields such as epidemic [1], medical treatment [2], +finance [3, 4] and industrial Internet [5]. Time series fore- +casting therefore attracts attention from researchers around +the world. Nevertheless, how to fully utilize observation +to generate accurate and reasonable predictions is still an +unsolved problem. +To realize accurate prediction of future, researchers de- +velop various kinds of solutions. RNN model has been +favored by researchers since it was proposed. Because of its +recurrent architecture design, RNN models can effectively +model long-term dependencies [11], therefore achieve an +effective understanding of temporal data [12, 13] as well. +However, RNN may encounter memory overflow due to +continuous storage of previous states and gradient vanish- +ing problems. To further make up for shortcomings of RNN, +an improved solution based on it is proposed which is called +LSTM [14]. Its core concepts are the memory cell states that +allow information to be passed on backwards, and the gate +• +Yuanpeng He is with Key Laboratory of High Confidence Software Tech- +nologies, Peking University, Peking, 100871, China; School of Computer +Science, Peking University, Peking, 100871, China. +E-mail: heyuanpengpku@gmail.com +Manuscript received; revised +structures that allow certain information to be added and +removed. Coincidentally, researchers also find that temporal +task can also benefit form LSTM’s characteristics [15, 16]. +In a quite long period of time, the model based on RNN +has played an important role in development of time series +forecasting. Recently, transformer-based models [6–9] have +been proposed enormously, which applied self-attention +mechanism to distill useful semantic information in time +series. However, there exists a doubt that transformer-like +structure is not suitable for the task of time series fore- +casting. Under certain circumstances, the performance of +the models can not even match ingeniously designed linear +model [10], which has shaken the position of transformer- +based models in time series forecasting. At present, the +controversy still continues. Besides, there also exist lots of +meaningful works trying to satisfy demand of time series +forecasting from other multiple aspects as well [17]. +Moreover, CNN-based models are also widely utilized +for prediction of temporal data. They are mainly divided +into two categories, one is the variation of causal and +dilated convolution [18], the other is algorithms using graph +convolutional neural network [19] to solve corresponding +problems. Generally, it can be concluded that transformer +and CNN models, the two well-established solutions in +the field of computer vision, also achieve excellent perfor- +mance in tasks of time series forecasting. Back to CNN- +based models, there have been many new CNN models in +recent years, for instance, temporal convolutional network +(TCN) [20], convolutionally low-rank model [21] and non- +pooling CNN [22]. Among them, TCN attracts the most +attention which is capable of large-scale parallel processing +arXiv:2301.13691v1 [cs.AI] 31 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +and managing a series of sequences of arbitrary length and +uniformly output sequences with the same length. Specif- +ically, the casual and dilated convolution introduced by it +enable CNN forecasting model to possess a larger receptive +field to better acquire information in a longer range under +strict time restrictions. Moreover, other effective models +[23–25] also improve performance by enlarging ranges of +data selection and more ingenious and flexible extraction of +relationships of adjacent and non-adjacent elements because +of similar considerations about demand of time series fore- +casting mentioned above. Nevertheless, it is noting that all +of the models still receive data in a relatively restricted way +without considering global data features. +To address the issue, we design a kind of data recon- +struction method referencing solution based on partitioned +universe of discourse [26] partly which transforms original +temporal data into difference sequence [27–29] ensuring +that the model is more likely dealing with steady-state +sequences and associates relative positional information of +data captured in the view of whole observation time se- +ries to reduce difficulty of model learning to some extent. +More than that, we choose to replace all of elements by +the last one only keeping their positional information as +subsidiaries to maximize timeliness of data without losing +too much semantic information of temporal data. Besides, +the relationship among converted information in different +subsections and time series are probably separate [30], so +there is a need to devise a convolution strategy with dif- +ferent directions and shapes to further mine underlying +information. Due to particularity of time series, traditional +squared convolution is not capable to manage complex +extraction of relationship of elements in temporal data, a +variation of asymmetric convolution [31, 32] which is called +semi-asymmetric convolution is designed accordingly. The +semi-asymmetric convolution is divided into horizontal and +vertical filters, and they probably possess different length +to retrieve interaction information at a more fine-grained +level in selected fragments from difference series. The ad- +vantage of this improvement is that it is able to effectively +obtain temporal features using adjustable scales [33, 34], +and speeding up the training and inference process of the +proposed model [35] at the same time. In general, the major +contributions of this work are summarized as follows: +1) +The input to proposed model is difference sequences +transformed by original observation time series to +enable model to learn more easily +2) +A new kind of method of data reconstruction is +designed to endow each elements with their corre- +sponding relative positional information +3) +A novel convolutional architecture called semi- +asymmetric convolution with flexible scales is de- +signed to acquire information at different levels. +The rest of this paper is organized as follows. In the sec- +ond section, some related concepts of the proposed model +are introduced. And the details of the proposed model are +presented in the third section. Besides, the fifth section pro- +vides experimental results and corresponding discussions +with respect to models. In the last section, conclusions and +outlook of future work are given. +2 +PRELIMINARY +In this section, related concepts about the proposed model +are briefly introduced. +2.1 +Difference of First Order +A first order difference is the difference between two con- +secutive adjacent terms in a discrete function. Assume there +exists a function y = f(x), y is defined only on the non- +negative integer value of x and when the independent +variable x is iterated through the non-negative integers in +turn, namely x = 0, 1, 2, ..., the corresponding values of +function can be defined as: +f(0), f(1), f(2), ... +(1) +it can be abbreviated as: +y0, y1, y2, ... +(2) +when the independent value changes from x to x + 1, the +variation of y = f(x) can be defined as: +∆yx = f(x + 1) − f(x), (x = 1, 2, 3, ...) +(3) +it’s called the first difference of the function y(x) at point x +which is usually denoted as: +∆yx = yx+1 − yx, (x = 1, 2, 3, ...) +(4) +2.2 +Asymmetric Convolution Architecture +CNN has embraced a quick development recently, it is +widely applied in different fields, such as time series and +computer vision [36–38] due to its stable and excellent +performance. For an operation of convolving, assume an +input ς ∈ RH×W and filter C, the process of generating +output λ ∈ RH′×W ′ can be defined as: +λ = C ∗ ς, +ς ∈ RH×W , λ ∈ RH′×W ′, C ∈ Rd×d +(5) +where ∗ is the 2D convolution operator. Moreover, asym- +metric convolution [35, 39] is considered as an economical +choice to approximate an existing square-kernel convolu- +tional layer for obtaining acceleration and compression. +Specifically, the original filter can be decomposed into hori- +zontal and vertical filters, Ch, Cv, respectively, which can be +defined as: +C ∗ ς = Cv ∗ (Ch ∗ ς), Cv ∈ Rd×1, Ch ∈ R1×d +(6) +compared with the original convolution utilizing d×d kernel +size, the time complexity changes from O((d2H′W ′) to +O(2dH′W ′). Due to efficiency of the asymmetric architec- +ture, it is widely applied in convolutional neural network +design [40, 41] and gains performance improvement gener- +ally. +2.3 +Atrous Algorithm +The atrous algorithm is proposed in [42, 43] which is also +known as dilated convolution. Assume there exists a one- +dimensional input α[s], the corresponding output β[s] of +dilated convolution via a filter ω[e] with length E can be +defined as: +β[s] = +E +� +e=1 +α[s + r · e]ω[e] +(7) + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +where rate parameter r is corresponding to the stride and +standard convolution is a special case for r = 1. Gener- +ally, atrous algorithm is designed to avoid precision loss +brought by reduction of feature map on account of multiple +convolutional and pooling layers on vision tasks and is +broadly utilized in many other import fields, such as audio +processing [44] and time series forecasting [20]. +2.4 +Naive Forecasting +The naive forecasting is the simplest prediction method +in the field of time series which regards the most recent +observation value as the prediction of future. Assume there +exists a time series T with a length n which can be defined +as: +T = {(t1, ζ1), (t2, ζ2), ..., (tn−1, ζn−1), (tn, ζn)} +(8) +where ζi, i ∈ [1, n] represents observation value at time +point i. For example, if there is a need to predict ζn+q +which is unknown, the value of ζn can be referenced as +the prediction value of ζn+q directly. Assume the prediction +value of ζn+q is ˆζn+q, then the method of forecasting can be +defined as: +ˆζn+q = ζn +(9) +under various circumstances, naive forecasting is an effec- +tive solution to tell the future trend of time series like stock +price prediction. Nevertheless, the method introduced is not +a satisfying solution in time series forecasting, but it can +provide a benchmark for other prediction methods. +2.5 +Fuzzy Time Series +One of concepts of fuzzy time series is introduced by Chen +[26] which is developed based on theories proposed in [45– +47]. The method of fuzzy time series could extract informa- +tion effectively by utilizing overall characteristics of time +series data and provide stable performance. Specifically, +given a time series T, and let ηmin and ηmax be minimum +and maximum value in T, the four steps to generate fuzzy +time series can be presented as: +Step 1: Select two proper positive numbers η1 and η2, a +universe of discourse U can be defined as [ηmin −η1, ηmax − +η2] +Step 2: Partition U into segments with equal length +{u1, u2, ...um} which is called fuzzy intervals +Step 3: Let Z1, Z2, ..., Zk be fuzzy sets and they are +defined on universe of discourse U as: +� +� +� +� +� +� +� +� +� +Z1 = z11/u1 + z12/u2 + ... + z1m/um, +Z2 = z21/u1 + z22/u2 + ... + z2m/um, +... +Zk = zk1/u1 + zk2/u2 + ... + zkm/um +(10) +where zij ∈ [0, 1], i ∈ [1, k], j ∈ [1, m] and the value of zij +represents the degree of membership of uj in fuzzy set Zi. +Step 4: The derived fuzzy logical relationships which +possess identical initial states are divided into the same +group. Then, the matches between actual values in time +series and groups of fuzzy logical relationships can be +acquired. +After these four steps, the original data is transformed +into fuzzy time series. +3 +ARCHITECTURE OF THE PROPOSED FORECAST- +ING MODEL +In this section, the proposed forecasting model based on +relevant concepts mentioned above is introduced. +3.1 +Difference Layer: Convert Time Series into First +Order Difference Sequence +First, a time series T is transformed into its first order +difference sequence T∆. The process can be given as: +� +� +� +� +� +� +� +� +� +� +� +T = {(t1, ζ1), (t2, ζ2), ..., (tn−1, ζn−1), (tn, ζn)} +⇓ +T− +∆ = {(t2,1, ζ2 − ζ1), ..., (tn,n−1, ζn − ζn−1)} +⇓ +T∆ = {α1, α2, ..., αh} +(11) +where h = n − 1 and αi, i ∈ [1, h] only contains observed +value without timestamp. Obviously, it can be obtained that +the length of T∆ is n − 1. In the next step, the input is T∆ +instead of T. +3.2 +Division layer: Divide Converted Time Series into +Sub-Series Based on Sliding Window +Second, series T∆ is divided by sliding window whose size +is W into sub-series, the segmented data fragments are: +TSeg +∆ += {Υ1, Υ2, ..., Υc}, c = h − W + 1 +(12) +where Υj = {αj, αj+1, ..., αj+W−1}, j ∈ [1, c]. +3.3 +Encoder of segmented sequences +3.3.1 +Relative +Positional +Encoding: +Reconstruct +Sub- +Series with View on Global Observation Temporal Data +Third, in the concept of fuzzy time series proposed in [26], +the two numbers η1 and η2 are selected intuitively, which +may lead to non-reproducibility of experiment results on +various datasets. As a result, η1 and η2 are uniformly set as +standard deviation of corresponding first order difference +sequence, which can be given as: +ϕ = η1 = η2 = σ(T∆) +(13) +where σ represents standard deviation. Then, universe of +discourse U of T∆ can be calculated as: +UT∆ = [αmin − ϕ, αmax + ϕ] = [βl, βu] +(14) +where αmin and αmax represent minimum and maximum +element contained in T∆ and βl and βu denote lower and +upper bound of UT∆. And the number of intervals, N, can +be confirmed as: +N = logh +2 − 1 +(15) +the partitioned universe of discourse can be given as: +UT∆ = [βl, βl + ξ, ..., βl + κ × ξ, ..., βu − ξ, βu] +(16) +where ξ = (βu − βl)/N and κ ∈ [1, N]. Then, integrate +each element contained in Υj into the partitioned universe + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽# +𝛽# +𝛼 +― +𝛽# +𝛼 +― +𝛼 +― +𝛼 +― +𝛼 +― +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽" +𝛽# +𝛽# +𝛽# +𝛽# +𝛽" +𝛽" +𝛽" +𝛽" +𝛼 +― +𝛽# +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛽" +𝛼 +― +𝛽# +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛼 +― +𝛽# +𝛽# +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛽" +𝛽# +𝛽# +𝛽# +𝛽# +𝛽# +𝛽# +𝛽# +𝛽# +𝛽# +𝛽# +𝛼 +― +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛼 +― +𝛽# +𝛼 +― +𝛽" +𝛼 +― +𝛽# +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛼 +― +𝛽# +𝛼 +― +𝛽" +𝛽" +𝛼 +― +𝛽# +𝛼 +― +𝛽" +𝛼 +― +𝛽# +𝛽# +𝛼 +― +𝛽" +𝛽" +Central-Bidirectional Atrous Algorithm +Padding Mechanism and Cropping +Fig. 1. Padding Mechanism, Cropping and Central-Bidirectional Atrous Algorithm with Dilation Factor d = 1 +of discourse UT∆ to create new sequences based on data +fragments captured by sliding window: +Υ +UT∆ +j += +� +������� +βl +... +βl + κ′ × ξ +αj +βl + (κ′ + 1) × ξ +... +βu +βl +... +βl + κ′′ × ξ +αj+1 +βl − (κ′′ + 1) × ξ +... +βu +... +... +... +... +... +... +... +βl +... +βl + κ′′′ × ξ +αj+W−1 +βl − (κ′′′ + 1) × ξ +... +βu +� +������� +(17) +the position of each integrated element is uniquely identi- +fied. Then, all the data from sliding window in the recon- +structed series is replaced with the last element in original +subset divided only keeping position information of former +elements: +ℓj = +� +���� +βl +... +βl + κ′ × ξ +αj+W−1 +βl + (κ′ + 1) × ξ +... +βu +βl +... +βl + κ′′ × ξ +αj+W−1 +βl + (κ′′ + 1) × ξ +... +βu +... +... +... +... +... +... +... +βl +... +βl + κ′′′ × ξ +αj+W−1 +βl + (κ′′′ + 1) × ξ +... +βu +� +���� +(18) +the simplified form of it can be given as: +ℓj = +� +���� +βl +... +x1 +α +˚x1 +... +βu +βl +... +x2 +α +˚x2 +... +βu +... +... +... +... +... +... +... +βl +... +xp +α +˚xp +... +βu +� +���� +(19) +where p ∈ [1, W] and the final input to the proposed +network is: +TInput = {ℓ1, ℓ2, ..., ℓc} +(20) +3.3.2 +Padding Mechanism and Cropping +Forth, one side of each row of input data is filled separately +so that the length of data on both sides of the last element +in original subset is the same. Assume data of row p in ℓℏ is +vector ⃗Λp, the process of padding can be given as: +⃗˘ +Λp = +� +Concat(rep(βl)D, ⃗Λ +βl⇒xp +p +), ||( ⃗Λ +βl⇒xp +p +)|| < ||( ⃗Λ +˚ +xp⇒βu +p +)|| +Concat( ⃗Λ +˚ +xp⇒βu +p +, rep(βu)D′), ||( ⃗Λ +βl⇒xp +p +)|| > ||( ⃗Λ +˚ +xp⇒βu +p +)|| +(21) +where ⃗Λ ˙o⇒¨o +p +represents a segmented vector which ranges +from element ˙o to ¨o, || ⃗Λ ˙o⇒¨o +p +|| is the length of ⃗Λ ˙o⇒¨o +p +and +Concat denotes the operation of concatenation of two vec- +tors. Besides, rep( ˙o)D means creating a vector containing D +copies of element ˙o and D = ||( ⃗Λ˚x⇒βu +p +)|| − ||( ⃗Λβl⇒x +p +)|| or +D′ = ||( ⃗Λβl⇒x +p +)|| − ||( ⃗Λ˚x⇒βu +p +)||. +Then, each row containing in ℓℏ is padded ensuring +lengths of two sides of the last element in original subset are +equal. However, the operation of padding brings a problem +that length of each row is not exactly the same which is +difficult for neural network to acquire information and cap- +ture features. As a result, there is a need to crop redundant +elements in each padded data row. Assume length of the +shortest padded vector is S and the operation of cropping +M elements which lie from both ends of the vector ˘Λp to its +centre is CropM, the process of cropping is defined as: +⃗Λp = CropM( ⃗˘Λp) +(22) +where M = (|| ⃗˘Λp|| − S)/2 and ⃗Λp is the cropped vector. +3.3.3 +Central-Bidirectional Atrous Algorithm +Fifth, the processed information needs to be further ex- +tracted so that subsequent networks can capture more useful +information and avoid unnecessary calculations. Because +of the unique nature of the reconstructed timing data, the +atrous algorithm is modified to obtain data from the centre +to both sides of each segment, which reserves the nearest +observation value and corresponding position distribution +information from the prediction object. Assume the leftmost +and rightmost element in ⃗Λp are ϵlp and ϵrp, the input which +is divided into two parts by the central element to central- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +bidirectional atrous algorithm (CBAA) is: +� +� +� +ℓl +j = [ ⃗Λ +˚x1⇒ϵr1 +1˚ +x1+g×d, ⃗Λ +˚x2⇒ϵr2 +2˚ +x2+g×d, ..., ⃗Λ +˚xp−1⇒ϵrp−1 +p−1˚ +xp−1+g×d, ⃗Λ +˚xp⇒ϵrp +p˚ +xp+g×d] +ℓr +j = [ +⃗ +Λ +ϵl1⇒x1 +1x1−g×d, +⃗ +Λ +ϵl2⇒x2 +2x2−g×d, ..., +⃗ +Λ +ϵlp−1⇒xp−1 +p−1xp−1−g×d, +⃗ +Λ +ϵlp⇒xp +pxp−g×d] +(23) +where ℓl +j and ℓr +j denote left and right part of cropped +vector ⃗Λp and the directions of filters on them are opposite. +Moreover, assume a filter f : {0, ..., v −1} and the operation +of CBAA, F, starting with elements ˚xp and xp is defined as: +F(α)j = Concat( +v−1 +� +g=0 +f(g) · ℓl +j, α, +v−1 +� +g=0 +f(g) · ℓr +j) +(24) +where · represents the operation of dilated convolution, d is +the factor of dilation, v means the filter size, ˚xp + g × d and +xp − g × d account for the direction of movement of filters. +When d = 1, the form of atrous algorithm degenerates +into regular convolution. A larger dilation factor enables the +algorithm to capture features at a longer range. In original +atrous algorithm, the operation of dilation is utilized to +enlarge the receptive field without reduce sizes of feature +maps. But in the CBAA, the dilated convolution is mainly +used to construct efficient maps with proper sizes contain- +ing underlying features of historic information via multiple +non-adjacent fuzzy intervals. +3.4 +Semi-Asymmetric Convolutional Architecture +Sixth, a semi-asymmetric convolutional neural network +(SACNN) is designed to aggregate information and produce +differential predictions. SACNN is made up of a stack of one +module which is called SAC block. The SAC block consists +of two parts, the first part is the batchnorm layer Bn which +is defined as: +˜Cj = Bn(F(α)j) = F(α)j − +¯ +F(α)j +� +σ(F(α)j) + ϵ ∗ γ + δ +(25) +where +¯ +F(α)j and σ(F(α)j) denotes mean and standard- +deviation of F(α)j, γ and δ are learnable parameter vectors +whose size is the number of channel of input. The output ˜Aj +is supposed to be sent into the next part, semi-asymmetric +convolutional layer Sa which consists of L combinations of +X horizontal and vertical filters ˇfV ∈ RV ×1 and ˇfH ∈ R1×H +. Assume the input ˜Cj ∈ RH′×V ′×Y with H′ × V ′ feature +map and Y channels, the process of generating output can +be defined as: +Bj = Sa( ˜Cj) = [ ˇfV ⋄( ˇfH ⋄ ˜Cj)]×L, Bj ∈ RH′′×V ′′×X (26) +where ⋄ +represents +semi-asymmetric +convolution +and +OUT = X/Y is the lifting factor of number of input’s to +output’s channels. When V = H, Sa degenerates into the +form of regular asymmetric convolution. Before outputting +the final values, the information is expected to be sent into +two linear layers: +Qj = SAC(F(α)j) = (BjAT + b)A′T + b′ +(27) +where A and A′ are the learnable weights of the module +of shape which is transposed to times the original input Bj +and b and b′ are the biases to be added. Then, Qj is the +prediction which the proposed model produce on the first +order difference sequence T∆. +BatchNorm2D +Horizontal Filter +Vertical Filter +Output +Linear1 +Linear2 +Encoder +Input +Restore +… +BatchNorm2D +Horizontal Filter +Vertical Filter +Linear1 +Linear2 +Encoder +BatchNorm2D +Horizontal Filter +Vertical Filter +Linear1 +Linear2 +Encoder +… +Difference Layer +Division Layer +Fig. 2. Details of the Proposed Model +3.5 +Restore Output of Network to Original Position and +Make Prediction +Seventh, restore the differential prediction Qj to the original +time series T. The process of a sliding window generating +corresponding prediction is given as: +ˆζj+W+1 = ζj+W + Qj +(28) +on the training data, the proposed model is expected to ap- +proximate trend of changes of time series. For the prediction +of value beyond the known time series data, the prediction +is made as: +ˆζn+1 = ζn + Qn−W +(29) +the process of producing prediction of the proposed model +is illustrated in Fig.2. +4 +EXPERIMENTS +In this section, multiple experiments are conducted to eval- +uate the effectiveness and validity of the proposed method. +4.1 +Datasets Description +In order to fully illustrate the performance of the proposed +model, the comparison experiments are conducted on 43 +datasets which are provided by monash time series forecast- +ing archive (MTSFA) [48]. Specifically, among them, there +are 27 univariate and 16 multivariate datasets and they +cover multiple domains, such as Tourism, Banking, En- +ergy, Sales, Economic, Transport, Nature, Web and Health. +Moreover, the datasets have different sampling rates such +as yearly, quarterly and monthly, which also correspond +disparate expected forecast horizons. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +TABLE 1 +MEAN MAE RESULTS OF UNIVARIATE DATASETS +Dataset +Naive +SES +Theta +TBATS +ETS +ARIMA +PR +CatBoost +FFNN +DeepAR +M1 Yearly +221512.32 +171353.41 +152799.26 +103006.90 +146110.11 +145608.87 +134246.38 +215904.20 +136238.80 +152084.40 +M1 Quarterly +3350.81 +2206.27 +1981.96 +2326.46 +2088.15 +2191.10 +1630.38 +1802.18 +1617.39 +1951.14 +M1 Monthly +2866.26 +2259.04 +2166.18 +2237.50 +1905.28 +2080.13 +2088.25 +2052.32 +2162.58 +1860.81 +M3 Yearly +1563.64 +1022.27 +957.40 +1192.85 +1031.40 +1416.31 +1018.48 +1163.36 +1082.03 +994.72 +M3 Quarterly +711.65 +571.96 +486.31 +561.77 +513.06 +559.40 +519.30 +593.29 +528.47 +519.35 +M3 Monthly +1002.94 +743.41 +623.71 +630.59 +626.46 +654.80 +692.97 +732.00 +692.48 +728.81 +M3 Other +452.11 +277.83 +215.35 +189.42 +194.98 +193.02 +234.43 +318.13 +240.17 +247.56 +M4 Yearly +1487.58 +1009.06 +890.51 +960.45 +920.66 +1067.16 +875.76 +929.06 +- +- +M4 Quarterly +838.19 +622.57 +574.34 +570.26 +573.19 +604.51 +610.51 +609.55 +631.01 +597.16 +M4 Monthly +835.69 +625.24 +563.58 +589.52 +582.60 +575.36 +596.19 +611.69 +612.52 +615.22 +M4 Weekly +480.94 +336.82 +333.32 +296.15 +335.66 +321.61 +293.21 +364.65 +338.37 +351.78 +M4 Daily +255.42 +178.27 +178.86 +176.60 +193.26 +179.67 +181.92 +231.36 +177.91 +299.79 +M4 Hourly +399.84 +1218.06 +1220.97 +386.27 +3358.10 +1310.85 +257.39 +285.35 +385.49 +886.02 +Tourism Yearly +117966.55 +95579.23 +90653.60 +94121.08 +94818.89 +95033.24 +82682.97 +79567.22 +79593.22 +71471.29 +Tourism Quarterly +13988.39 +15014.19 +7656.49 +9972.42 +8925.52 +10475.47 +9092.58 +10267.97 +8981.04 +9511.37 +Tourism Monthly +3019.44 +5302.10 +2069.96 +2940.08 +2004.51 +2536.77 +2187.28 +2537.04 +2022.21 +1871.69 +CIF 2016 +650535.53 +581875.97 +714818.58 +855578.40 +642421.42 +469059.49 +563205.57 +603551.30 1495923.44 3200418.00 +Aus. Electricity Demand +241.77 +659.60 +665.04 +370.74 +1282.99 +1045.92 +247.18 +241.77 +258.76 +302.41 +Dominick +5.86 +5.70 +5.86 +7.08 +5.81 +7.10 +8.19 +8.09 +5.85 +5.23 +Bitcoin +6.57×1017 5.33×1018 +5.33×1018 +9.9×1017 +1.1×1018 +3.62×1018 +6.66×1017 +1.93×1018 1.45×1018 1.95×1018 +Pedestrian Counts +65.59 +170.87 +170.94 +222.38 +216.50 +635.16 +44.18 +43.41 +46.41 +44.78 +Vehicle Trips +13.37 +29.98 +30.76 +21.21 +30.95 +30.07 +27.24 +22.61 +22.93 +22.00 +KDD Cup +72.17 +42.04 +42.06 +39.20 +44.88 +52.20 +36.85 +34.82 +37.16 +48.98 +Weather +2.79 +2.24 +2.51 +2.30 +2.35 +2.45 +8.17 +2.51 +2.09 +2.02 +Sunspot +0.14 +4.93 +4.93 +2.57 +4.93 +2.57 +3.83 +2.27 +7.97 +0.77 +Saugeen River Flow +12.49 +21.50 +21.49 +22.26 +30.69 +22.38 +25.24 +21.28 +22.98 +23.51 +US Births +1497.36 +1192.20 +586.93 +399.00 +419.73 +526.33 +574.93 +441.70 +557.87 +424.93 +Dataset +N-BEATS +WaveNet Transformer +MSS∗ +FEDformer∗ +NetAtt∗ +Pyraformer∗ +PFSD∗ +Informer +Ours +M1 Yearly +173300.20 +284953.90 +164637.90 +59228.64 +124729.30 +66409.64 +127110.48 +51417.35 +- +66062.77 +M1 Quarterly +1820.25 +1855.89 +1864.08 +1686.22 +1683.57 +1727.60 +1721.32 +1231.13 +- +1234.77 +M1 Monthly +1820.37 +2184.42 +2723.88 +2063.19 +2394.66 +1720.12 +2421.01 +1952.81 +- +1620.47 +M3 Yearly +962.33 +987.28 +924.47 +933.80 +873.74 +906.63 +891.88 +858.70 +- +530.78 +M3 Quarterly +494.85 +523.04 +719.62 +538.85 +623.58 +591.25 +711.46 +473.84 +- +307.63 +M3 Monthly +648.60 +699.30 +798.38 +1127.37 +728.60 +1014.96 +693.24 +912.28 +- +547.31 +M3 Other +221.85 +245.29 +239.24 +229.01 +217.03 +297.44 +196.81 +210.80 +- +92.47 +M4 Yearly +- +- +- +792.87 +730.24 +967.37 +757.92 +528.36 +- +415.63 +M4 Quarterly +580.44 +596.78 +637.60 +560.72 +594.24 +617.30 +608.55 +445.09 +- +382.57 +M4 Monthly +578.48 +655.51 +780.47 +644.51 +688.95 +781.42 +694.29 +608.31 +- +353.34 +M4 Weekly +277.73 +359.46 +378.89 +301.26 +317.16 +322.59 +295.60 +250.68 +- +222.42 +M4 Daily +190.44 +189.47 +201.08 +173.20 +167.05 +207.44 +161.36 +103.28 +- +62.54 +M4 Hourly +425.75 +393.63 +320.54 +1355.21 +246.33 +1841.90 +228.87 +999.83 +- +94.06 +Tourism Yearly +70951.80 +69905.47 +74316.52 +- +- +- +- +- +- +53029.08 +Tourism Quarterly +8640.56 +9137.12 +9521.67 +- +- +- +- +- +- +7799.49 +Tourism Monthly +2003.02 +2095.13 +2146.98 +- +- +- +- +- +- +2227.20 +CIF 2016 +679034.80 5998224.62 4057973.04 +- +- +- +- +- +- +226103.58 +Aus. Electricity Demand +213.83 +227.50 +231.45 +- +- +- +- +- +- +42.48 +Dominick +8.28 +5.10 +5.18 +5.39 +5.10 +6.02 +5.16 +4.80 +- +4.43 +Bitcoin +1.06×1018 2.46×1018 +2.61×1018 +- +- +- +- +- +- +3.77×1017 +Pedestrian Counts +66.84 +46.46 +47.29 +- +- +- +- +- +- +66.66 +Vehicle Trips +28.16 +24.15 +28.01 +- +- +- +- +- +- +12.36 +KDD Cup +49.10 +37.08 +44.46 +- +- +- +- +- +- +5.92 +Weather +2.34 +2.29 +2.03 +- +- +- +- +- +- +1.87 +Sunspot +14.47 +0.17 +0.13 +- +- +- +- +- +19.43 +0.14 +Saugeen River Flow +27.92 +22.17 +28.06 +- +- +- +- +- +28.59 +8.77 +US Births +422.00 +504.40 +452.87 +- +- +- +- +- +609.43 +538.37 +4.2 +Baseline Methods for Comparison +To demonstrate the performance improvement gained by +the proposed model, we compare it with baseline methods, +such as Naive (Forecasting)1, Simple Exponential Smooth- +ing (SES) [49], Theta [50], Trigonometric Box-Cox ARMA +Trend Seasonal Model (TBATS) [51], Exponential Smooth- +ing (ETS) [52], (Dynamic Harmonic Regression-)ARIMA +1. The results produced by naive forecasting dose not participate in +the comparison with experimental results of other models because of its +particularity in forecasting strategy which is provided only for a simple +reference. For example, on Solar 10 Minutes dataset, naive forecasting +achieve surprising results whose error is 0.00, which is unintuitive and +unreasonable. +[53, 54], Pooled Regression Model (PR) [55], CatBoost [56], +Feed-Forward Neural Network (FFNN) [57], DeepAR [58], +N-BEATS [59], WaveNet [60], Transformer [61], MSS∗ [62], +FEDformer∗ [6], NetAtt∗ [63], Pyraformer∗ [7], PFSD∗ [64] +and Informer [8]. The experimental results of these methods +except naive forecasting are acquired from MTSFA and +PFSD. Besides, the results of experiments of Naive (Fore- +casting) are generated by following the experimental rules +given by MTSFA strictly. +4.3 +Evaluation Metrics +Measurement of model performance is an important objec- +tive of the experiments. Mean Absolute Error (MAE) and + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +TABLE 2 +MEAN RMSE RESULTS OF UNIVARIATE DATASETS +Dataset +Naive +SES +Theta +TBATS +ETS +ARIMA +PR +CatBoost +FFNN +DeepAR +M1 Yearly +237288.10 +193829.49 +171458.07 +116850.90 +167739.02 +175343.75 +152038.68 +237644.50 +154309.80 +173075.10 +M1 Quarterly +3798.89 +2545.73 +2282.65 +2673.91 +2408.47 +2538.45 +1909.31 +2161.01 +1871.85 +2313.32 +M1 Monthly +3533.38 +2725.83 +2564.88 +2594.48 +2263.96 +2450.61 +2478.88 +2461.68 +2527.03 +2202.19 +M3 Yearly +1729.92 +1172.85 +1106.05 +1386.33 +1189.21 +1662.17 +1181.81 +1341.70 +1256.21 +1157.88 +M3 Quarterly +804.54 +670.56 +567.70 +653.61 +598.73 +650.76 +605.50 +697.96 +621.73 +606.56 +M3 Monthly +1193.11 +893.88 +753.99 +765.20 +755.26 +790.76 +830.04 +874.20 +833.15 +873.71 +M3 Other +479.26 +309.68 +242.13 +216.95 +224.08 +220.77 +262.31 +349.90 +268.99 +277.74 +M4 Yearly +1612.24 +1154.49 +1020.48 +1099.95 +1052.12 +1230.35 +1000.18 +1065.02 +- +- +M4 Quarterly +955.55 +732.82 +673.15 +672.74 +674.27 +709.99 +711.93 +714.21 +735.84 +700.32 +M4 Monthly +1002.72 +755.45 +683.72 +743.41 +705.70 +702.06 +720.46 +734.79 +743.47 +740.26 +M4 Weekly +553.29 +412.60 +405.17 +356.74 +408.50 +386.30 +350.29 +420.84 +399.10 +422.18 +M4 Daily +293.15 +209.75 +210.37 +208.36 +229.97 +212.64 +213.01 +263.13 +209.44 +343.48 +M4 Hourly +477.27 +1476.81 +1483.70 +469.87 +3830.44 +1563.05 +312.98 +344.62 +467.89 +1095.10 +Tourism Yearly +130104.42 +106665.20 +99914.21 +105799.40 +104700.51 +106082.60 +89645.61 +87489.00 +87931.79 +78470.68 +Tourism Quarterly +17050.68 +17270.57 +9254.63 +12001.48 +10812.34 +12564.77 +11746.85 +12787.97 +12182.57 +11761.96 +Tourism Monthly +3873.31 +7039.35 +2701.96 +3661.51 +2542.96 +3132.40 +2739.43 +3102.76 +2584.10 +2359.87 +CIF 2016 +712332.30 +657112.42 +804654.19 +940099.90 +722397.37 +526395.02 +648890.31 +705273.30 1629741.53 3532475.00 +Aus. Electricity Demand +340.70 +766.27 +771.51 +446.59 +1404.02 +1234.76 +319.98 +300.55 +330.91 +357.00 +Dominick +8.31 +6.48 +6.74 +8.03 +6.59 +7.96 +9.44 +9.15 +6.79 +6.67 +Bitcoin +8.27×1017 5.35×1018 +5.35×1018 +1.16×1018 +1.22×1018 +3.96×1018 +8.29×1018 +2.02×1018 1.57×1018 2.02×1018 +Pedestrian Counts +94.29 +228.14 +228.20 +261.25 +278.26 +820.28 +61.84 +60.78 +67.17 +65.77 +Vehicle Trips +18.13 +36.53 +37.44 +25.69 +37.61 +34.95 +31.69 +27.28 +27.88 +26.46 +KDD Cup +111.97 +73.81 +73.83 +71.21 +76.71 +82.66 +68.20 +65.71 +68.43 +80.19 +Weather +3.80 +2.85 +3.27 +2.89 +2.96 +3.07 +9.08 +3.09 +2.81 +2.74 +Sunspot +0.53 +4.95 +4.95 +2.97 +4.95 +2.96 +3.95 +2.38 +8.43 +1.14 +Saugeen River Flow +22.30 +39.79 +39.79 +42.58 +50.39 +43.23 +47.70 +39.32 +40.64 +45.28 +US Births +1921.21 +1369.50 +735.51 +606.54 +607.20 +705.51 +732.09 +618.38 +726.72 +683.99 +Dataset +N-BEATS +WaveNet Transformer +MSS∗ +FEDformer∗ +NetAtt∗ +Pyraformer∗ +PFSD∗ +Informer +Ours +M1 Yearly +192489.80 +312821.80 +182850.60 +68119.81 +143607.73 +81092.33 +145991.89 +59867.94 +- +80553.54 +M1 Quarterly +2267.27 +2271.68 +2231.50 +1977.00 +1992.56 +2057.60 +2026.49 +1458.75 +- +1519.10 +M1 Monthly +2183.37 +2578.93 +3129.84 +2427.46 +2918.05 +2024.08 +2957.84 +2369.96 +- +2085.94 +M3 Yearly +1117.37 +1147.62 +1084.75 +1079.09 +1019.83 +1061.72 +1054.66 +981.94 +- +655.87 +M3 Quarterly +582.83 +606.75 +819.18 +636.68 +735.21 +693.52 +810.20 +568.22 +- +384.34 +M3 Monthly +796.91 +845.30 +948.40 +1311.49 +877.76 +1193.29 +836.84 +1079.11 +- +697.11 +M3 Other +248.53 +276.97 +271.02 +260.48 +245.08 +335.88 +227.20 +247.66 +- +115.68 +M4 Yearly +- +- +- +898.74 +787.35 +1173.95 +816.41 +606.06 +- +516.66 +M4 Quarterly +684.65 +696.96 +739.06 +662.18 +691.68 +715.13 +712.44 +514.54 +- +476.51 +M4 Monthly +705.21 +787.94 +902.38 +778.20 +831.55 +902.91 +853.13 +720.67 +- +471.81 +M4 Weekly +330.78 +437.26 +456.90 +354.97 +379.04 +388.03 +337.62 +320.38 +- +286.07 +M4 Daily +221.69 +220.45 +233.63 +205.22 +192.67 +249.70 +183.30 +118.88 +- +84.87 +M4 Hourly +501.19 +468.09 +391.22 +1643.46 +304.69 +2124.99 +284.09 +1209.48 +- +136.42 +Tourism Yearly +78241.67 +77581.31 +80089.25 +- +- +- +- +- +- +62680.50 +Tourism Quarterly +11305.95 +11546.58 +11724.14 +- +- +- +- +- +- +10014.33 +Tourism Monthly +2596.21 +2694.22 +2660.06 +- +- +- +- +- +- +2932.16 +CIF 2016 +772924.30 6085242.41 4625974.00 +- +- +- +- +- +- +288763.10 +Aus. Electricity Demand +268.37 +286.48 +295.22 +- +- +- +- +- +- +62.16 +Dominick +9.78 +6.81 +6.63 +7.39 +6.97 +7.02 +6.89 +6.56 +- +6.60 +Bitcoin +1.26×1018 2.55×1018 +2.67×1018 +- +4.39×1017 +Pedestrian Counts +99.33 +67.99 +70.17 +- +- +- +- +- +- +98.13 +Vehicle Trips +33.56 +28.99 +32.98 +- +- +- +- +- +- +23.58 +KDD Cup +80.39 +68.87 +76.21 +- +- +- +- +- +- +10.16 +Weather +3.09 +2.98 +2.81 +- +- +- +- +- +- +2.70 +Sunspot +14.52 +0.66 +0.52 +- +- +- +- +- +20.31 +0.53 +Saugeen River Flow +48.91 +42.99 +49.12 +- +- +- +- +- +44.42 +13.36 +US Births +627.74 +768.81 +686.51 +- +- +- +- +- +734.44 +679.99 +Root Mean Square Error (RMSE) are selected to evaluate +the accuracy of forecasting of chosen comparative models +whose definitions are defined as: +MAE = +�N +i=1 |ˆyi − yi| +N +(30) +RMSE = +��N +i=1 |ˆyi − yi|2 +N +(31) +where ˆyi represents the value of forecasting. +4.4 +Implementation Details +The proposed model is realized using the code framework +provided by Pytorch 1.13.0. The experimental is conducted +with CPU AMD 5900X, GPU NVIDIA RTX 3090, 64GB +memory and SSD 2TB. The model is trained for 500 epochs +using optimizer NAdam, scheduler ReduceLROnPlateau +with factor 0.5, eps 1e-5, threshold 1e-5 and patience 5 and +loss function L1Loss without any data augmentation. +4.5 +Discussion on Experimental Results +The experimental results of univariate and multivariate +datasets are provided in Table 1, 2 and Table 3, 4 respec- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +TABLE 3 +MEAN MAE RESULTS OF MULTIVARIATE DATASETS +Dataset +Naive +SES +Theta +TBATS +ETS +ARIMA +PR +CatBoost +FFNN +DeepAR +NN5 Daily +4.63 +6.63 +3.80 +3.70 +3.72 +4.41 +5.47 +4.22 +4.06 +3.94 +NN5 Weekly +19.44 +15.66 +15.30 +14.98 +15.70 +15.38 +14.94 +15.29 +15.02 +14.69 +Web Traffic Daily +484.67 +363.43 +358.73 +415.40 +403.23 +340.36 +- +- +- +- +Web Traffic Weekly +2756.28 +2337.11 +2373.98 +2241.84 +2668.28 +3115.03 +4051.75 +10715.36 +2025.23 +2272.58 +Solar 10 Minutes +0.00 +3.28 +3.29 +8.77 +3.28 +2.37 +3.28 +5.69 +3.28 +3.28 +Solar Weekly +998.99 +1202.39 +1210.83 +908.65 +1131.01 +839.88 +1044.98 +1513.49 +1050.84 +721.59 +Electricity Hourly +279.78 +845.97 +846.03 +574.30 +1344.61 +868.20 +537.38 +407.14 +354.39 +329.75 +Electricity Weekly +99675.88 +74149.18 +74111.14 +24347.24 +67737.82 +28457.18 +44882.52 +34518.43 +27451.83 +50312.05 +Carparts +0.66 +0.55 +0.53 +0.58 +0.56 +0.56 +0.41 +0.53 +0.39 +0.39 +FRED-MD +5607.17 +2798.22 +3492.84 +1989.97 +2041.42 +2957.11 +8921.94 +2475.68 +2339.57 +4263.36 +Traffic Hourly +0.01 +0.03 +0.03 +0.04 +0.03 +0.04 +0.02 +0.02 +0.01 +0.01 +Traffic Weekly +1.25 +1.12 +1.13 +1.17 +1.14 +1.22 +1.13 +1.17 +1.15 +1.18 +Rideshare +1.61 +6.29 +7.62 +6.45 +6.29 +3.37 +6.30 +6.07 +6.59 +6.28 +Hospital +32.29 +21.76 +18.54 +17.43 +17.97 +19.60 +19.24 +19.17 +22.86 +18.25 +COVID Deaths +310.84 +353.71 +321.32 +96.29 +85.59 +85.77 +347.98 +475.15 +144.14 +201.98 +Temperature Rain +6.66 +8.18 +8.22 +7.14 +8.21 +7.19 +6.13 +6.76 +5.56 +5.37 +Dataset +N-BEATS +WaveNet +Transformer +MSS∗ +FEDformer∗ +NetAtt∗ +Pyraformer∗ +PFSD∗ +Informer +Ours +NN5 Daily +4.92 +3.97 +4.16 +- +- +- +- +- +4.07 +4.92 +NN5 Weekly +14.19 +19.34 +20.34 +- +- +- +- +- +19.45 +15.09 +Web Traffic Daily +- +- +- +- +- +- +- +- +- +217.47 +Web Traffic Weekly +2051.30 +2025.50 +3100.32 +- +- +- +- +- +- +1437.49 +Solar 10 Minutes +3.52 +- +3.28 +3.36 +3.18 +3.93 +3.22 +2.17 +3.67 +1.60 +Solar Weekly +1172.64 +1996.89 +576.35 +841.69 +479.30 +1247.77 +513.24 +649.22 +2360.71 +700.31 +Electricity Hourly +350.37 +286.56 +398.80 +- +- +- +- +- +441.77 +203.39 +Electricity Weekly +32991.72 +61429.32 +76382.47 +- +- +- +- +- +47773.67 +15699.48 +Carparts +0.98 +0.40 +0.39 +- +- +- +- +- +- +0.53 +FRED-MD +2557.80 +2508.40 +4666.04 +- +- +- +- +- +32700.73 +596.54 +Traffic Hourly +0.02 +0.02 +0.01 +- +- +- +- +- +0.02 +0.008 +Traffic Weekly +1.11 +1.20 +1.42 +- +- +- +- +- +1.42 +1.10 +Rideshare +5.55 +2.75 +6.29 +- +- +- +- +- +- +0.79 +Hospital +20.18 +19.35 +36.19 +- +- +- +- +- +38.82 +16.40 +COVID Deaths +158.81 +1049.48 +408.66 +- +- +- +- +- +- +8.84 +Temperature Rain +7.28 +5.81 +5.24 +- +- +- +- +- +- +4.56 +TABLE 4 +MEAN RMSE RESULTS OF MULTIVARIATE DATASETS +Dataset +Naive +SES +Theta +TBATS +ETS +ARIMA +PR +CatBoost +FFNN +DeepAR +NN5 Daily +6.68 +8.23 +5.28 +5.20 +5.22 +6.05 +7.26 +5.73 +5.79 +5.50 +NN5 Weekly +24.27 +18.82 +18.65 +18.53 +18.82 +18.55 +18.62 +18.67 +18.29 +18.53 +Web Traffic Daily +911.51 +590.11 +583.32 +740.74 +650.43 +595.43 +- +- +- +- +Web Traffic Weekly +4020.90 +2970.78 +3012.39 +2951.87 +3369.64 +3777.28 +4750.26 +14040.64 +2719.65 +2981.91 +Solar 10 Minutes +0.00 +7.23 +7.23 +10.71 +7.23 +5.55 +7.23 +8.73 +7.21 +7.22 +Solar Weekly +1350.79 +1331.26 +1341.55 +1049.01 +1264.43 +967.87 +1168.18 +1754.22 +1231.54 +873.62 +Electricity Hourly +414.29 +1026.29 +1026.36 +743.35 +1524.87 +1082.44 +689.85 +582.66 +519.06 +477.99 +Electricity Weekly +104510.94 +77067.87 +76935.58 +28039.73 +70368.97 +32594.81 +47802.08 +37289.74 +30594.15 +53100.26 +Carparts +1.17 +0.78 +0.78 +0.84 +0.80 +0.81 +0.73 +0.79 +0.74 +0.74 +FRED-MD +6333.09 +3103.00 +3898.72 +2295.74 +2341.72 +3312.46 +9736.93 +2679.38 +2631.4 +4638.71 +Traffic Hourly +0.02 +0.04 +0.04 +0.05 +0.04 +0.04 +0.03 +0.03 +0.02 +0.02 +Traffic Weekly +1.63 +1.51 +1.53 +1.53 +1.53 +1.54 +1.50 +1.50 +1.55 +1.51 +Rideshare +1.98 +7.17 +8.60 +7.35 +7.17 +4.80 +7.18 +6.95 +7.14 +7.15 +Hospital +39.54 +26.55 +22.59 +21.28 +22.02 +23.68 +23.48 +23.45 +27.77 +22.01 +COVID Deaths +313.04 +403.41 +370.14 +113.00 +102.08 +100.46 +394.07 +607.92 +173.14 +230.47 +Temperature Rain +10.15 +10.34 +10.36 +9.20 +10.38 +9.22 +9.83 +8.71 +8.89 +9.11 +Dataset +N-BEATS +WaveNet +Transformer +MSS∗ +FEDformer∗ +NetAtt∗ +Pyraformer∗ +PFSD∗ +Informer +Ours +NN5 Daily +6.47 +5.75 +5.92 +- +- +- +- +- +5.52 +6.52 +NN5 Weekly +17.35 +24.16 +24.02 +- +- +- +- +- +23.03 +18.98 +Web Traffic Daily +- +- +- +- +- +- +- +- +- +465.84 +Web Traffic Weekly +2820.62 +2719.37 +3815.38 +- +- +- +- +- +- +2197.40 +Solar 10 Minutes +6.62 +- +7.23 +6.94 +6.91 +7.97 +7.18 +5.28 +6.41 +1.60 +Solar Weekly +1307.78 +2569.26 +693.84 +972.45 +609.94 +2493.06 +672.54 +776.15 +2623.95 +863.46 +Electricity Hourly +510.91 +489.91 +514.68 +- +- +- +- +- +629.88 +302.56 +Electricity Weekly +35576.83 +63916.89 +78894.67 +- +- +- +- +- +54022.60 +22540.95 +Carparts +1.11 +0.74 +0.74 +- +- +- +- +- +- +0.78 +FRED-MD +2812.97 +2779.48 +5098.91 +- +- +- +- +- +32867.61 +708.38 +Traffic Hourly +0.02 +0.03 +0.02 +- +- +- +- +- +0.04 +0.01 +Traffic Weekly +1.44 +1.61 +1.94 +- +- +- +- +- +1.76 +1.50 +Rideshare +6.23 +3.51 +7.17 +- +- +- +- +- +- +1.04 +Hospital +24.18 +23.38 +40.48 +- +- +- +- +- +44.25 +20.45 +COVID Deaths +186.54 +1135.41 +479.96 +- +- +- +- +- +- +13.27 +Temperature Rain +11.03 +9.07 +9.01 +- +- +- +- +- +- +7.05 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +2 +4 +6 +8 +10 +SES +Theta +TBATS +ETS +ARIMA +PR +CatBoost +FFNN +DeepAR +N-BEATS +WaveNet +Transformer +Ours +(a) +MAE Comparison Among Models Without ∗ +2 +4 +6 +8 +10 +SES +Theta +TBATS +ETS +ARIMA +PR +CatBoost +FFNN +DeepAR +N-BEATS +WaveNet +Transformer +Ours +(b) +RMSE Comparison Among Models Without ∗ +1 +2 +3 +4 +5 +6 +MSS +FEDformer +NetAtt +Pyraformer +PFSD +Ours +(c) +MAE Comparison Among Models With ∗ +1 +2 +3 +4 +5 +6 +MSS +FEDformer +NetAtt +Pyraformer +PFSD +Ours +(d) +RMSE Comparison Among Models With ∗ +Fig. 3. Friedman Test Figure: the Performance Comparison Based MAE and RMSE Among Models From the Perspective of Nemenyi Test. +tively. Generally, the proposed model obtains state-of-the- +art results on most of the experimental time series datasets. +However, the results of RMSE fail to remain consistent with +MAE, it demonstrates that the proposed model’s ability in +handling abnormal prediction values is relatively lacking. +We argue that the main reason for this phenomenon is that +the proposed model pays much more attention to the global +information distributed to the elements captured by the +sliding window and ignores the influence of the original +values on the future trend to a certain extent due to the +strategies of data encoding and utilization of information +processed of the proposed model. Especially, our proposed +model outperforms transformer-based methods which at- +tract lots of researchers’ attention recently on almost all of +the datasets, we consider that temporal data is not similar +to images and videos in which there are enormous amount +of semantic information needed to be extracted . +4.6 +Overall Performance Comparison Between Pro- +posed and Comparative Models +In order to comprehensively demonstrate superiority of +the proposed model, we utilize Nemenyi test with CD = +q0.05 +� +k(k+1) +6Nd +in which k is the number of algorithms partic- +ipating in the comparison and Nis the number of datasets. +Due to lack of some results of models with superscript ∗ +on certain datasets, the Nemenyi test is divided into two +groups to ensure fairness of comparison and the evaluation +results are shown in Friedman test figure at Fig.3. It can be +easily concluded that the proposed model acquire the most +excellent integrated performance on experimental datasets +provided. +4.7 +Parameter Study +Different datasets have their corresponding optimal param- +eter setting for the proposed model. We selected four uni- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +2 +4 +6 +8 +10 +OUT +5 +10 +15 +20 +WINDOW SIZE +8.8 +9 +9.2 +9.4 +9.6 +9.8 +(a) Saugeenday Dataset +2 +4 +6 +8 +10 +OUT +5 +10 +15 +WINDOW SIZE +2400 +2600 +2800 +3000 +3200 +3400 +3600 +(b) Tourism Monthly Dataset +(c) Tourism Quarterly Dataset +2 +4 +6 +8 +10 +OUT +5 +10 +15 +20 +WINDOW SIZE +600 +700 +800 +900 +(d) US Births Dataset +Fig. 4. MAE Variations When Parameter OUT and Window Size Vary on Univariate Datasets +(a) Electricity Weekly Dataset +2 +4 +6 +8 +10 +OUT +5 +10 +15 +20 +WINDOW SIZE +17 +18 +19 +(b) Hospital Dataset +2 +4 +6 +8 +10 +OUT +5 +10 +15 +20 +WINDOW SIZE +1.12 +1.14 +1.16 +(c) Traffic Weekly Dataset +2 +4 +6 +8 +10 +OUT +5 +10 +15 +20 +WINDOW SIZE +800 +900 +1000 +1100 +(d) Solar Weekly Dataset +Fig. 5. MAE Variations When Parameter OUT and Window Size Vary on Multivariate Datasets +variate and four multivariate data sets for a brief analysis. +In Fig.4 and 5, it can be obtained that the performance +of the proposed model benefits from a larger window +size. And lifting factor OUT has limited influence on the +model capability and can reduce the error in some cases. +Besides, synthesizing conditions of Fig.6 and 7, increasing +the window size dose not necessarily improve model’s per- +formance, but larger window sizes can help capture more +information and establish the foundation of precise predic- +tions in general. Specifically, on multiple datasets such as +M4 Monthly, Quarterly, KDD Cup 2018 and Covid Deaths +datasets, error increases considerably when window size +equals 2. The main reason probably is that the size sacrifices +timeliness of data to some extent and is not capable of +providing sufficient semantic information to the model so +that the proposed model encounters difficulty in producing +accurate predictions. +5 +CONCLUSION +In this paper, a novel time series forecasting model is pro- +posed which consists of encoder part and semi-asymmetric +convolutional architecture. The main role of devised data +encoder is assigning elements in original observation time + +X10° +1.5 +1.3 +MAE +1.1 +0.9 +8 +5 +10 +15 +20 +WINDOW SIZE1000 +MAE +800 +600 +8 +6 +? +5 +10 +15 +20 +SIZE +WINDOW1.7 +MAE +1.65 +1.6 +5 +10 +15 +20 +SIZE +WINDOW20 +19 +MAE +18 +17 +8 +6 +1 +OUT +5 +10 +15 +20 +WINDOW SIZE1.17 +AE +1.15 +M +1.13 +1.11 +8 +6 +1 +OU +5 +10 +15 +20 +SIZE +WINDOW1100 +MAE +950 +800 +8 +6 +? +5 +2 +10 +15 +20 +WINDOW SIZE10 +MAE +9.5 +9 +5 +10 +15 +20 +SIZE +WINDOW4000 +3500 +MAE +3000 +2500 +8 +6 +OUT +4. +8 +12 +16 +WINDOW SIZEJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +2 +4 +6 +8 +10 +WINDOW SIZE +1600 +1650 +1700 +1750 +1800 +MAE +(a) M1 Monthly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +1000 +1500 +2000 +2500 +MAE +(b) M1 Quarterly Dataset +(c) M1 Yearly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +540 +560 +580 +600 +620 +640 +MAE +(d) M3 Monthly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +90 +95 +100 +105 +110 +MAE +(e) M3 Other Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +300 +320 +340 +360 +380 +MAE +(f) M3 Quarterly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +500 +600 +700 +800 +900 +1000 +MAE +(g) M3 Yearly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +62.4 +62.6 +62.8 +63 +63.2 +63.4 +MAE +(h) M4 Daily Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +90 +100 +110 +120 +130 +140 +MAE +(i) M4 Hourly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +380 +390 +400 +410 +420 +MAE +(j) M4 Quarterly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +220 +225 +230 +235 +240 +245 +MAE +(k) M4 Weekly Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +400 +500 +600 +700 +MAE +(l) M4 Yearly Dataset +Fig. 6. MAE Variations on Different Window Sizes on Univariate Datasets +2 +4 +6 +8 +10 +WINDOW SIZE +0.55 +0.56 +0.57 +0.58 +MAE +(a) Car Parts Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +9 +10 +11 +12 +13 +14 +MAE +(b) Covid Deaths Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +5 +5.5 +6 +6.5 +7 +MAE +(c) NN5 Daily Dataset +2 +4 +6 +8 +10 +WINDOW SIZE +0.795 +0.8 +0.805 +0.81 +0.815 +MAE +(d) Rideshare Dataset +Fig. 7. MAE Variations on Different Window Sizes on Multivariate Datasets +230136 +230436 +230736 +5000 +6000 +7000 +8000 +9000 +Input +Actual +Prediction +230136 +230436 +230736 +4000 +5000 +6000 +7000 +Input +Actual +Prediction +231672 +231972 +232272 +4000 +5000 +6000 +7000 +Input +Actual +Prediction +230184 +230484 +230784 +750 +1250 +1750 +2250 +Input +Actual +Prediction +Fig. 8. Qualitative Prediction Results by the Proposed Method on Aus. Electricity Demand Dataset + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +12 +series with positional information so that the model could +possess a global view on observation sequences. Based on +processed data, a novel architecture is designed referencing +asymmetric convolution and considering variability of time +series which enables the model to obtain information at +flexible scales on different time series. Capturing features +with separate range helps model to learn underlying rela- +tionship among elements with effective understanding of +associated positional information. Both of them contributes +to the outstanding performance of the proposed model. +To comprehensively demonstrate the performance of the +model proposed in this paper, we conduct experiments on +27 univariate and 16 multivariate datasets. The experimen- +tal results illustrate that the proposed model outperforms +comparative methods on most of forecasting tasks. Specifi- +cally, the proposed model achieves the highest rank on all +competition datasets such as M series, KDD Cup and Web +Traffic. In addition to these intuitive results, Nemenyi test +also strongly demonstrates the excellent performance of the +proposed model. Besides, we also investigate the influences +of the two main parameters on 24 datasets to further explain +settings of the proposed model. +Nevertheless, the proposed model achieves relatively +satisfying performance in most of forecasting experiments, +there are still some potentials in it which can be further +explored. We think we may be able to improve the model +in two possible directions: 1): The attention mechanism can +be introduced into the model to help the model better un- +derstand semantic information in time series, 2): A recurrent +architecture of convolutional neural network is expected to +be developed to better memory past information. +ACKNOWLEDGMENT +REFERENCES +[1] +R. R. Sharma, M. Kumar, S. 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Xiao, “Time series forecasting based on fuzzy cognitive visibility +graph and weighted multi-subgraph similarity,” IEEE Transactions on Fuzzy +Systems, 2022. + diff --git a/XdFRT4oBgHgl3EQf-jgZ/content/tmp_files/load_file.txt b/XdFRT4oBgHgl3EQf-jgZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a75a369d06d063caaadb2292ed5cf023d87b1394 --- /dev/null +++ b/XdFRT4oBgHgl3EQf-jgZ/content/tmp_files/load_file.txt @@ -0,0 +1,2657 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf,len=2656 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 1 Time Series Forecasting via Semi-Asymmetric Convolutional Architecture with Global Atrous Sliding Window Yuanpeng He Abstract—The proposed method in this paper is designed to address the problem of time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Although some exquisitely designed models achieve excellent prediction performances, how to extract more useful information and make accurate predictions is still an open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Most of modern models only focus on a short range of information, which are fatal for problems such as time series forecasting which needs to capture long-term information characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' As a result, the main concern of this work is to further mine relationship between local and global information contained in time series to produce more precise predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In this paper, to satisfactorily realize the purpose, we make three main contributions that are experimentally verified to have performance advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Firstly, original time series is transformed into difference sequence which serves as input to the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' And secondly, we introduce the global atrous sliding window into the forecasting model which references the concept of fuzzy time series to associate relevant global information with temporal data within a time period and utilizes central-bidirectional atrous algorithm to capture underlying-related features to ensure validity and consistency of captured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Thirdly, a variation of widely-used asymmetric convolution which is called semi-asymmetric convolution is devised to more flexibly extract relationships in adjacent elements and corresponding associated global features with adjustable ranges of convolution on vertical and horizontal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The proposed model in this paper achieves state-of-the-art on most of time series datasets provided compared with competitive modern models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Index Terms—Local and global information, Difference sequence, Global atrous sliding window, Semi-asymmetric convolution !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 1 INTRODUCTION T IME series is a sequence taken at successive equally spaced points in time which is also known as dynamic series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Precise prediction of time series has close connections to human society, for instance, it may help people format schedules and company make adjustments on investment strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Moreover, foreseeing future behaviour based on analysis of known historical data is of great importance in lots of fields such as epidemic [1], medical treatment [2], finance [3, 4] and industrial Internet [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Time series fore- casting therefore attracts attention from researchers around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Nevertheless, how to fully utilize observation to generate accurate and reasonable predictions is still an unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' To realize accurate prediction of future, researchers de- velop various kinds of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' RNN model has been favored by researchers since it was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Because of its recurrent architecture design, RNN models can effectively model long-term dependencies [11], therefore achieve an effective understanding of temporal data [12, 13] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' However, RNN may encounter memory overflow due to continuous storage of previous states and gradient vanish- ing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' To further make up for shortcomings of RNN, an improved solution based on it is proposed which is called LSTM [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Its core concepts are the memory cell states that allow information to be passed on backwards, and the gate Yuanpeng He is with Key Laboratory of High Confidence Software Tech- nologies, Peking University, Peking, 100871, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' School of Computer Science, Peking University, Peking, 100871, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' E-mail: heyuanpengpku@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='com Manuscript received;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' revised structures that allow certain information to be added and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Coincidentally, researchers also find that temporal task can also benefit form LSTM’s characteristics [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In a quite long period of time, the model based on RNN has played an important role in development of time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Recently, transformer-based models [6–9] have been proposed enormously, which applied self-attention mechanism to distill useful semantic information in time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' However, there exists a doubt that transformer-like structure is not suitable for the task of time series fore- casting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Under certain circumstances, the performance of the models can not even match ingeniously designed linear model [10], which has shaken the position of transformer- based models in time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' At present, the controversy still continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, there also exist lots of meaningful works trying to satisfy demand of time series forecasting from other multiple aspects as well [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Moreover, CNN-based models are also widely utilized for prediction of temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' They are mainly divided into two categories, one is the variation of causal and dilated convolution [18], the other is algorithms using graph convolutional neural network [19] to solve corresponding problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Generally, it can be concluded that transformer and CNN models, the two well-established solutions in the field of computer vision, also achieve excellent perfor- mance in tasks of time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Back to CNN- based models, there have been many new CNN models in recent years, for instance, temporal convolutional network (TCN) [20], convolutionally low-rank model [21] and non- pooling CNN [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Among them, TCN attracts the most attention which is capable of large-scale parallel processing arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='13691v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='AI] 31 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 2 and managing a series of sequences of arbitrary length and uniformly output sequences with the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Specif- ically, the casual and dilated convolution introduced by it enable CNN forecasting model to possess a larger receptive field to better acquire information in a longer range under strict time restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Moreover, other effective models [23–25] also improve performance by enlarging ranges of data selection and more ingenious and flexible extraction of relationships of adjacent and non-adjacent elements because of similar considerations about demand of time series fore- casting mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Nevertheless, it is noting that all of the models still receive data in a relatively restricted way without considering global data features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' To address the issue, we design a kind of data recon- struction method referencing solution based on partitioned universe of discourse [26] partly which transforms original temporal data into difference sequence [27–29] ensuring that the model is more likely dealing with steady-state sequences and associates relative positional information of data captured in the view of whole observation time se- ries to reduce difficulty of model learning to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' More than that, we choose to replace all of elements by the last one only keeping their positional information as subsidiaries to maximize timeliness of data without losing too much semantic information of temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, the relationship among converted information in different subsections and time series are probably separate [30], so there is a need to devise a convolution strategy with dif- ferent directions and shapes to further mine underlying information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Due to particularity of time series, traditional squared convolution is not capable to manage complex extraction of relationship of elements in temporal data, a variation of asymmetric convolution [31, 32] which is called semi-asymmetric convolution is designed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The semi-asymmetric convolution is divided into horizontal and vertical filters, and they probably possess different length to retrieve interaction information at a more fine-grained level in selected fragments from difference series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The ad- vantage of this improvement is that it is able to effectively obtain temporal features using adjustable scales [33, 34], and speeding up the training and inference process of the proposed model [35] at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' the major ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='contributions of this work are summarized as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='The input to proposed model is difference sequences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='transformed by original observation time series to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='enable model to learn more easily ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='A new kind of method of data reconstruction is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='designed to endow each elements with their corre- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='sponding relative positional information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='A novel convolutional architecture called semi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='asymmetric convolution with flexible scales is de- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='signed to acquire information at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In the sec- ond section, some related concepts of the proposed model are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' And the details of the proposed model are presented in the third section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, the fifth section pro- vides experimental results and corresponding discussions with respect to models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In the last section, conclusions and outlook of future work are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 2 PRELIMINARY In this section, related concepts about the proposed model are briefly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='1 Difference of First Order A first order difference is the difference between two con- secutive adjacent terms in a discrete function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume there exists a function y = f(x), y is defined only on the non- negative integer value of x and when the independent variable x is iterated through the non-negative integers in turn, namely x = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', the corresponding values of function can be defined as: f(0), f(1), f(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' (1) it can be abbreviated as: y0, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' (2) when the independent value changes from x to x + 1, the variation of y = f(x) can be defined as: ∆yx = f(x + 1) − f(x), (x = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=') (3) it’s called the first difference of the function y(x) at point x which is usually denoted as: ∆yx = yx+1 − yx, (x = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=') (4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2 Asymmetric Convolution Architecture CNN has embraced a quick development recently, it is widely applied in different fields, such as time series and computer vision [36–38] due to its stable and excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' For an operation of convolving, assume an input ς ∈ RH×W and filter C, the process of generating output λ ∈ RH′×W ′ can be defined as: λ = C ∗ ς, ς ∈ RH×W , λ ∈ RH′×W ′, C ∈ Rd×d (5) where ∗ is the 2D convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Moreover, asym- metric convolution [35, 39] is considered as an economical choice to approximate an existing square-kernel convolu- tional layer for obtaining acceleration and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Specifically, the original filter can be decomposed into hori- zontal and vertical filters, Ch, Cv, respectively, which can be defined as: C ∗ ς = Cv ∗ (Ch ∗ ς), Cv ∈ Rd×1, Ch ∈ R1×d (6) compared with the original convolution utilizing d×d kernel size, the time complexity changes from O((d2H′W ′) to O(2dH′W ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Due to efficiency of the asymmetric architec- ture, it is widely applied in convolutional neural network design [40, 41] and gains performance improvement gener- ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3 Atrous Algorithm The atrous algorithm is proposed in [42, 43] which is also known as dilated convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume there exists a one- dimensional input α[s], the corresponding output β[s] of dilated convolution via a filter ω[e] with length E can be defined as: β[s] = E � e=1 α[s + r · e]ω[e] (7) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 3 where rate parameter r is corresponding to the stride and standard convolution is a special case for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Gener- ally, atrous algorithm is designed to avoid precision loss brought by reduction of feature map on account of multiple convolutional and pooling layers on vision tasks and is broadly utilized in many other import fields, such as audio processing [44] and time series forecasting [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='4 Naive Forecasting The naive forecasting is the simplest prediction method in the field of time series which regards the most recent observation value as the prediction of future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume there exists a time series T with a length n which can be defined as: T = {(t1, ζ1), (t2, ζ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', (tn−1, ζn−1), (tn, ζn)} (8) where ζi, i ∈ [1, n] represents observation value at time point i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' For example, if there is a need to predict ζn+q which is unknown, the value of ζn can be referenced as the prediction value of ζn+q directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume the prediction value of ζn+q is ˆζn+q, then the method of forecasting can be defined as: ˆζn+q = ζn (9) under various circumstances, naive forecasting is an effec- tive solution to tell the future trend of time series like stock price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Nevertheless, the method introduced is not a satisfying solution in time series forecasting, but it can provide a benchmark for other prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='5 Fuzzy Time Series One of concepts of fuzzy time series is introduced by Chen [26] which is developed based on theories proposed in [45– 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The method of fuzzy time series could extract informa- tion effectively by utilizing overall characteristics of time series data and provide stable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Specifically, given a time series T, and let ηmin and ηmax be minimum and maximum value in T, the four steps to generate fuzzy time series can be presented as: Step 1: Select two proper positive numbers η1 and η2, a universe of discourse U can be defined as [ηmin −η1, ηmax − η2] Step 2: Partition U into segments with equal length {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='um} which is called fuzzy intervals Step 3: Let Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', Zk be fuzzy sets and they are defined on universe of discourse U as: � � � � � � � � � Z1 = z11/u1 + z12/u2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' + z1m/um, Z2 = z21/u1 + z22/u2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' + z2m/um, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Zk = zk1/u1 + zk2/u2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' + zkm/um (10) where zij ∈ [0, 1], i ∈ [1, k], j ∈ [1, m] and the value of zij represents the degree of membership of uj in fuzzy set Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Step 4: The derived fuzzy logical relationships which possess identical initial states are divided into the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Then, the matches between actual values in time series and groups of fuzzy logical relationships can be acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' After these four steps, the original data is transformed into fuzzy time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3 ARCHITECTURE OF THE PROPOSED FORECAST- ING MODEL In this section, the proposed forecasting model based on relevant concepts mentioned above is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='1 Difference Layer: Convert Time Series into First Order Difference Sequence First, a time series T is transformed into its first order difference sequence T∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The process can be given as: � � � � � � � � � � � T = {(t1, ζ1), (t2, ζ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', (tn−1, ζn−1), (tn, ζn)} ⇓ T− ∆ = {(t2,1, ζ2 − ζ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', (tn,n−1, ζn − ζn−1)} ⇓ T∆ = {α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', αh} (11) where h = n − 1 and αi, i ∈ [1, h] only contains observed value without timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Obviously, it can be obtained that the length of T∆ is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In the next step, the input is T∆ instead of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2 Division layer: Divide Converted Time Series into Sub-Series Based on Sliding Window Second, series T∆ is divided by sliding window whose size is W into sub-series, the segmented data fragments are: TSeg ∆ = {Υ1, Υ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', Υc}, c = h − W + 1 (12) where Υj = {αj, αj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', αj+W−1}, j ∈ [1, c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3 Encoder of segmented sequences 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='1 Relative Positional Encoding: Reconstruct Sub- Series with View on Global Observation Temporal Data Third, in the concept of fuzzy time series proposed in [26], the two numbers η1 and η2 are selected intuitively, which may lead to non-reproducibility of experiment results on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' As a result, η1 and η2 are uniformly set as standard deviation of corresponding first order difference sequence, which can be given as: ϕ = η1 = η2 = σ(T∆) (13) where σ represents standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Then, universe of discourse U of T∆ can be calculated as: UT∆ = [αmin − ϕ, αmax + ϕ] = [βl, βu] (14) where αmin and αmax represent minimum and maximum element contained in T∆ and βl and βu denote lower and upper bound of UT∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' And the number of intervals, N, can be confirmed as: N = logh 2 − 1 (15) the partitioned universe of discourse can be given as: UT∆ = [βl, βl + ξ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', βl + κ × ξ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', βu − ξ, βu] (16) where ξ = (βu − βl)/N and κ ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Then, integrate each element contained in Υj into the partitioned universe JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' AUGUST 2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='― ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='𝛽" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='Central-Bidirectional Atrous Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='Padding Mechanism and Cropping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Padding Mechanism, Cropping and Central-Bidirectional Atrous Algorithm with Dilation Factor d = 1 of discourse UT∆ to create new sequences based on data fragments captured by sliding window: Υ UT∆ j = � ������� βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl + κ′ × ξ αj βl + (κ′ + 1) × ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl + κ′′ × ξ αj+1 βl − (κ′′ + 1) × ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl + κ′′′ × ξ αj+W−1 βl − (κ′′′ + 1) × ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu � ������� (17) the position of each integrated element is uniquely identi- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Then, all the data from sliding window in the recon- structed series is replaced with the last element in original subset divided only keeping position information of former elements: ℓj = � ���� βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl + κ′ × ξ αj+W−1 βl + (κ′ + 1) × ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl + κ′′ × ξ αj+W−1 βl + (κ′′ + 1) × ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl + κ′′′ × ξ αj+W−1 βl + (κ′′′ + 1) × ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu � ���� (18) the simplified form of it can be given as: ℓj = � ���� βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' x1 α ˚x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' x2 α ˚x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' xp α ˚xp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' βu � ���� (19) where p ∈ [1, W] and the final input to the proposed network is: TInput = {ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', ℓc} (20) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2 Padding Mechanism and Cropping Forth, one side of each row of input data is filled separately so that the length of data on both sides of the last element in original subset is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume data of row p in ℓℏ is vector ⃗Λp, the process of padding can be given as: ⃗˘ Λp = � Concat(rep(βl)D, ⃗Λ βl⇒xp p ), ||( ⃗Λ βl⇒xp p )|| < ||( ⃗Λ ˚ xp⇒βu p )|| Concat( ⃗Λ ˚ xp⇒βu p , rep(βu)D′), ||( ⃗Λ βl⇒xp p )|| > ||( ⃗Λ ˚ xp⇒βu p )|| (21) where ⃗Λ ˙o⇒¨o p represents a segmented vector which ranges from element ˙o to ¨o, || ⃗Λ ˙o⇒¨o p || is the length of ⃗Λ ˙o⇒¨o p and Concat denotes the operation of concatenation of two vec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, rep( ˙o)D means creating a vector containing D copies of element ˙o and D = ||( ⃗Λ˚x⇒βu p )|| − ||( ⃗Λβl⇒x p )|| or D′ = ||( ⃗Λβl⇒x p )|| − ||( ⃗Λ˚x⇒βu p )||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Then, each row containing in ℓℏ is padded ensuring lengths of two sides of the last element in original subset are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' However, the operation of padding brings a problem that length of each row is not exactly the same which is difficult for neural network to acquire information and cap- ture features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' As a result, there is a need to crop redundant elements in each padded data row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume length of the shortest padded vector is S and the operation of cropping M elements which lie from both ends of the vector ˘Λp to its centre is CropM, the process of cropping is defined as: ⃗Λp = CropM( ⃗˘Λp) (22) where M = (|| ⃗˘Λp|| − S)/2 and ⃗Λp is the cropped vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3 Central-Bidirectional Atrous Algorithm Fifth, the processed information needs to be further ex- tracted so that subsequent networks can capture more useful information and avoid unnecessary calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Because of the unique nature of the reconstructed timing data, the atrous algorithm is modified to obtain data from the centre to both sides of each segment, which reserves the nearest observation value and corresponding position distribution information from the prediction object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume the leftmost and rightmost element in ⃗Λp are ϵlp and ϵrp, the input which is divided into two parts by the central element to central- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 5 bidirectional atrous algorithm (CBAA) is: � � � ℓl j = [ ⃗Λ ˚x1⇒ϵr1 1˚ x1+g×d, ⃗Λ ˚x2⇒ϵr2 2˚ x2+g×d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', ⃗Λ ˚xp−1⇒ϵrp−1 p−1˚ xp−1+g×d, ⃗Λ ˚xp⇒ϵrp p˚ xp+g×d] ℓr j = [ ⃗ Λ ϵl1⇒x1 1x1−g×d, ⃗ Λ ϵl2⇒x2 2x2−g×d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', ⃗ Λ ϵlp−1⇒xp−1 p−1xp−1−g×d, ⃗ Λ ϵlp⇒xp pxp−g×d] (23) where ℓl j and ℓr j denote left and right part of cropped vector ⃗Λp and the directions of filters on them are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Moreover, assume a filter f : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=', v −1} and the operation of CBAA, F, starting with elements ˚xp and xp is defined as: F(α)j = Concat( v−1 � g=0 f(g) · ℓl j, α, v−1 � g=0 f(g) · ℓr j) (24) where · represents the operation of dilated convolution, d is the factor of dilation, v means the filter size, ˚xp + g × d and xp − g × d account for the direction of movement of filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' When d = 1, the form of atrous algorithm degenerates into regular convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' A larger dilation factor enables the algorithm to capture features at a longer range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In original atrous algorithm, the operation of dilation is utilized to enlarge the receptive field without reduce sizes of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' But in the CBAA, the dilated convolution is mainly used to construct efficient maps with proper sizes contain- ing underlying features of historic information via multiple non-adjacent fuzzy intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='4 Semi-Asymmetric Convolutional Architecture Sixth, a semi-asymmetric convolutional neural network (SACNN) is designed to aggregate information and produce differential predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' SACNN is made up of a stack of one module which is called SAC block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The SAC block consists of two parts, the first part is the batchnorm layer Bn which is defined as: ˜Cj = Bn(F(α)j) = F(α)j − ¯ F(α)j � σ(F(α)j) + ϵ ∗ γ + δ (25) where ¯ F(α)j and σ(F(α)j) denotes mean and standard- deviation of F(α)j, γ and δ are learnable parameter vectors whose size is the number of channel of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The output ˜Aj is supposed to be sent into the next part, semi-asymmetric convolutional layer Sa which consists of L combinations of X horizontal and vertical filters ˇfV ∈ RV ×1 and ˇfH ∈ R1×H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Assume the input ˜Cj ∈ RH′×V ′×Y with H′ × V ′ feature map and Y channels, the process of generating output can be defined as: Bj = Sa( ˜Cj) = [ ˇfV ⋄( ˇfH ⋄ ˜Cj)]×L, Bj ∈ RH′′×V ′′×X (26) where ⋄ represents semi-asymmetric convolution and OUT = X/Y is the lifting factor of number of input’s to output’s channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' When V = H, Sa degenerates into the form of regular asymmetric convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Before outputting the final values, the information is expected to be sent into two linear layers: Qj = SAC(F(α)j) = (BjAT + b)A′T + b′ (27) where A and A′ are the learnable weights of the module of shape which is transposed to times the original input Bj and b and b′ are the biases to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Then, Qj is the prediction which the proposed model produce on the first order difference sequence T∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' BatchNorm2D Horizontal Filter Vertical Filter Output Linear1 Linear2 Encoder Input Restore … BatchNorm2D Horizontal Filter Vertical Filter Linear1 Linear2 Encoder BatchNorm2D Horizontal Filter Vertical Filter Linear1 Linear2 Encoder … Difference Layer Division Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Details of the Proposed Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='5 Restore Output of Network to Original Position and Make Prediction Seventh, restore the differential prediction Qj to the original time series T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The process of a sliding window generating corresponding prediction is given as: ˆζj+W+1 = ζj+W + Qj (28) on the training data, the proposed model is expected to ap- proximate trend of changes of time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' For the prediction of value beyond the known time series data, the prediction is made as: ˆζn+1 = ζn + Qn−W (29) the process of producing prediction of the proposed model is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, multiple experiments are conducted to eval- uate the effectiveness and validity of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='1 Datasets Description In order to fully illustrate the performance of the proposed model, the comparison experiments are conducted on 43 datasets which are provided by monash time series forecast- ing archive (MTSFA) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Specifically, among them, there are 27 univariate and 16 multivariate datasets and they cover multiple domains, such as Tourism, Banking, En- ergy, Sales, Economic, Transport, Nature, Web and Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Moreover, the datasets have different sampling rates such as yearly, quarterly and monthly, which also correspond disparate expected forecast horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 6 TABLE 1 MEAN MAE RESULTS OF UNIVARIATE DATASETS Dataset Naive SES Theta TBATS ETS ARIMA PR CatBoost FFNN DeepAR M1 Yearly 221512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='32 171353.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='80 152084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='40 M1 Quarterly 3350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='81 2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='27 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='96 2326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='46 2088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} 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+page_content='77 US Births 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='00 504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='40 452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='87 609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='43 538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2 Baseline Methods for Comparison To demonstrate the performance improvement gained by the proposed model, we compare it with baseline methods, such as Naive (Forecasting)1, Simple Exponential Smooth- ing (SES) [49], Theta [50], Trigonometric Box-Cox ARMA Trend Seasonal Model (TBATS) [51], Exponential Smooth- ing (ETS) [52], (Dynamic Harmonic Regression-)ARIMA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The results produced by naive forecasting dose not participate in the comparison with experimental results of other models because of its particularity in forecasting strategy which is provided only for a simple reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' For example, on Solar 10 Minutes dataset, naive forecasting achieve surprising results whose error is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='00, which is unintuitive and unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' [53, 54], Pooled Regression Model (PR) [55], CatBoost [56], Feed-Forward Neural Network (FFNN) [57], DeepAR [58], N-BEATS [59], WaveNet [60], Transformer [61], MSS∗ [62], FEDformer∗ [6], NetAtt∗ [63], Pyraformer∗ [7], PFSD∗ [64] and Informer [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The experimental results of these methods except naive forecasting are acquired from MTSFA and PFSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, the results of experiments of Naive (Fore- casting) are generated by following the experimental rules given by MTSFA strictly.' metadata={'source': 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are selected to evaluate the accuracy of forecasting of chosen comparative models whose definitions are defined as: MAE = �N i=1 |ˆyi − yi| N (30) RMSE = ��N i=1 |ˆyi − yi|2 N (31) where ˆyi represents the value of forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='4 Implementation Details The proposed model is realized using the code framework provided by Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The experimental is conducted with CPU AMD 5900X, GPU NVIDIA RTX 3090, 64GB memory and SSD 2TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The model is trained for 500 epochs using optimizer NAdam, scheduler ReduceLROnPlateau with factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='5, eps 1e-5, threshold 1e-5 and patience 5 and loss function L1Loss without any data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='5 Discussion on Experimental Results The experimental results of univariate and multivariate datasets are provided in Table 1, 2 and Table 3, 4 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='MSS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='FEDformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='NetAtt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='Pyraformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='PFSD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='Ours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='RMSE Comparison Among Models With ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Friedman Test Figure: the Performance Comparison Based MAE and RMSE Among Models From the Perspective of Nemenyi Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Generally, the proposed model obtains state-of-the- art results on most of the experimental time series datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' However, the results of RMSE fail to remain consistent with MAE, it demonstrates that the proposed model’s ability in handling abnormal prediction values is relatively lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' We argue that the main reason for this phenomenon is that the proposed model pays much more attention to the global information distributed to the elements captured by the sliding window and ignores the influence of the original values on the future trend to a certain extent due to the strategies of data encoding and utilization of information processed of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Especially, our proposed model outperforms transformer-based methods which at- tract lots of researchers’ attention recently on almost all of the datasets, we consider that temporal data is not similar to images and videos in which there are enormous amount of semantic information needed to be extracted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='6 Overall Performance Comparison Between Pro- posed and Comparative Models In order to comprehensively demonstrate superiority of the proposed model, we utilize Nemenyi test with CD = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='05 � k(k+1) 6Nd in which k is the number of algorithms partic- ipating in the comparison and Nis the number of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Due to lack of some results of models with superscript ∗ on certain datasets, the Nemenyi test is divided into two groups to ensure fairness of comparison and the evaluation results are shown in Friedman test figure at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' It can be easily concluded that the proposed model acquire the most excellent integrated performance on experimental datasets provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='7 Parameter Study Different datasets have their corresponding optimal param- eter setting for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' We selected four uni- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 10 2 4 6 8 10 OUT 5 10 15 20 WINDOW SIZE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='8 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='8 (a) Saugeenday Dataset 2 4 6 8 10 OUT 5 10 15 WINDOW SIZE 2400 2600 2800 3000 3200 3400 3600 (b) Tourism Monthly Dataset (c) Tourism Quarterly Dataset 2 4 6 8 10 OUT 5 10 15 20 WINDOW SIZE 600 700 800 900 (d) US Births Dataset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' MAE Variations When Parameter OUT and Window Size Vary on Univariate Datasets (a) Electricity Weekly Dataset 2 4 6 8 10 OUT 5 10 15 20 WINDOW SIZE 17 18 19 (b) Hospital Dataset 2 4 6 8 10 OUT 5 10 15 20 WINDOW SIZE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='16 (c) Traffic Weekly Dataset 2 4 6 8 10 OUT 5 10 15 20 WINDOW SIZE 800 900 1000 1100 (d) Solar Weekly Dataset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' MAE Variations When Parameter OUT and Window Size Vary on Multivariate Datasets variate and four multivariate data sets for a brief analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='4 and 5, it can be obtained that the performance of the proposed model benefits from a larger window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' And lifting factor OUT has limited influence on the model capability and can reduce the error in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, synthesizing conditions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='6 and 7, increasing the window size dose not necessarily improve model’s per- formance, but larger window sizes can help capture more information and establish the foundation of precise predic- tions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Specifically, on multiple datasets such as M4 Monthly, Quarterly, KDD Cup 2018 and Covid Deaths datasets, error increases considerably when window size equals 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The main reason probably is that the size sacrifices timeliness of data to some extent and is not capable of providing sufficient semantic information to the model so that the proposed model encounters difficulty in producing accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 5 CONCLUSION In this paper, a novel time series forecasting model is pro- posed which consists of encoder part and semi-asymmetric convolutional architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The main role of devised data encoder is assigning elements in original observation time X10° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='3 MAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='9 8 5 10 15 20 WINDOW SIZE1000 MAE 800 600 8 6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 5 10 15 20 SIZE WINDOW1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='7 MAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='6 5 10 15 20 SIZE WINDOW20 19 MAE 18 17 8 6 1 OUT 5 10 15 20 WINDOW SIZE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='17 AE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='15 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content='11 8 6 1 OU 5 10 15 20 SIZE WINDOW1100 MAE 950 800 8 6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 5 2 10 15 20 WINDOW SIZE10 MAE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} 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230784 750 1250 1750 2250 Input Actual Prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Qualitative Prediction Results by the Proposed Method on Aus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Electricity Demand Dataset JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' 8, AUGUST 2015 12 series with positional information so that the model could possess a global view on observation sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Based on processed data, a novel architecture is designed referencing asymmetric convolution and considering variability of time series which enables the model to obtain information at flexible scales on different time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Capturing features with separate range helps model to learn underlying rela- tionship among elements with effective understanding of associated positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Both of them contributes to the outstanding performance of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' To comprehensively demonstrate the performance of the model proposed in this paper, we conduct experiments on 27 univariate and 16 multivariate datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' The experimen- tal results illustrate that the proposed model outperforms comparative methods on most of forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Specifi- cally, the proposed model achieves the highest rank on all competition datasets such as M series, KDD Cup and Web Traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' In addition to these intuitive results, Nemenyi test also strongly demonstrates the excellent performance of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Besides, we also investigate the influences of the two main parameters on 24 datasets to further explain settings of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Nevertheless, the proposed model achieves relatively satisfying performance in most of forecasting experiments, there are still some potentials in it which can be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' We think we may be able to improve the model in two possible directions: 1): The attention mechanism can be introduced into the model to help the model better un- derstand semantic information in time series, 2): A recurrent architecture of convolutional neural network is expected to be developed to better memory past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' ACKNOWLEDGMENT REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Sharma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' Maheshwari, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQf-jgZ/content/2301.13691v1.pdf'} +page_content=' P.' metadata={'source': 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a/XdFRT4oBgHgl3EQfNjdq/content/tmp_files/2301.13510v1.pdf.txt b/XdFRT4oBgHgl3EQfNjdq/content/tmp_files/2301.13510v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..efe77d94a19a07c6c27efa603ea469b1df3f29df --- /dev/null +++ b/XdFRT4oBgHgl3EQfNjdq/content/tmp_files/2301.13510v1.pdf.txt @@ -0,0 +1,1181 @@ +Published as a conference paper at ICLR 2023 +SDF-FORMER: MONOCULAR SCENE RECONSTRUC- +TION WITH 3D SDF TRANSFORMERS +Weihao Yuan, Xiaodong Gu, Heng Li, Zilong Dong, Siyu Zhu +Alibaba Group +{qianmu.ywh, dadong.gxd, baoshu.lh, list.dzl, siting.zsy} +@alibaba-inc.com +ABSTRACT +Monocular scene reconstruction from posed images is challenging due to the com- +plexity of a large environment. Recent volumetric methods learn to directly pre- +dict the TSDF volume and have demonstrated promising results in this task. How- +ever, most methods focus on how to extract and fuse the 2D features to a 3D fea- +ture volume, but none of them improve the way how the 3D volume is aggregated. +In this work, we propose an SDF transformer network, which replaces the role +of 3D CNN for better 3D feature aggregation. To reduce the explosive compu- +tation complexity of the 3D multi-head attention, we propose a sparse window +attention module, where the attention is only calculated between the non-empty +voxels within a local window. Then a top-down-bottom-up 3D attention network +is built for 3D feature aggregation, where a dilate-attention structure is proposed +to prevent geometry degeneration, and two global modules are employed to equip +with global receptive fields. The experiments on multiple datasets show that this +3D transformer network generates a more accurate and complete reconstruction, +which outperforms previous methods by a large margin. Remarkably, the mesh ac- +curacy is improved by 41.8%, and the mesh completeness is improved by 25.3% +on the ScanNet dataset. 1 +1 +INTRODUCTION +Monocular 3D reconstruction is a classical task in computer vision and is essential for numerous +applications like autonomous navigation, robotics, and augmented/virtual reality. Such a vision task +aims to reconstruct an accurate and complete dense 3D shape of an unstructured scene from only a +sequence of monocular RGB images. While the camera poses can be estimated accurately with the +state-of-the-art SLAM (Campos et al., 2021) or SfM systems (Schonberger & Frahm, 2016), a dense +3D scene reconstruction from these posed images is still a challenging problem due to the complex +geometry of a large-scale environment, such as the various objects, flexible lighting, reflective sur- +faces, and diverse cameras of different focus, distortion, and sensor noise. Many previous methods +reconstruct the scenario in a multi-view depth manner (Yao et al., 2018; Chen et al., 2019; Duzceker +et al., 2021). They predict the dense depth map of each target frame, which can estimate accurate +local geometry but need additional efforts in fusing these depth maps (Murez et al., 2020; Sun et al., +2021), e.g., solving the inconsistencies between different views. +Recently, some methods have tried to directly regress the complete 3D surface of the entire +scene (Murez et al., 2020; Sun et al., 2021) from a truncated signed distance function (TSDF) +representation. They first extract the 2D features with 2D convolutional neural networks (CNN), +and then back-project the features to 3D space. Afterward, the 3D feature volume is processed by +a 3D CNN network to output a TSDF volume prediction, which is extracted to a surface mesh by +marching cubes (Lorensen & Cline, 1987). This way of reconstruction is end-to-end trainable, and +is demonstrated to output accurate, coherent, and complete meshes. In this paper, we follow this +volume-based 3D reconstruction path and directly regress the TSDF volume. +Inspired by recent successes of vision transformer (Vaswani et al., 2017; Dosovitskiy et al., 2020), +some approaches (Bozic et al., 2021; Stier et al., 2021) have adopted this structure in 3D recon- +1Project page: https://weihaosky.github.io/sdfformer +1 +arXiv:2301.13510v1 [cs.CV] 31 Jan 2023 + +Published as a conference paper at ICLR 2023 +Figure 1: The overview of the 3D reconstruction framework. The input images are extracted to features by +a 2D backbone network, then the 2D features are back-projected and fused to 3D feature volumes, which are +aggregated by our 3D SDF transformer and generate the reconstruction in a coarse-to-fine manner. +struction, but their usages are all limited to fusing the 2D features from different views while the +aggregation of the 3D feature volumes is still performed by the 3D CNN. In this paper, we claim +that the aggregation of 3D feature volume is also critical, and the evolution from 3D CNN to 3D +multi-head attention could further improve both the accuracy and completeness of the reconstruc- +tion. Obviously, the limited usage of 3D multi-head attention in 3D feature volume aggregation +is mainly due to its explosive computation. Specifically, the attention between each voxel and any +other voxel needs to be calculated, which is hard to be realized in a general computing platform. +This is also the reason why there are only a few applications of 3D transformers in solving 3D tasks. +In this work, to address the above challenges and make the 3D transformer practical for 3D scene +reconstruction, we propose a sparse window multi-head attention structure. Inspired by the sparse +CNN (Yan et al., 2018), we first sparsify the 3D feature volume with predicted occupancy, in which +way the number of the voxels is reduced to only the occupied ones. Then, to compute the attention +score of a target voxel, we define a local window centered on this voxel, within which the non-empty +voxels are considered for attention computing. In this way, the computation complexity of the 3D +multi-head attention can be reduced by orders of magnitude, and this module can be embedded into +a network for 3D feature aggregation. Therefore, with this module, we build the first 3D transformer +based top-down-bottom-up network, where a dilate-attention module and its inverse are used to +downsample and upsample the 3D feature volume. In addition, to make up for the local receptive +field of the sparse window attention, we add a global attention module and a global context module +at the bottom of this network since the size of the volume is very small at the bottom level. With +this network, the 3D shape is estimated in a coarse-to-fine manner of three levels, as is displayed in +Figure 1. To the best of our knowledge, this is the first paper employing the 3D transformer for 3D +scene reconstruction from a TSDF representation. +In the experiments, our method is demonstrated to outperform previous methods by a significant +margin on multiple datasets. Specifically, the accuracy metric of the mesh on the ScanNet dataset +is reduced by 41.8%, from 0.055 to 0.032, and the completeness metric is reduced by 25.3%, from +0.083 to 0.062. In the qualitative results, the meshes reconstructed by our method are dense, accu- +rate, and complete. The main contributions of this work are then summarized as follows: +• We propose a sparse window multi-head attention module, with which the computation complexity +of the 3D transformer is reduced significantly and becomes feasible. +• We propose a dilate-attention structure to avoid geometry degeneration in the downsampling, with +which we build the first top-down-bottom-up 3D transformer network for 3D feature aggregation. +This network is further improved with a bottom-level global attention module and a global context +encoding module. +• This 3D transformer is employed to aggregate the 3D features back-projected from the 2D features +of an image sequence in a coarse-to-fine manner, and predict TSDF values for accurate and complete +3D reconstruction. This framework shows a significant improvement on multiple datasets. +2 +RELATED WORK +Depth-based 3D Reconstruction. In traditional methods, reconstructing a 3D model of a scene +usually involves depth estimating for a series of images, and then fusing these depths together into +a 3D data structure (Sch¨onberger et al., 2016). After the rising of deep learning, many works have +2 + +V2 +V2 +3D SDF +F2 +Fusion +02 +Former +3D SDF +F1 +V1 +Fusion +V1 +V1 +01 +Former +2D +F +3D SDF +Backbone +Fusion +Vo +VO +So +Former +2D +DPublished as a conference paper at ICLR 2023 +tried to estimate accurate and dense depth maps with deep neural networks (Yao et al., 2018; Wang +& Shen, 2018; Chen et al., 2019; Im et al., 2019; Yuan et al., 2021; 2022; Long et al., 2021). They +usually estimate the depth map of the reference image by constructing a 3D cost volume from sev- +eral frames in a local window. Also, to leverage the information in the image sequence, some other +methods try to propagate the message from previously predicted depths utilizing probabilistic filter- +ing (Liu et al., 2019), Gaussian process (Hou et al., 2019a), or recurrent neural networks (Duzceker +et al., 2021). Although the predicted depth maps are increasingly accurate, there is still a gap be- +tween these single-view depths and the complete 3D shape. Post mesh generation like Poisson +reconstruction (Kazhdan & Hoppe, 2013), Delaunay triagulation (Labatut et al., 2009), and TSDF +fusion (Newcombe et al., 2011) are proposed to solve this problem, but the inconsistency between +different views is still a challenge. +Volume-based 3D Reconstruction. To avoid the depth estimation and fusion in 3D reconstruction, +some methods try to directly regress a volumetric data structure end-to-end. SurfaceNet (Ji et al., +2017) encodes the camera parameters together with the images to predict a 3D surface occupancy +volume with 3D convolutional networks. Afterward, Atlas (Murez et al., 2020) back-projects the +2D features of all images into a 3D feature volume with the estimated camera poses, and then feeds +this 3D volume into a 3D U-Net to predict a TSDF volume. Then NeuralRecon (Sun et al., 2021) +improves the efficiency by doing this within a local window and then fusing the prediction together +using a GRU module. Recently, to improve the accuracy of the reconstruction, some methods also +introduce transformers to do the fusion of 2D features from different views (Bozic et al., 2021; Stier +et al., 2021). However, their transformers are all limited in 2D space and used to process 2D features, +which is not straightforward in the 3D reconstruction task. +There are also some methods for object 3D shape prediction, which can infer the 3D shape of objects +with only a few views (Xie et al., 2020; Wang et al., 2021a). But the network of these methods can +only infer the shape of one category of small objects. Lately, some works represent the 3D shape +with an implicit network, and optimize the implicit representation by neural rendering (Yariv et al., +2020; Wang et al., 2021b; Yariv et al., 2021). These methods could obtain a fine surface of an object +with iterative optimization, but with the cost of a long-time reconstruction. +Transformers in 3D Vision. The transformer structure (Vaswani et al., 2017) has attracted a lot of +attention and achieved many successes in vision tasks (Dosovitskiy et al., 2020; Liu et al., 2021). +Most of them, nevertheless, are used for 2D feature extraction and aggregation. Even in 2D feature +processing, the computation complexity is already quite high, so many works are proposed to reduce +the resource-consuming (Dosovitskiy et al., 2020; Liu et al., 2021). Directly extending the trans- +former from 2D to 3D would cause catastrophic computation. Thus most works are only carefully +performed on resource-saving feature extraction, e.g., the one-off straightforward feature mapping +without any downsampling or upsampling (Wang et al., 2021a), where the size of the feature vol- +ume remains unchanged, or the top-down tasks with only downsampling (Mao et al., 2021), where +the size of the feature volume is reduced gradually. In 3D reconstruction, however, a top-down- +bottom-up structure is more reasonable for feature extraction and shape generation, as in most of the +3D-CNN-based structures (Murez et al., 2020; Sun et al., 2021; Stier et al., 2021). So in this work, +we design the first 3D transformer based top-down-bottom-up structure for improving the quality +of 3D reconstruction. In addition, a sparse window multi-head attention mechanism is proposed to +save the computation cost. Although the sparse structure can handle the highly-sparse data, like the +object detection of Lidar points (Mao et al., 2021), it is not suitable for processing a relatively-dense +data, like a mesh of an indoor scene. Therefore, a sparse window structure is needed in 3D scene +reconstruction, where a dense surface within a window could be sufficiently aggregated. +3 +METHOD +3.1 +OVERVIEW +The overview framework of our method is illustrated in Figure 1. Given a sequence of images +{Ii}N +i=1 of a scene and the corresponding camera intrinsics {Ki}N +i=1 and extrinsics {Pi}N +i=1, we +first extract the image features {Fi}N +i=1 in 2D space in three levels, and then back project these 2D +features to 3D space, which are fused to three feature volumes in the coarse, medium, and fine levels, +respectively. Afterward, these three feature volumes are aggregated by our SDF 3D transformer in +a coarse-to-fine manner. At the coarse and medium levels, the output of the 3D transformer is +3 + +Published as a conference paper at ICLR 2023 +Figure 2: (a) Illustration of the sparse window attention. For calculating the attention of the current voxel (in +orange), we first sparsify the volume using the occupancy prediction from the coarser level, and then search the +occupied voxels (in dark blue) within a small window. The attention is hence computed based on only these +neighbor occupied voxels. (b) Illustration of the dilate-attention in a 2D slice. We dilate the occupied voxels +and calculate the attention of these dilated voxels (in yellow) to maintain the geometry structure. +two occupancy volumes O2, O1, while at the fine level, the output is the predicted TSDF volume +S0. The coarse occupancy volume O2 and the medium occupancy volume O1 store the occupancy +values o ∈ [0, 1] of the voxels, which are used to sparsify the finer level. Therefore, the feature +volumes could be processed sparsely to reduce the computation complexity. Finally, the predicted +mesh is extracted using marching cubes (Lorensen & Cline, 1987) from the TSDF volume S0. +3.2 +FEATURE VOLUME CONSTRUCTION +The 2D features {Fl +i}N +i=1 in three levels l = 0, 1, 2 are extracted by a feature pyramid network (Lin +et al., 2017) with the MnasNet-B1 (Tan et al., 2019) as the backbone. The resolution of the features +at these three levels are 1 +4, 1 +8, 1 +16, respectively. Then following Murez et al. (2020), we back project +the 2D features to 3D space with the camera parameters {Ki}N +i=1 and {Pi}N +i=1, generating 3D +feature volumes {Vl +i}N +i=1 of size NX × NY × NZ. +In previous work, usually the fusion of these feature volumes from different views is computed by +taking the average (Murez et al., 2020; Sun et al., 2021). However, the back-projected features +from different views contribute differently to the 3D shape, e.g., the view with a bad viewing angle +and the voxels far from the surface. Therefore, a weighted average is more reasonable than taking +the average. To compute these weights, for each voxel we calculate the variance of the features of +different views by +Varl +i = (Vl +i − V +l)2, +(1) +where V +l is the average of the features of all views. Then we feed the features and the variance into +a small MLP to calculate the weights Wi, which are used to compute a weighted average of the +features from different views as +Vl +w = 1 +N +� +i +Vl +i × SoftMax(Wi), +(2) +where × denotes element-wise multiplication. +Inspired by Yao et al. (2018), we also calculate the total variance of all feature volumes and then +concatenate it with the weighted average to the final feature volumes, as +Vl = {Vl +w, 1 +N +� +i +Varl +i}, +(3) +3.3 +SPARSE WINDOW MULTI-HEAD ATTENTION +The multi-head attention structure has been shown to be effective in many vision tasks (Dosovitskiy +et al., 2020; Liu et al., 2021). Most of them, however, are limited to 2D feature processing rather +than 3D feature processing. This is because the computation complexity of the multi-head attention +is usually higher than convolutional networks, which problem is further enlarged in 3D features. To +compute this for a 3D feature volume, the attentions between a voxel and any other voxels need to +be computed, i.e., NX ×NY ×NZ attentions for one voxel and NX ×NY ×NZ ×NX ×NY ×NZ +attentions for all voxels, which is extremely large and hard to be realized in regular GPUs. +4 + +Published as a conference paper at ICLR 2023 +Figure 3: The structure of the SDF transformer. “S-W- +Attn” denotes sparse window attention. +To deal with this problem and make the multi- +head attention of 3D volumes feasible, we pro- +pose to use a sparse window structure to cal- +culate the attention. +As is displayed in Fig- +ure 1, in the medium and the fine level, we +sparsify the volumes using the occupancy pre- +diction O2, O1, and only compute the attention +of the non-empty voxels. In addition, consider- +ing that the nearby voxels contribute more to +the shape of the current voxel and the distant +voxels contribute less, we only calculate the at- +tention within a local window of each voxel, +as is shown in Figure 2. +Therefore, we are +able to only calculate the multi-head attention +of the occupied voxels within a small window, +in which way the computation complexity is re- +duced significantly. +Specifically, for any non-empty voxel vi in the +feature volume V , we first search all non-empty +voxels within a n × n × n window centered on +this voxel and get the neighbor voxels {vj, j ∈ +Ω(i)}. Then the query, key, and value embeddings are calculated as +Qi = Lq(V (vi)), Kj = Lk(V (vj)), Vj = Lv(V (vj)), +(4) +where Lq, Lk, Lv are the linear projection layers. +For the position embedding P, we hope to block the influence from the scale of the 3D world +coordinates. Hence we compute it based on the relative voxel position in the volume rather than +based on the real-world coordinates (Mao et al., 2021), as +Pj = Lp(vj − vi). +(5) +Then the attention is calculated as +Attention(vi) = +� +j∈Ω(i) +SoftMax(Qi(Kj + Pj)/ +√ +d)(Vj + Pj). +(6) +In this case, the computation complexity is reduced from +O3D-Attn = NX × NY × NZ × NX × NY × NZ × O(ij), +(7) +to +OSW-3D-Attn = Noccu × noccu × O(ij), +(8) +where O(ij) is the complexity of one attention computation between voxel vi and vj, Noccu is the +number of occupied voxels in the volume, and noccu is the number of occupied voxels within the +local window. Assuming that the occupancy rate of the volume is 10% and the window size is +1 +10 of the volume size, the computation complexity of the sparse window attention would be only +n3/10 +10NXNY NZ = +1 +100000 of the dense 3D attention. +3.4 +SDF 3D TRANSFORMER +Limited by the high resource-consuming of the multi-head attention, most of the previous works +related to 3D transformers are only carefully performed on resource-saving feature processing, e.g., +the one-off straightforward feature mapping without any downsampling or upsampling (Wang et al., +2021a), where the size of feature volumes remains unchanged, or the top-down tasks with only +downsampling (Mao et al., 2021), where the size of feature volumes is reduced gradually. In 3D +reconstruction, however, a top-down-bottom-up structure is more reasonable for feature extraction +and prediction generation, as in most of the 3D-CNN-based structures (Murez et al., 2020; Sun et al., +2021; Stier et al., 2021). So in this work, we design the first 3D transformer based top-down-bottom- +up structure, as is shown in Figure 3. +5 + +V +TSDF +Down +Dilate Attn +Inv +Dilate Attn +3D S-W-Attn +3D S-W-Attn +Down +Dilate Attn +Inv +3D S-W-Attn +3D S-W-Attn +Down +Dilate Attn +Inv +3D S-W-Attn +3D S-W-Attn +Down +Dilate Attn +Inv +3D S-W-Attn +3D Global Attn +Global EncodePublished as a conference paper at ICLR 2023 +Baseline ++ SDF Transformer ++ Post Dilate Attention +Ground Truth +Figure 4: Ablation study on the ScanNet dataset. +Taking the network for the fine volume (V 0 in Figure 1) as an example, there are four feature levels +in total, i.e. +1 +2, 1 +4, 1 +8, 1 +16, as shown in Figure 3. In the encoder part, at each level, a combination +of downsampling and dilate-attention is proposed to downsample the feature volume. Then two +blocks of the sparse window multi-head attention are used to aggregate the feature volumes. At +the bottom level, a global attention block is employed to make up the small receptive field of the +window attention, and a global context encoding block is utilized to extract the global information. +In the decoder part, we use the inverse sparse 3D CNN to upsample the feature volume, i.e., we store +the mapping of the down flow and now restore the spatial structure by inversing the sparse 3D CNN +in the dilate-attention. Therefore, the final shape after the up flow should be the same as the input. +Similar to FPN (Lin et al., 2017), the features in the down flow are also added to the upsampled +features in the corresponding level. To enable the deformation ability, a post-dilate-attention block +is equipped after the down-up flow. Finally, a submanifold 3D CNN head with Tanh activation is +appended to output the TSDF prediction. For the coarse volume V 2 and medium volume V 1, two +and three-level of similar structures with Sigmoid activation are adopted. +Dilate-attention. The direct downsampling of a sparse structure is prone to losing geometry struc- +ture. To deal with this, between each level we first downsample the feature volume, and then dilate +the volume with a sparse 3D CNN with the kernel size of 3, which calculates the output if any voxel +within its kernel is non-empty. The dilation operation alone may also harm the geometry, since it +may add some wrong voxels into the sparse structure. Thus we calculate the sparse window atten- +tion of the dilated voxels, such that the voxels far from the surface would get low scores and do +not contribute to the final shape. The dilated voxels are then joined to the downsampled volume by +concatenating the voxels together. With this dilate-attention module, the 3D shape is prevented from +collapsing. Without this module, the network performs badly and only generates a degraded shape. +Global attention and global context encoding. Since the attention blocks in the top-down flow +are all local-window based, there could be a lack of the global receptive field. Considering the +resolution of the bottom level is not high, we equip with a global attention block at the bottom level, +i.e., we calculate the attention between each non-empty voxel and any other non-empty voxel in +the volume. This could build the long-range dependency missing in the sparse window attention +blocks. In addition, we use the multi-scale global averaging pooling (Zhao et al., 2017) of scales +1, 2, 3 to extract the global context code of the scene. This encoding module could aggregate the +global information and explain the illumination, global texture, and global geometry style. +3.5 +LOSS FUNCTION +The final TSDF prediction S0 is supervised by the log L1 distance between the prediction and the +ground truth as L0 = | log S0 − log �S|. +To supervise the occupancy predictions O2, O1 in the coarse and medium levels, we generate the +occupancy volumes based on the TSDF values. Specifically, the voxels with TSDF of −1 ∼ 1 are +regarded as occupied, and the values are set to 1, otherwise set to 0. Then a binary cross-entropy +loss is calculated between the prediction and the ground truth as: Ll = −� +Ol log Ol, l = 1, 2. +To supervise the averaging weights Wl +i, we use the occupancy in the back-projection following +Stier et al. (2021). Intuitively, when the feature is back-projected from a 2D image to the 3D space +along the camera ray using multiple depth values, we hope the voxels close to the mesh surface have +bigger weights in the fusion. Therefore, the 3D position is regarded as occupied if the difference +between the project depth and the true depth from the depth map is smaller than the TSDF truncation +distance. Then the cross entropy loss is applied to the weights and the occupancy: +Ll +w = −� +Ol +i log σ(Wl +i), l = 1, 2, 3, +(9) +where σ denotes Sigmoid, and � +Ol +i is the ground truth occupancy in the back-projection of image Ii. +6 + +Published as a conference paper at ICLR 2023 +Method +Acc ↓ +Comp ↓ +Chamfer ↓ +Prec ↑ +Recall ↑ +F-score ↑ +DeepVideoMVS (Duzceker et al., 2021) +0.079 +0.133 +0.106 +0.521 +0.454 +0.474 +Atlas (Murez et al., 2020) +0.068 +0.098 +0.083 +0.640 +0.539 +0.583 +NeuralRecon (Sun et al., 2021) +0.054 +0.128 +0.091 +0.684 +0.479 +0.562 +VoRTX (Stier et al., 2021) +0.054 +0.090 +0.072 +0.708 +0.588 +0.641 +Ours +0.049 +0.068 +0.058 +0.754 +0.664 +0.705 +Colmap (Sch¨onberger et al., 2016) +0.102 +0.119 +0.111 +0.509 +0.474 +0.489 +MVDepthNet (Wang & Shen, 2018) +0.129 +0.083 +0.106 +0.443 +0.487 +0.460 +GP-MVS (Hou et al., 2019a) +0.129 +0.080 +0.105 +0.453 +0.510 +0.477 +DPSNet (Im et al., 2019) +0.119 +0.076 +0.098 +0.474 +0.519 +0.492 +ESTDepth (Long et al., 2021) +0.127 +0.075 +0.101 +0.456 +0.542 +0.491 +DeepVideoMVS (Duzceker et al., 2021) +0.107 +0.069 +0.088 +0.541 +0.592 +0.563 +Atlas (Murez et al., 2020) +0.072 +0.076 +0.074 +0.675 +0.605 +0.636 +NeuralRecon (Sun et al., 2021) +0.051 +0.091 +0.071 +0.630 +0.612 +0.619 +TransformerFusion (Bozic et al., 2021) +0.055 +0.083 +0.069 +0.728 +0.600 +0.655 +Ours +0.032 +0.062 +0.047 +0.829 +0.694 +0.754 +Table 1: Evaluation of the 3D meshes on ScanNet. The upper part follows the evaluation in Sun et al. (2021) +while the lower part follows Bozic et al. (2021). The metric definitions are explained in the appendix. +Method Abs Rel ↓ Abs Diff ↓ Sq Rel ↓ RMSE ↓ δ-1.25 ↑ δ-1.252 ↑ δ-1.253 ↑ +Colmap (Sch¨onberger et al., 2016) +0.137 +0.264 +0.138 +0.502 +0.834 +− +− +MVDepthNet (Wang & Shen, 2018) +0.098 +0.191 +0.061 +0.293 +0.896 +0.977 +0.994 +GP-MVS (Hou et al., 2019a) +0.130 +0.239 +0.339 +0.472 +0.906 +0.967 +0.980 +DPSNet (Im et al., 2019) +0.087 +0.158 +0.035 +0.232 +0.925 +0.984 +0.995 +Atlas (Murez et al., 2020) +0.065 +0.124 +0.043 +0.251 +0.936 +0.971 +0.986 +NeuralRecon (Sun et al., 2021) +0.065 +0.106 +0.031 +0.195 +0.948 +0.961 +0.975 +Vortx (Stier et al., 2021) +0.061 +0.096 +0.038 +0.205 +0.943 +0.973 +0.987 +Ours +0.051 +0.086 +0.033 +0.199 +0.958 +0.980 +0.990 +Table 2: Evaluation of the 2D depth maps on the ScanNet dataset. The upper part shows the results of depth- +based methods, while the lower part shows volumetric methods, whose depths are rendered from the meshes. +4 +EXPERIMENTS +4.1 +EXPERIMENTS SETUP +Our work is implemented in Pytorch and trained on Nvidia V100 GPUs. The network is optimized +with the Adam optimizer (β1 = 0.9, β2 = 0.999) with learning rate of 1 × 10−4. For a fair +comparison with previous methods, the voxel size of the fine level is set to 4cm, and the TSDF +truncation distance is set to triple the voxel size. Thus the voxel size of the medium and the coarse +levels are 8 cm and 16 cm, respectively. For the balance of efficiency and receptive field, the window +size of the sparse window attention is set to 10. For the view selection, we first follow Hou et al. +(2019b) to remove the redundant views, i.e., a new incoming frame is added to the system only if +its relative translation is greater than 0.1 m and the relative rotation angle is greater than 15 degree. +Then if the number of the remaining views exceeds the upper limit, a random selection is adopted +for memory efficiency. The view limit is set to 20 in the training, which means twenty images are +input to the network for one iteration, while the limit for testing is set to 150. Our framework runs +at an online speed of 75 FPS for the keyframes. Detailed efficiency experiments are reported in the +supplemental materials. +ScanNet (Dai et al., 2017) is a large-scale indoor dataset composed of 1613 RGB-D videos of 806 +indoor scenes. We follow the official train/test split, where there are 1513 scans used for training +and 100 scans used for testing. TUM-RGBD (Sturm et al., 2012) and ICL-NUIM (Handa et al., +2014) are also two datasets composed of RGB-D videos but with small-number scenes. Therefore, +following previous methods (Stier et al., 2021), we only perform the generalization evaluation of the +model trained on ScanNet on these two datasets, where 13 scenes of TUM-RGBD and 8 scenes of +ICL-NUIM are used. +4.2 +EVALUATION +To compare with previous methods, we evaluate the proposed method on the ScanNet test set. The +quantitative results are presented in Table 1 and the qualitative comparison are displayed in Figure 5. +7 + +Published as a conference paper at ICLR 2023 +Figure 5: The qualitative results on the ScanNet dataset. Texture-less rendering is displayed in the appendix. +We first directly evaluate the reconstructed meshes with the ground-truth meshes, and obtain a sig- +nificant improvement from previous methods, improving from F-score = 0.641 to F-score = 0.705, +as shown in Table 1. Then following Bozic et al. (2021), we add the same occlusion mask at evalua- +tion to avoid penalizing a more complete reconstruction, which is because the ground-truth meshes +are incomplete due to unobserved and occluded regions, while our method could reconstruct a more +complete 3D shape, as shown in Figure 5. This results in a more reasonable evaluation, as in the sec- +ond part of Table 1. The improvement is further enlarged, from F-score = 0.655 to F-score = 0.754 +compared to previous best method. The accuracy error is decreased from 0.055 m to 0.032 m, which +is almost half (41.8%) of the previous best method, while the completeness error is decreased by +25.3%, from 0.083 m to 0.062 m. This owes to the feature aggregating ability of the proposed 3D +SDF transformer, which can predict a more accurate 3D shape. This is also demonstrated in the +generalization experiments on ICL-NUIM and TUM-RGBD datasets, as shown in Figure 3. +After evaluating the reconstructed meshes, we also evaluate the depth accuracy of our method. Since +our method does not predict the depth maps explicitly, we render the predicted 3D shape to the +8 + +Published as a conference paper at ICLR 2023 +image planes and get the depth maps, following previous methods (Murez et al., 2020). The results +are shown in Table 2, from which we can see our method decreases the error a lot from previous +methods. The relative error is reduced by 16.4%, from 0.061 to 0.051. The accuracy of the depth +maps also demonstrates the accurate feature analysis ability of the proposed 3D SDF transformer. +Method Acc ↓ Comp ↓ Prec ↑ Recall ↑ F-score ↑ +ICL +Atlas 0.175 +0.314 +0.280 +0.194 +0.229 +NeuralRecon 0.215 +1.031 +0.214 +0.036 +0.058 +VoRTX 0.102 +0.146 +0.449 +0.375 +0.408 +Ours 0.083 0.142 0.522 0.390 +0.447 +TUM +Atlas 0.208 +2.344 +0.360 +0.089 +0.132 +NeuralRecon 0.130 +2.528 +0.382 +0.075 +0.115 +Vortx 0.175 +0.314 +0.280 +0.194 +0.229 +Ours 0.129 +0.455 +0.406 +0.173 +0.254 +Table 3: Generalization experiments on the ICL-NUIM and +TUM-RGBD datasets. +From the qualitative visualization in Fig- +ure 5, we can see our method can predict +a complete and accurate 3D shape. Pre- +vious methods which can recover a com- +plete mesh usually reconstruct a smooth +3D shape with losing some details (Murez +et al., 2020). However, our method could +predict a more complete mesh than the +ground truth, while the details of the 3D +shapes are better recovered. Please note +that for a fair comparison, the voxel size is +set to 4 cm, such that it is hard to recon- +struct the geometry details less than 4 cm. +4.3 +ABLATION STUDY +Method Acc ↓ Comp ↓ Prec ↑ Recall ↑ F-score ↑ +Baseline 0.056 +0.089 +0.698 +0.587 +0.636 ++ Var Fusion 0.054 +0.090 +0.713 +0.594 +0.647 ++ SDF Former 0.036 +0.065 +0.807 +0.671 +0.732 ++ Global 0.033 +0.064 +0.823 +0.676 +0.741 ++ Post-Dila-Attn 0.032 +0.062 +0.829 +0.694 +0.754 +Window Size +1 0.052 +0.086 +0.721 +0.604 +0.656 +3 0.044 +0.078 +0.768 +0.636 +0.695 +5 0.037 +0.069 +0.799 +0.660 +0.730 +8 0.033 +0.065 +0.822 +0.682 +0.746 +10 0.032 +0.062 +0.829 +0.694 +0.754 +Table 4: Ablation study on the ScanNet dataset. Components +are added one by one in the upper part. +SDF transformer. +To verify the effec- +tiveness of the proposed SDF transformer, +we first build a baseline model with the +same structure as Figure 1, but the 3D +SDF transformer is replaced by a UNet +structure of 3D CNN. Adding the variance +fusion would improve the mesh in some +clutter areas and slightly increase the per- +formance. +Then we add a base version +of the SDF transformer, which does not +include the global module and the post- +dilate-attention module. The performance is significantly improved with this module, as is shown in +Table 4 and Figure 4. The reconstructed meshes possess much more geometry details compared to +the baseline. +Global module. We next add the global module, including the bottom-level global attention and the +global context code. The sparse window attention block can only obtain the long-range dependency +within a local window. Thus it may have problems when it can not get enough information within +this local window, e.g., the texture-free regions. Also, the global module could reason the global +information like the illumination and the texture style. +Dilate attention. The dilate attention module is crucial in the SDF transformer, so we can not re- +move all the dilate attention blocks. That will destroy the whole framework and generate a degraded +3D shape. Therefore, we only ablate the post dilate attention block after the down-up flow. This +block could deform the shape and make it more complete, e.g., making up the crack as shown in +Figure 4. From the quantitative results in Table 4, we can also see the improvement of completeness. +Window size. As shown in Table 4, we study the impact of the window size of the attention. It +is expected that a larger window size would generate a better result, since the range of the depen- +dency is longer, but with the cost of more resource consumption. We choose 10 as the default size, +considering that the performance improvement is minor after that. +5 +CONCLUSION +We propose the first top-down-bottom-up 3D transformer for 3D scene reconstruction. A sparse win- +dow attention module is proposed to reduce the computation, a dilate attention module is proposed +to avoid geometry degeneration, and a global module at the bottom level is employed to extract the +global information. This structure could be used to aggregate any 3D feature volume, thus it could +be applied to more 3D tasks in the future, such as 3D segmentation. +9 + +Published as a conference paper at ICLR 2023 +REFERENCES +Aljaz Bozic, Pablo Palafox, Justus Thies, Angela Dai, and Matthias Nießner. Transformerfusion: +Monocular rgb scene reconstruction using transformers. Advances in Neural Information Pro- +cessing Systems, 34:1403–1414, 2021. +Carlos Campos, Richard Elvira, Juan J G´omez Rodr´ıguez, Jos´e MM Montiel, and Juan D Tard´os. +Orb-slam3: An accurate open-source library for visual, visual–inertial, and multimap slam. IEEE +Transactions on Robotics, 37(6):1874–1890, 2021. +Rui Chen, Songfang Han, Jing Xu, and Hao Su. Point-based multi-view stereo network. 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The ground-truth occupancy volumes are generated based on the ground-truth +TSDF volumes. The voxels with TSDF value of [−1, 1] are regarded as occupied and set to 1, +otherwise set to 0. In the training or the inference, after the occupancy volume is predicted in the +coarser level, the voxels of occupancy value less than 0.5 are regarded as empty and discarded, while +the remaining voxels are regarded as occupied and transmitted to the next level. The sparse volume +is stored in a hash table, where the key of the table is the hash value of the voxel, and the value of +the table stores the corresponding feature. In the coarsest level, all voxels are regarded as non-empty +and stored in the hash table, which does not consume much memory because the size of the volume +is small. +Training and inference. The training and inference are performed in a similar way. For a given +sequence of images, first a view selection is performed to select images with translation greater than +0.1m and rotation greater than 15 degrees. Then a random selection is adopted from the remaining +images if the number exceeds the upper limit. These images are then fed to the 2D backbone for +feature extraction, after which the features are fused to a 3D volume and fed to the 3D part to produce +the TSDF volume prediction. The final mesh is extracted by marching cubes from the TSDF volume. +This process is the same as previous methods like Atlas, TransformerFusion or VORTX. +For the number of the upper limit of the images, actually any number for the sequence length is +okay for our framework, although more images lead to a better reconstruction of a scene. In our +experiments, the number for training is set to 20 and the number for inference is set to 150. +Method +Per Frame Time +Per Scene Time +FPS +Atlas (Murez et al., 2020) +71 ms +840 ms +14 +NeuralRecon (Sun et al., 2021) +30 ms +0 ms +33 +TransformerFusion (Bozic et al., 2021) +130.5 ms +243.3 ms +7 +VoRTX (Stier et al., 2021) +71.4 ms +231.7 ms +14 +Ours +13.3 ms +286.4 ms +75 +Table 5: Efficiency experiments. +A.2 +EFFICIENCY +The runtime analysis is presented in Table 5. For a fair comparison to previous methods, the time is +tested on a chunk of size 1.5×1.5×1.5 m3 with an Nvidia RTX 3090 GPU. Our framework consists +of two parts: one is the per-frame part, including the feature extraction of the 2D images; the other +one is the per-scene part, including the feature fusion, 3D feature processing, and mesh generation. +The per-frame model runs for every keyframe, i.e., it keeps running whenever a new keyframe +comes. Differently, the per-scene model runs only once for generating a mesh reconstruction of a +scene, i.e., it only works after all frames are fed, or when we need to output a mesh. Therefore, the +online speed of a normal running is 75 FPS, which only performs the mesh generation once at the +end. +A.3 +METRICS +The definitions of the 2D metrics and 3D metrics used for evaluation are explained in Table 6. +A.4 +LIMITATIONS +Due to the volume representation, our framework is limited by the trade-off between the resolution +of the volume and the memory consumption. A smaller voxel size would cost much more memory. +The voxel size is set to 4 cm, such that the geometry details less than 4 cm are hard to be recovered. +13 + +Published as a conference paper at ICLR 2023 +Metrics +Definition +Abs Rel +1 +n +� |d − d∗|/d∗ +Abs Diff +1 +n +� |d − d∗| +Sq Rel +1 +n +� |d − d∗|2/d∗ +RMSE +� +1 +n +� |d − d∗|2 +δ − 1.25i +1 +n +�(max( d +d∗ , d∗ +d ) < 1.25i) +Acc +meanp∈P (minp∗∈P ∗ ||p − p∗||) +Comp +meanp∗∈P ∗(minp∈ ||p − p∗||) +Chamfer distance +Acc + Comp +2 +Prec +meanp∈P (minp∗∈P ∗ ||p − p∗|| < 0.05) +Recall +meanp∗∈P ∗(minp∈ ||p − p∗|| < 0.05) +F-score +2×Prec×Recall +Prec + Recall +Table 6: Metric definitions. n denotes the number of pixels with both valid ground truth and prediction, d and +d∗ denote the predicted and the ground-truth depths, p and p∗ denote the predicted and the ground-truth point +clouds. +A.5 +ROBUSTNESS TO THE POSE NOISE +Our method is based on the given accurate camera poses, which is the same as previous state-of- +the-art methods like Atlas (Murez et al., 2020), NeuralRecon (Sun et al., 2021), and Transformer- +Fusion (Bozic et al., 2021), where the camera poses are obtained by the standard SfM or SLAM +systems. To inspect the robustness of our method to the pose errors, we add the Gaussian noise to +the camera poses. A translation noise [Nx, Ny, Nz] of N = Gauss{0, σT } is added to the trans- +lation of the pose, while a rotation noise [Nroll, Npitch, Nyaw] of N = Gauss{0, σR} is added to +the three angles of the pose. The metrics following NeuralRecon (Sun et al., 2021) are reported +in Table 7. From the results, we can see our system can handle some translation errors but cannot +handle the rotation errors well. But if the poses of only some frames are miscalculated, e.g., 10% of +all frames, the performance decrease would be under control. +Ratio σT (cm) σR (deg) Acc ↓ Comp ↓ Prec ↑ Recall ↑ F-score ↑ +0 +0 +0 +0.049 +0.068 +0.754 +0.664 +0.705 +100% +0.5 +0 +0.050 +0.068 +0.745 +0.658 +0.698 +100% +1 +0 +0.055 +0.073 +0.708 +0.626 +0.663 +100% +0 +0.5 +0.084 +0.117 +0.525 +0.446 +0.480 +100% +0 +1 +0.109 +0.185 +0.406 +0.314 +0.351 +100% +0.5 +0.5 +0.084 +0.117 +0.516 +0.435 +0.471 +100% +1 +1 +0.114 +0.187 +0.380 +0.296 +0.330 +10% +0.5 +0.5 +0.055 +0.074 +0.715 +0.629 +0.668 +10% +1 +1 +0.064 +0.087 +0.662 +0.577 +0.616 +Table 7: Experiments with pose noise following NeuralRecon (Sun et al., 2021) metrics. +A.6 +RESULTS WITH SMALLER VOXEL SIZE +As expected, a smaller voxel size leads to a more accurate reconstruction but consumes much more +GPU memory. We have trained the models with voxel sizes of 2cm and 3cm, but it is hard to +evaluate the models in the large scene of ScanNet test set, because the model of 2cm requires too +much memory of the GPU. Thus we only compare them on a medium scene, i.e., Scene-709, as +reported in Table 8. The per-frame time remains unchanged while the per-scene time increases. +14 + +Published as a conference paper at ICLR 2023 +Voxel Size Acc ↓ Comp ↓ Prec ↑ Recall ↑ F-score ↑ Per Scene Time +4cm +0.033 +0.053 +0.837 +0.730 +0.780 +286.4 ms +3cm +0.025 +0.061 +0.882 +0.756 +0.814 +435.6 ms +2cm +0.019 +0.062 +0.913 +0.764 +0.832 +891.2 ms +Table 8: Evaluation of different voxel sizes on Scene-709. +Figure 6: Texture-less rendering of the ground-truth meshes for the qualitative comparison on the ScanNet +dataset. +A.7 +MORE RESULTS +The texture-less rendering of the ground-truth meshes is shown in Figure 6. More results are pre- +sented in Figure 7 and Figure 8. +15 + +NeuralRecon +Atlas +Ours +Ground truthPublished as a conference paper at ICLR 2023 +Figure 7: More qualitative results. +16 + +Published as a conference paper at ICLR 2023 +Figure 8: More qualitative results. +17 + diff --git a/XdFRT4oBgHgl3EQfNjdq/content/tmp_files/load_file.txt b/XdFRT4oBgHgl3EQfNjdq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d47ccb8cb48fbb9dbf009a52cab3cbca62799ca8 --- /dev/null +++ b/XdFRT4oBgHgl3EQfNjdq/content/tmp_files/load_file.txt @@ -0,0 +1,941 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf,len=940 +page_content='Published as a conference paper at ICLR 2023 SDF-FORMER: MONOCULAR SCENE RECONSTRUC- TION WITH 3D SDF TRANSFORMERS Weihao Yuan, Xiaodong Gu, Heng Li, Zilong Dong, Siyu Zhu Alibaba Group {qianmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='ywh, dadong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='gxd, baoshu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='lh, list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='dzl, siting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='zsy} @alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='com ABSTRACT Monocular scene reconstruction from posed images is challenging due to the com- plexity of a large environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Recent volumetric methods learn to directly pre- dict the TSDF volume and have demonstrated promising results in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' How- ever, most methods focus on how to extract and fuse the 2D features to a 3D fea- ture volume, but none of them improve the way how the 3D volume is aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In this work, we propose an SDF transformer network, which replaces the role of 3D CNN for better 3D feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To reduce the explosive compu- tation complexity of the 3D multi-head attention, we propose a sparse window attention module, where the attention is only calculated between the non-empty voxels within a local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then a top-down-bottom-up 3D attention network is built for 3D feature aggregation, where a dilate-attention structure is proposed to prevent geometry degeneration, and two global modules are employed to equip with global receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The experiments on multiple datasets show that this 3D transformer network generates a more accurate and complete reconstruction, which outperforms previous methods by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Remarkably, the mesh ac- curacy is improved by 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='8%, and the mesh completeness is improved by 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3% on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 1 1 INTRODUCTION Monocular 3D reconstruction is a classical task in computer vision and is essential for numerous applications like autonomous navigation, robotics, and augmented/virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Such a vision task aims to reconstruct an accurate and complete dense 3D shape of an unstructured scene from only a sequence of monocular RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' While the camera poses can be estimated accurately with the state-of-the-art SLAM (Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) or SfM systems (Schonberger & Frahm, 2016), a dense 3D scene reconstruction from these posed images is still a challenging problem due to the complex geometry of a large-scale environment, such as the various objects, flexible lighting, reflective sur- faces, and diverse cameras of different focus, distortion, and sensor noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Many previous methods reconstruct the scenario in a multi-view depth manner (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Duzceker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' They predict the dense depth map of each target frame, which can estimate accurate local geometry but need additional efforts in fusing these depth maps (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', solving the inconsistencies between different views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Recently, some methods have tried to directly regress the complete 3D surface of the entire scene (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) from a truncated signed distance function (TSDF) representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' They first extract the 2D features with 2D convolutional neural networks (CNN), and then back-project the features to 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Afterward, the 3D feature volume is processed by a 3D CNN network to output a TSDF volume prediction, which is extracted to a surface mesh by marching cubes (Lorensen & Cline, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This way of reconstruction is end-to-end trainable, and is demonstrated to output accurate, coherent, and complete meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In this paper, we follow this volume-based 3D reconstruction path and directly regress the TSDF volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Inspired by recent successes of vision transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020), some approaches (Bozic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) have adopted this structure in 3D recon- 1Project page: https://weihaosky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='io/sdfformer 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='13510v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='CV] 31 Jan 2023 Published as a conference paper at ICLR 2023 Figure 1: The overview of the 3D reconstruction framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The input images are extracted to features by a 2D backbone network, then the 2D features are back-projected and fused to 3D feature volumes, which are aggregated by our 3D SDF transformer and generate the reconstruction in a coarse-to-fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' struction, but their usages are all limited to fusing the 2D features from different views while the aggregation of the 3D feature volumes is still performed by the 3D CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In this paper, we claim that the aggregation of 3D feature volume is also critical, and the evolution from 3D CNN to 3D multi-head attention could further improve both the accuracy and completeness of the reconstruc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Obviously, the limited usage of 3D multi-head attention in 3D feature volume aggregation is mainly due to its explosive computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Specifically, the attention between each voxel and any other voxel needs to be calculated, which is hard to be realized in a general computing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This is also the reason why there are only a few applications of 3D transformers in solving 3D tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In this work, to address the above challenges and make the 3D transformer practical for 3D scene reconstruction, we propose a sparse window multi-head attention structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Inspired by the sparse CNN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2018), we first sparsify the 3D feature volume with predicted occupancy, in which way the number of the voxels is reduced to only the occupied ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then, to compute the attention score of a target voxel, we define a local window centered on this voxel, within which the non-empty voxels are considered for attention computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In this way, the computation complexity of the 3D multi-head attention can be reduced by orders of magnitude, and this module can be embedded into a network for 3D feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, with this module, we build the first 3D transformer based top-down-bottom-up network, where a dilate-attention module and its inverse are used to downsample and upsample the 3D feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In addition, to make up for the local receptive field of the sparse window attention, we add a global attention module and a global context module at the bottom of this network since the size of the volume is very small at the bottom level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' With this network, the 3D shape is estimated in a coarse-to-fine manner of three levels, as is displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To the best of our knowledge, this is the first paper employing the 3D transformer for 3D scene reconstruction from a TSDF representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In the experiments, our method is demonstrated to outperform previous methods by a significant margin on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Specifically, the accuracy metric of the mesh on the ScanNet dataset is reduced by 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='8%, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='055 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='032, and the completeness metric is reduced by 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3%, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='083 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In the qualitative results, the meshes reconstructed by our method are dense, accu- rate, and complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The main contributions of this work are then summarized as follows: We propose a sparse window multi-head attention module, with which the computation complexity of the 3D transformer is reduced significantly and becomes feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We propose a dilate-attention structure to avoid geometry degeneration in the downsampling, with which we build the first top-down-bottom-up 3D transformer network for 3D feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This network is further improved with a bottom-level global attention module and a global context encoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This 3D transformer is employed to aggregate the 3D features back-projected from the 2D features of an image sequence in a coarse-to-fine manner, and predict TSDF values for accurate and complete 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This framework shows a significant improvement on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 2 RELATED WORK Depth-based 3D Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In traditional methods, reconstructing a 3D model of a scene usually involves depth estimating for a series of images, and then fusing these depths together into a 3D data structure (Sch¨onberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' After the rising of deep learning, many works have 2 V2 V2 3D SDF F2 Fusion 02 Former 3D SDF F1 V1 Fusion V1 V1 01 Former 2D F 3D SDF Backbone Fusion Vo VO So Former 2D DPublished as a conference paper at ICLR 2023 tried to estimate accurate and dense depth maps with deep neural networks (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Wang & Shen, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' They usually estimate the depth map of the reference image by constructing a 3D cost volume from sev- eral frames in a local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Also, to leverage the information in the image sequence, some other methods try to propagate the message from previously predicted depths utilizing probabilistic filter- ing (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2019), Gaussian process (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2019a), or recurrent neural networks (Duzceker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Although the predicted depth maps are increasingly accurate, there is still a gap be- tween these single-view depths and the complete 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Post mesh generation like Poisson reconstruction (Kazhdan & Hoppe, 2013), Delaunay triagulation (Labatut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2009), and TSDF fusion (Newcombe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2011) are proposed to solve this problem, but the inconsistency between different views is still a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Volume-based 3D Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To avoid the depth estimation and fusion in 3D reconstruction, some methods try to directly regress a volumetric data structure end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' SurfaceNet (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017) encodes the camera parameters together with the images to predict a 3D surface occupancy volume with 3D convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Afterward, Atlas (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020) back-projects the 2D features of all images into a 3D feature volume with the estimated camera poses, and then feeds this 3D volume into a 3D U-Net to predict a TSDF volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then NeuralRecon (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) improves the efficiency by doing this within a local window and then fusing the prediction together using a GRU module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Recently, to improve the accuracy of the reconstruction, some methods also introduce transformers to do the fusion of 2D features from different views (Bozic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' However, their transformers are all limited in 2D space and used to process 2D features, which is not straightforward in the 3D reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' There are also some methods for object 3D shape prediction, which can infer the 3D shape of objects with only a few views (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' But the network of these methods can only infer the shape of one category of small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Lately, some works represent the 3D shape with an implicit network, and optimize the implicit representation by neural rendering (Yariv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Yariv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' These methods could obtain a fine surface of an object with iterative optimization, but with the cost of a long-time reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Transformers in 3D Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The transformer structure (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017) has attracted a lot of attention and achieved many successes in vision tasks (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Most of them, nevertheless, are used for 2D feature extraction and aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Even in 2D feature processing, the computation complexity is already quite high, so many works are proposed to reduce the resource-consuming (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Directly extending the trans- former from 2D to 3D would cause catastrophic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Thus most works are only carefully performed on resource-saving feature extraction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', the one-off straightforward feature mapping without any downsampling or upsampling (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021a), where the size of the feature vol- ume remains unchanged, or the top-down tasks with only downsampling (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), where the size of the feature volume is reduced gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In 3D reconstruction, however, a top-down- bottom-up structure is more reasonable for feature extraction and shape generation, as in most of the 3D-CNN-based structures (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' So in this work, we design the first 3D transformer based top-down-bottom-up structure for improving the quality of 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In addition, a sparse window multi-head attention mechanism is proposed to save the computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Although the sparse structure can handle the highly-sparse data, like the object detection of Lidar points (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), it is not suitable for processing a relatively-dense data, like a mesh of an indoor scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, a sparse window structure is needed in 3D scene reconstruction, where a dense surface within a window could be sufficiently aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 3 METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='1 OVERVIEW The overview framework of our method is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Given a sequence of images {Ii}N i=1 of a scene and the corresponding camera intrinsics {Ki}N i=1 and extrinsics {Pi}N i=1, we first extract the image features {Fi}N i=1 in 2D space in three levels, and then back project these 2D features to 3D space, which are fused to three feature volumes in the coarse, medium, and fine levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Afterward, these three feature volumes are aggregated by our SDF 3D transformer in a coarse-to-fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' At the coarse and medium levels, the output of the 3D transformer is 3 Published as a conference paper at ICLR 2023 Figure 2: (a) Illustration of the sparse window attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For calculating the attention of the current voxel (in orange), we first sparsify the volume using the occupancy prediction from the coarser level, and then search the occupied voxels (in dark blue) within a small window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The attention is hence computed based on only these neighbor occupied voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (b) Illustration of the dilate-attention in a 2D slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We dilate the occupied voxels and calculate the attention of these dilated voxels (in yellow) to maintain the geometry structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' two occupancy volumes O2, O1, while at the fine level, the output is the predicted TSDF volume S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The coarse occupancy volume O2 and the medium occupancy volume O1 store the occupancy values o ∈ [0, 1] of the voxels, which are used to sparsify the finer level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, the feature volumes could be processed sparsely to reduce the computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Finally, the predicted mesh is extracted using marching cubes (Lorensen & Cline, 1987) from the TSDF volume S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='2 FEATURE VOLUME CONSTRUCTION The 2D features {Fl i}N i=1 in three levels l = 0, 1, 2 are extracted by a feature pyramid network (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017) with the MnasNet-B1 (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2019) as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The resolution of the features at these three levels are 1 4, 1 8, 1 16, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then following Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2020), we back project the 2D features to 3D space with the camera parameters {Ki}N i=1 and {Pi}N i=1, generating 3D feature volumes {Vl i}N i=1 of size NX × NY × NZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In previous work, usually the fusion of these feature volumes from different views is computed by taking the average (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' However, the back-projected features from different views contribute differently to the 3D shape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', the view with a bad viewing angle and the voxels far from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, a weighted average is more reasonable than taking the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To compute these weights, for each voxel we calculate the variance of the features of different views by Varl i = (Vl i − V l)2, (1) where V l is the average of the features of all views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then we feed the features and the variance into a small MLP to calculate the weights Wi, which are used to compute a weighted average of the features from different views as Vl w = 1 N � i Vl i × SoftMax(Wi), (2) where × denotes element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Inspired by Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2018), we also calculate the total variance of all feature volumes and then concatenate it with the weighted average to the final feature volumes, as Vl = {Vl w, 1 N � i Varl i}, (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3 SPARSE WINDOW MULTI-HEAD ATTENTION The multi-head attention structure has been shown to be effective in many vision tasks (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Most of them, however, are limited to 2D feature processing rather than 3D feature processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This is because the computation complexity of the multi-head attention is usually higher than convolutional networks, which problem is further enlarged in 3D features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To compute this for a 3D feature volume, the attentions between a voxel and any other voxels need to be computed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', NX ×NY ×NZ attentions for one voxel and NX ×NY ×NZ ×NX ×NY ×NZ attentions for all voxels, which is extremely large and hard to be realized in regular GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 4 Published as a conference paper at ICLR 2023 Figure 3: The structure of the SDF transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' “S-W- Attn” denotes sparse window attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To deal with this problem and make the multi- head attention of 3D volumes feasible, we pro- pose to use a sparse window structure to cal- culate the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' As is displayed in Fig- ure 1, in the medium and the fine level, we sparsify the volumes using the occupancy pre- diction O2, O1, and only compute the attention of the non-empty voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In addition, consider- ing that the nearby voxels contribute more to the shape of the current voxel and the distant voxels contribute less, we only calculate the at- tention within a local window of each voxel, as is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, we are able to only calculate the multi-head attention of the occupied voxels within a small window, in which way the computation complexity is re- duced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Specifically, for any non-empty voxel vi in the feature volume V , we first search all non-empty voxels within a n × n × n window centered on this voxel and get the neighbor voxels {vj, j ∈ Ω(i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then the query, key, and value embeddings are calculated as Qi = Lq(V (vi)), Kj = Lk(V (vj)), Vj = Lv(V (vj)), (4) where Lq, Lk, Lv are the linear projection layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For the position embedding P, we hope to block the influence from the scale of the 3D world coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Hence we compute it based on the relative voxel position in the volume rather than based on the real-world coordinates (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), as Pj = Lp(vj − vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (5) Then the attention is calculated as Attention(vi) = � j∈Ω(i) SoftMax(Qi(Kj + Pj)/ √ d)(Vj + Pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (6) In this case, the computation complexity is reduced from O3D-Attn = NX × NY × NZ × NX × NY × NZ × O(ij), (7) to OSW-3D-Attn = Noccu × noccu × O(ij), (8) where O(ij) is the complexity of one attention computation between voxel vi and vj, Noccu is the number of occupied voxels in the volume, and noccu is the number of occupied voxels within the local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Assuming that the occupancy rate of the volume is 10% and the window size is 1 10 of the volume size, the computation complexity of the sparse window attention would be only n3/10 10NXNY NZ = 1 100000 of the dense 3D attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='4 SDF 3D TRANSFORMER Limited by the high resource-consuming of the multi-head attention, most of the previous works related to 3D transformers are only carefully performed on resource-saving feature processing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', the one-off straightforward feature mapping without any downsampling or upsampling (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021a), where the size of feature volumes remains unchanged, or the top-down tasks with only downsampling (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), where the size of feature volumes is reduced gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In 3D reconstruction, however, a top-down-bottom-up structure is more reasonable for feature extraction and prediction generation, as in most of the 3D-CNN-based structures (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' So in this work, we design the first 3D transformer based top-down-bottom- up structure, as is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 5 V TSDF Down Dilate Attn Inv Dilate Attn 3D S-W-Attn 3D S-W-Attn Down Dilate Attn Inv 3D S-W-Attn 3D S-W-Attn Down Dilate Attn Inv 3D S-W-Attn 3D S-W-Attn Down Dilate Attn Inv 3D S-W-Attn 3D Global Attn Global EncodePublished as a conference paper at ICLR 2023 Baseline + SDF Transformer + Post Dilate Attention Ground Truth Figure 4: Ablation study on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Taking the network for the fine volume (V 0 in Figure 1) as an example, there are four feature levels in total, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 1 2, 1 4, 1 8, 1 16, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In the encoder part, at each level, a combination of downsampling and dilate-attention is proposed to downsample the feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then two blocks of the sparse window multi-head attention are used to aggregate the feature volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' At the bottom level, a global attention block is employed to make up the small receptive field of the window attention, and a global context encoding block is utilized to extract the global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In the decoder part, we use the inverse sparse 3D CNN to upsample the feature volume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', we store the mapping of the down flow and now restore the spatial structure by inversing the sparse 3D CNN in the dilate-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, the final shape after the up flow should be the same as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Similar to FPN (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017), the features in the down flow are also added to the upsampled features in the corresponding level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To enable the deformation ability, a post-dilate-attention block is equipped after the down-up flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Finally, a submanifold 3D CNN head with Tanh activation is appended to output the TSDF prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For the coarse volume V 2 and medium volume V 1, two and three-level of similar structures with Sigmoid activation are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Dilate-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The direct downsampling of a sparse structure is prone to losing geometry struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To deal with this, between each level we first downsample the feature volume, and then dilate the volume with a sparse 3D CNN with the kernel size of 3, which calculates the output if any voxel within its kernel is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The dilation operation alone may also harm the geometry, since it may add some wrong voxels into the sparse structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Thus we calculate the sparse window atten- tion of the dilated voxels, such that the voxels far from the surface would get low scores and do not contribute to the final shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The dilated voxels are then joined to the downsampled volume by concatenating the voxels together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' With this dilate-attention module, the 3D shape is prevented from collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Without this module, the network performs badly and only generates a degraded shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Global attention and global context encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Since the attention blocks in the top-down flow are all local-window based, there could be a lack of the global receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Considering the resolution of the bottom level is not high, we equip with a global attention block at the bottom level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', we calculate the attention between each non-empty voxel and any other non-empty voxel in the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This could build the long-range dependency missing in the sparse window attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In addition, we use the multi-scale global averaging pooling (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017) of scales 1, 2, 3 to extract the global context code of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This encoding module could aggregate the global information and explain the illumination, global texture, and global geometry style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5 LOSS FUNCTION The final TSDF prediction S0 is supervised by the log L1 distance between the prediction and the ground truth as L0 = | log S0 − log �S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To supervise the occupancy predictions O2, O1 in the coarse and medium levels, we generate the occupancy volumes based on the TSDF values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Specifically, the voxels with TSDF of −1 ∼ 1 are regarded as occupied, and the values are set to 1, otherwise set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then a binary cross-entropy loss is calculated between the prediction and the ground truth as: Ll = −� Ol log Ol, l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To supervise the averaging weights Wl i, we use the occupancy in the back-projection following Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Intuitively, when the feature is back-projected from a 2D image to the 3D space along the camera ray using multiple depth values, we hope the voxels close to the mesh surface have bigger weights in the fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, the 3D position is regarded as occupied if the difference between the project depth and the true depth from the depth map is smaller than the TSDF truncation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then the cross entropy loss is applied to the weights and the occupancy: Ll w = −� Ol i log σ(Wl i), l = 1, 2, 3, (9) where σ denotes Sigmoid, and � Ol i is the ground truth occupancy in the back-projection of image Ii.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='754 Table 1: Evaluation of the 3D meshes on ScanNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The upper part follows the evaluation in Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2021) while the lower part follows Bozic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The metric definitions are explained in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Method Abs Rel ↓ Abs Diff ↓ Sq Rel ↓ RMSE ↓ δ-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='25 ↑ δ-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='252 ↑ δ-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='253 ↑ Colmap (Sch¨onberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='137 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='975 Vortx (Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='987 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='990 Table 2: Evaluation of the 2D depth maps on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The upper part shows the results of depth- based methods, while the lower part shows volumetric methods, whose depths are rendered from the meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 4 EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='1 EXPERIMENTS SETUP Our work is implemented in Pytorch and trained on Nvidia V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The network is optimized with the Adam optimizer (β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='999) with learning rate of 1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For a fair comparison with previous methods, the voxel size of the fine level is set to 4cm, and the TSDF truncation distance is set to triple the voxel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Thus the voxel size of the medium and the coarse levels are 8 cm and 16 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For the balance of efficiency and receptive field, the window size of the sparse window attention is set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For the view selection, we first follow Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2019b) to remove the redundant views, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', a new incoming frame is added to the system only if its relative translation is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='1 m and the relative rotation angle is greater than 15 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then if the number of the remaining views exceeds the upper limit, a random selection is adopted for memory efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The view limit is set to 20 in the training, which means twenty images are input to the network for one iteration, while the limit for testing is set to 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Our framework runs at an online speed of 75 FPS for the keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Detailed efficiency experiments are reported in the supplemental materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' ScanNet (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2017) is a large-scale indoor dataset composed of 1613 RGB-D videos of 806 indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We follow the official train/test split, where there are 1513 scans used for training and 100 scans used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' TUM-RGBD (Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2012) and ICL-NUIM (Handa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2014) are also two datasets composed of RGB-D videos but with small-number scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, following previous methods (Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), we only perform the generalization evaluation of the model trained on ScanNet on these two datasets, where 13 scenes of TUM-RGBD and 8 scenes of ICL-NUIM are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='2 EVALUATION To compare with previous methods, we evaluate the proposed method on the ScanNet test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The quantitative results are presented in Table 1 and the qualitative comparison are displayed in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 7 Published as a conference paper at ICLR 2023 Figure 5: The qualitative results on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Texture-less rendering is displayed in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We first directly evaluate the reconstructed meshes with the ground-truth meshes, and obtain a sig- nificant improvement from previous methods, improving from F-score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='641 to F-score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='705, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then following Bozic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' (2021), we add the same occlusion mask at evalua- tion to avoid penalizing a more complete reconstruction, which is because the ground-truth meshes are incomplete due to unobserved and occluded regions, while our method could reconstruct a more complete 3D shape, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This results in a more reasonable evaluation, as in the sec- ond part of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The improvement is further enlarged, from F-score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='655 to F-score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='754 compared to previous best method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The accuracy error is decreased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='055 m to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='032 m, which is almost half (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='8%) of the previous best method, while the completeness error is decreased by 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3%, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='083 m to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='062 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This owes to the feature aggregating ability of the proposed 3D SDF transformer, which can predict a more accurate 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This is also demonstrated in the generalization experiments on ICL-NUIM and TUM-RGBD datasets, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' After evaluating the reconstructed meshes, we also evaluate the depth accuracy of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Since our method does not predict the depth maps explicitly, we render the predicted 3D shape to the 8 Published as a conference paper at ICLR 2023 image planes and get the depth maps, following previous methods (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The results are shown in Table 2, from which we can see our method decreases the error a lot from previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The relative error is reduced by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='4%, from 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='229 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='455 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='254 Table 3: Generalization experiments on the ICL-NUIM and TUM-RGBD datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' From the qualitative visualization in Fig- ure 5, we can see our method can predict a complete and accurate 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Pre- vious methods which can recover a com- plete mesh usually reconstruct a smooth 3D shape with losing some details (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' However, our method could predict a more complete mesh than the ground truth, while the details of the 3D shapes are better recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Please note that for a fair comparison, the voxel size is set to 4 cm, such that it is hard to recon- struct the geometry details less than 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3 ABLATION STUDY Method 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='754 Table 4: Ablation study on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Components are added one by one in the upper part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' SDF transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To verify the effec- tiveness of the proposed SDF transformer, we first build a baseline model with the same structure as Figure 1, but the 3D SDF transformer is replaced by a UNet structure of 3D CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Adding the variance fusion would improve the mesh in some clutter areas and slightly increase the per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then we add a base version of the SDF transformer, which does not include the global module and the post- dilate-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The performance is significantly improved with this module, as is shown in Table 4 and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The reconstructed meshes possess much more geometry details compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Global module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We next add the global module, including the bottom-level global attention and the global context code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The sparse window attention block can only obtain the long-range dependency within a local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Thus it may have problems when it can not get enough information within this local window, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', the texture-free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Also, the global module could reason the global information like the illumination and the texture style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Dilate attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The dilate attention module is crucial in the SDF transformer, so we can not re- move all the dilate attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' That will destroy the whole framework and generate a degraded 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, we only ablate the post dilate attention block after the down-up flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This block could deform the shape and make it more complete, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', making up the crack as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' From the quantitative results in Table 4, we can also see the improvement of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' As shown in Table 4, we study the impact of the window size of the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' It is expected that a larger window size would generate a better result, since the range of the depen- dency is longer, but with the cost of more resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We choose 10 as the default size, considering that the performance improvement is minor after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 5 CONCLUSION We propose the first top-down-bottom-up 3D transformer for 3D scene reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A sparse win- dow attention module is proposed to reduce the computation, a dilate attention module is proposed to avoid geometry degeneration, and a global module at the bottom level is employed to extract the global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This structure could be used to aggregate any 3D feature volume, thus it could be applied to more 3D tasks in the future, such as 3D segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 9 Published as a conference paper at ICLR 2023 REFERENCES Aljaz Bozic, Pablo Palafox, Justus Thies, Angela 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Volume rendering of neural implicit surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:4805–4815, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Weihao Yuan, Yazhan Zhang, Bingkun Wu, Siyu Zhu, Ping Tan, Michael Yu Wang, and Qifeng Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Stereo matching by self-supervision of multiscopic vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 5702–5709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Weihao Yuan, Xiaodong Gu, Zuozhuo Dai, Siyu Zhu, and Ping Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Neural window fully-connected crfs for monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 3916–3925, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Pyramid scene parsing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 2881–2890, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 12 Published as a conference paper at ICLR 2023 A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='1 MORE DETAILS Volume sparsify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The ground-truth occupancy volumes are generated based on the ground-truth TSDF volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The voxels with TSDF value of [−1, 1] are regarded as occupied and set to 1, otherwise set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In the training or the inference, after the occupancy volume is predicted in the coarser level, the voxels of occupancy value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5 are regarded as empty and discarded, while the remaining voxels are regarded as occupied and transmitted to the next level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The sparse volume is stored in a hash table, where the key of the table is the hash value of the voxel, and the value of the table stores the corresponding feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In the coarsest level, all voxels are regarded as non-empty and stored in the hash table, which does not consume much memory because the size of the volume is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The training and inference are performed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For a given sequence of images, first a view selection is performed to select images with translation greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='1m and rotation greater than 15 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Then a random selection is adopted from the remaining images if the number exceeds the upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' These images are then fed to the 2D backbone for feature extraction, after which the features are fused to a 3D volume and fed to the 3D part to produce the TSDF volume prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The final mesh is extracted by marching cubes from the TSDF volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' This process is the same as previous methods like Atlas, TransformerFusion or VORTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For the number of the upper limit of the images, actually any number for the sequence length is okay for our framework, although more images lead to a better reconstruction of a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' In our experiments, the number for training is set to 20 and the number for inference is set to 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Method Per Frame Time Per Scene Time FPS Atlas (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020) 71 ms 840 ms 14 NeuralRecon (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) 30 ms 0 ms 33 TransformerFusion (Bozic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5 ms 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3 ms 7 VoRTX (Stier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='4 ms 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='7 ms 14 Ours 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3 ms 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='4 ms 75 Table 5: Efficiency experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='2 EFFICIENCY The runtime analysis is presented in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' For a fair comparison to previous methods, the time is tested on a chunk of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5 m3 with an Nvidia RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Our framework consists of two parts: one is the per-frame part, including the feature extraction of the 2D images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' the other one is the per-scene part, including the feature fusion, 3D feature processing, and mesh generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The per-frame model runs for every keyframe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', it keeps running whenever a new keyframe comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Differently, the per-scene model runs only once for generating a mesh reconstruction of a scene, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', it only works after all frames are fed, or when we need to output a mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Therefore, the online speed of a normal running is 75 FPS, which only performs the mesh generation once at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='3 METRICS The definitions of the 2D metrics and 3D metrics used for evaluation are explained in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='4 LIMITATIONS Due to the volume representation, our framework is limited by the trade-off between the resolution of the volume and the memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A smaller voxel size would cost much more memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The voxel size is set to 4 cm, such that the geometry details less than 4 cm are hard to be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 13 Published as a conference paper at ICLR 2023 Metrics Definition Abs Rel 1 n � |d − d∗|/d∗ Abs Diff 1 n � |d − d∗| Sq Rel 1 n � |d − d∗|2/d∗ RMSE � 1 n � |d − d∗|2 δ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='25i 1 n �(max( d d∗ , d∗ d ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='25i) Acc meanp∈P (minp∗∈P ∗ ||p − p∗||) Comp meanp∗∈P ∗(minp∈ ||p − p∗||) Chamfer distance Acc + Comp 2 Prec meanp∈P (minp∗∈P ∗ ||p − p∗|| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='05) Recall meanp∗∈P ∗(minp∈ ||p − p∗|| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='05) F-score 2×Prec×Recall Prec + Recall Table 6: Metric definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' n denotes the number of pixels with both valid ground truth and prediction, d and d∗ denote the predicted and the ground-truth depths, p and p∗ denote the predicted and the ground-truth point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='5 ROBUSTNESS TO THE POSE NOISE Our method is based on the given accurate camera poses, which is the same as previous state-of- the-art methods like Atlas (Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2020), NeuralRecon (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), and Transformer- Fusion (Bozic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021), where the camera poses are obtained by the standard SfM or SLAM systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' To inspect the robustness of our method to the pose errors, we add the Gaussian noise to the camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A translation noise [Nx, Ny, Nz] of N = Gauss{0, σT } is added to the trans- lation of the pose, while a rotation noise [Nroll, Npitch, Nyaw] of N = Gauss{0, σR} is added to the three angles of the pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The metrics following NeuralRecon (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) are reported in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' From the results, we can see our system can handle some translation errors but cannot handle the rotation errors well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' But if the poses of only some frames are miscalculated, e.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', 2021) metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='6 RESULTS WITH SMALLER VOXEL SIZE As expected, a smaller voxel size leads to a more accurate reconstruction but consumes much more GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' We have trained the models with voxel sizes of 2cm and 3cm, but it is hard to evaluate the models in the large scene of ScanNet test set, because the model of 2cm requires too much memory of the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Thus we only compare them on a medium scene, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=', Scene-709, as reported in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' The per-frame time remains unchanged while the per-scene time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 Voxel Size Acc ↓ Comp ↓ Prec ↑ Recall ↑ F-score ↑ Per Scene Time 4cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='033 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='832 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='2 ms Table 8: Evaluation of different voxel sizes on Scene-709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' Figure 6: Texture-less rendering of the ground-truth meshes for the qualitative comparison on the ScanNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content='7 MORE RESULTS The texture-less rendering of the ground-truth meshes is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' More results are pre- sented in Figure 7 and Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 15 NeuralRecon Atlas Ours Ground truthPublished as a conference paper at ICLR 2023 Figure 7: More qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 16 Published as a conference paper at ICLR 2023 Figure 8: More qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFRT4oBgHgl3EQfNjdq/content/2301.13510v1.pdf'} diff --git a/XtE3T4oBgHgl3EQf1gsc/content/tmp_files/2301.04746v1.pdf.txt b/XtE3T4oBgHgl3EQf1gsc/content/tmp_files/2301.04746v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96ee2f1575f221ee125aef4fb487b6e89d8353f5 --- /dev/null +++ b/XtE3T4oBgHgl3EQf1gsc/content/tmp_files/2301.04746v1.pdf.txt @@ -0,0 +1,902 @@ +Switchable Lightweight Anti-symmetric Processing (SLAP) +with CNN to Reduce Sample Size and Speed up Learning – +Application in Gomoku Reinforcement Learning +Chi-Hang Suen1 + +Abstract. To replace data augmentation, this paper proposed a +method called SLAP to intensify experience to speed up machine +learning and reduce the sample size. SLAP is a model-independent +protocol/function to produce the same output given different +transformation variants. SLAP improved the convergence speed of +convolutional neural network learning by 83% in the experiments +with Gomoku game states, with only one eighth of the sample size +compared with data augmentation. In reinforcement learning for +Gomoku, using AlphaGo Zero/AlphaZero algorithm with data +augmentation as baseline, SLAP reduced the number of training +samples by a factor of 8 and achieved similar winning rate against +the same evaluator, but it was not yet evident that it could speed up +reinforcement learning. The benefits should at least apply to domains +that are invariant to symmetry or certain transformations. As future +work, SLAP may aid more explainable learning and transfer learning +for domains that are not invariant to symmetry, as a small step +towards artificial general intelligence. + + +1 +Introduction +1.1 Problem +Convolutional neural network (CNN) is now the mainstream family +of models for computer vision, thanks to its weight sharing +mechanism to efficiently share learning across the same plane by so- +called kernels, achieving local translational invariance. But CNN is +not reflection and rotation invariant. Typically it can be addressed by +data augmentation to inputs by reflection and rotation if necessary, +but the sample size would increase substantially. [1] criticised CNN +that it could not learn spatial relationships such as orientation, +position and hierarchy and advocated their novel capsule to replace +CNN. [2] improved capsule using routing by agreement mechanism +and outperformed CNN at recognising overlapping images, but they +also admitted that it tended to account for everything in the structure. +This implies capsule is too heavy in computation. Inspired by the +idea of capturing orientation information in capsule network [2], this +paper proposed a novel method called Switchable Lightweight Anti- +symmetric Process (SLAP), a protocol to produce the same output +given different transformation variants, with the main research +question: can symmetry variants be exploited directly by SLAP to +improve and combine with CNN for machine learning? +Very often, we know in advance if a certain machine learning task is +invariant to certain types of transformation, such as rotation and +reflection. E.g. in Gomoku, the state is rotation (perpendicularly) and +reflection (horizontally and vertically) invariant in terms of winning +probability, and “partially” translation invariant. Symmetry is often +exploited by data augmentation for deep learning. But this greatly +increases the dataset size if all symmetry variants are included – e.g. +there are 8 such variants for each Gomoku state. SLAP was invented +in this paper to avoid such expansion (see 1.2). + On the other hand, reinforcement learning is notorious for lengthy +training time and large sample size required. Data augmentation may +help improve performance in reinforcement learning, but it would +increase the sample size. This research tried to kill two birds by one +stone, SLAP, by applying with CNN in reinforcement learning (of + +1 City, University of London, Department of Computer Science, +email: chi.suen@city.ac.uk +Gomoku), challenging the widely used practice of data augmentation, +aiming at reducing the sample size and improving the learning speed. + +1.2 Switchable Lightweight Anti-symmetric Process (SLAP) +SLAP is a model-independent protocol and function to always +produce or choose the same variant regardless of which +transformation variant (by specified symmetry) is given, and if +required also output the corresponding transformation. It can be used +upon any function or model to produce outputs that are invariant with +regard to specified symmetric properties of the inputs. If some (type) +of the outputs are not invariant but follow the same transformation, +the corresponding transformation information from SLAP may be +used to transform these outputs back. It can be viewed as +standardization of symmetry, as opposed to standardization of scale. +After processing, symmetric variants are filtered out – that’s why it +is named ‘anti-symmetric process’. Ironically, with this anti- +symmetric process, the function or model (e.g. CNN) to be fed would +look as if it is symmetric with regard to whichever the symmetry +variant is the input, and the same output is produced. It is a novel +method to exploit symmetry variants in machine learning without +increasing the number of training samples by data augmentation. The +motivation is to concentrate experience to speed up learning, without +enlarging the sample size by data augmentation. See details in 3.1. +1.3 Gomoku +Gomoku, or Five in a Row, is a 2-player board game, traditionally +played with Go pieces (black and white stones) on a Go board +(19x19), nowadays on 15x15 board. For experiments in this +research, mini board 8x8 was used instead to save computation, +and the rule of freestyle version was adopted: +• +Black and white place stones of his colour alternatively at an +unoccupied intersection point of the board. Black first. +• +Winner: the one who first forms an unbroken chain of 5 stones +of his colour in a straight line (horizonal, vertical or diagonal). +• +Draw happens if there is no winner when the board is full. +Gomoku was chosen to demonstrate the benefit of SLAP because: +• +Gomoku has huge number of state representations (3225 ~= +2x10107), which justify the use of neural network for learning. +• +Gomoku is rotation and reflection invariant, but only “partially” +translation invariant, so ideal to test different transformations. +• +Gomoku is Markov Decision Process, meeting the basic +mathematical assumption of reinforcement learning. +• +[4] and [5] showed a general effective reinforcement learning +algorithm for board games and Gomoku is simple to implement. + +2 +Background +2.1 CNN +CNN (convolutional neural network) has been widely used for +computer vision but it is known that CNN is weak to deal with + +changes by rotation/orientation unless with much larger sample size +by data augmentation. To address this problem, [1] proposed that +neural network should make use of their then novel capsule, learning +to recognize an implicitly defined visual entity and output probability +of its existence and instantiation parameters such as pose; they +showed that a transforming auto-encoder could be learnt to force the +output (which is a vector instead of scalar) of a capsule to represent +an image property that one might want to manipulate. [2] showed +that a discriminatively trained, multi-layer capsule system achieves +state-of-the-art performance on MNIST and was considerably better +than CNN at recognizing highly overlapping digits, using the so- +called routing by agreement mechanism, and yet [2] admitted that +one drawback was the tendency of capsule to account for everything +in an image. It implies that the capsule might be too “heavy” for +computation and so a lightweight method is required. But the capsule +network with routing by agreement algorithm has been proved not to +be a universal approximator [3], i.e. not fit to all kinds of problems. +As such, this research did not attempt to replace CNN by capsule, +but simply created SLAP to combine with CNN. Instead of forcing +the output to represent certain transformation information (e.g. +orientation angle), SLAP forces the input of different variants (e.g. +different rotation angle) to give the same output variant (and output +the transformation information e.g. angle, if needed). Nevertheless, +the invention of SLAP was inspired by [1] & [2] trying to address +the weakness of CNN. +2.2 Groupoid in Gomoku +There are different Gomoku states of the same groupoid (see Fig. 1), +which means having local symmetry but not necessarily global +symmetry of the whole structure [6]. Groupoid is more challenging +than symmetry or group, as some groupoids may not have the same +status, e.g. see Fig. 1. But the potential for learning is huge as there +are much more variants, e.g. 156 variants by translation in Fig. 1. + + +Fig. 1: Gomoku groupoid. Black can stop white win in C, but not in A or B. + +2.3 AlphaGo Zero / Alpha Zero +For reinforcement learning of Gomoku in this research, the baseline +algorithm was chosen to follow that of AlphaGo Zero [4] and Alpha +Zero [5] papers because domain knowledge was not required. The +algorithm was concisely summarized by [7] as follows: +Neural network + The neural network feature extractor is a type of CNN. It takes +state st as input and yields value of state 𝑣𝜃(𝑠𝑡) ∈ [−1, 1] and +policy 𝑝𝜃 +⃗⃗⃗⃗ (st) as probability vector over all possible actions. It has +the following loss function (excl. regularization terms): + loss = ∑ (𝑣𝜃(𝑠𝑡) − +𝑡 +𝑧𝑡)2 – 𝜋⃗ t . log(𝑝𝜃 +⃗⃗⃗⃗ (st)) +, where 𝑧𝑡, 𝜋⃗ t are final outcome {-1,0,1} and estimate (to be +discussed below) of policy from state st respectively, with 1, 0, -1 +representing win, draw, lose respectively for current player. +Monte Carlo Tree Search (MCTS) as policy improvement operator + At each node, action is chosen by maximizing U(s, a), the upper +confidence bound of Q-value Q(s, a), calculated by: +U(s, a) = Q(s, a) + C * P(s, a) * +√∑ 𝑁(𝑠,𝑏) +𝑏 +1+𝑁(𝑠,𝑎) +where N(s, a) = no. of times taking action a from state s in MCTS +simulation, P(s, .) = 𝑝𝜃 +⃗⃗⃗⃗ (s), and the policy estimate of probability is +improved by using 𝜋⃗ t = N(s, .) / ∑ 𝑁(𝑠, 𝑏) + When a new node (not visited before from parent node) is +reached, instead of rollout, the value of new node is obtained from +neural network and propagated up the search path. Unless the new +node is terminal, the new node is expanded to have child nodes. +Self-play training as policy evaluation operator + In each turn, a fixed number of MCTS simulations are conducted +from the state st, and action is selected by sampling from the policy +estimate of probabilities improved by MCTS, thus generating +training sample data. At the end of an iteration, the neural network +is updated by learning from the training sample data. + The evaluation metric would be based on winning and drawing +percentages of the AI against an independent evaluation agent. There +are differences among AlphaGo Zero and AlphaZero, see Fig. 2: + + +AlphaGo Zero [4] +AlphaZero [5] +Pitting +models +Yes, model with new weights +plays against previous one; +new weights are adopted +only if it wins 55% or above +No, always use new +weights after each +iteration of neural +network learning +Symmetry +Data +augmentation +by +rotation and reflection to +increase sample size by 8 +times for training; transform +to one of 8 variants randomly +in self-play for inference +Not exploited, as it +is +intended +for +generalization +Action in +self-play +Sampled proportional to visit +count in MCTS in first 30 +moves, then selected greedily +by +max +visit +count +(asymptotically with highest +winning chance) in MCTS +Sampled +proportional to visit +count in MCTS +Outcome +prediction +Assume +binary +win/loss, +estimate & optimise winning +probability +Also consider draw +or other outcomes, +estimate & optimise +expected outcome +Fig. 2: Differences between AlphaGo Zero and AlphaZero. + +2.4 Other Related Works, Symmetry and AGI +On lightweight capsule, DSC-CapsNet was proposed as lightweight +capsule network, which focused on computing efficiency and +reducing number of parameters [8]; [9] proposed dense capsule +network with fewer parameters – neither had the novel structure +proposed in this study. On symmetric CNN, [10] proposed to impose +symmetry in neural network parameters by repeating some +parameters and achieved 25% reduction in number of parameters +with only 0.2% loss in accuracy using ResNet-101, a type of CNN; +but unlike SLAP, symmetry was not imposed in the inputs. [11] +incorporated symmetry into neural network by creating symmetry +(of specific type) invariant features, but no implementation or idea +similar to SLAP was used. Studies have shown rotation based +augmentation performed better than many other augmentation +techniques [19]. The type of data augmentation used as baseline in +this research was rotation and reflection based (also the type used by +AlphaGo Zero[4]). The novelty lies in the fact that SLAP is opposite +to the practice of data augmentation – decreasing the variety of +variants in the data instead for machine learning, though also +exploiting symmetry. + Symmetry is one of the natures of the real world. Animals can +detect the same object or the same prey being moved (translated), or +even rotated after being slapped (the novel method was deliberately +abbreviated as SLAP). Recognising symmetry can also speed up +learning patterns, a typical trick used for playing some board games. +To facilitate research exploiting symmetry in machine learning, [12] +connected symmetry transformations to vector representations by the +formalism of group and representation theory to arrive at the first +formal definition of disentangled representations, expected to benefit + +StateA +StateB +Stateclearning from separating out (disentangling) the underlying structure +of the world into disjoint parts of its representation. Upon this work, +[13] showed by theory and experiments that Symmetry-Based +Disentangled Representation Learning (SBDRL) could not only be +based on static observations: agents should interact with the +environment to discover its symmetries. They emphasized that the +representation should use transitions rather than still observations for +SBDRL. This was taken into account for designing the Gomoku +representation for reinforcement learning in this research. + One may expect that an artificial general intelligence (AGI) system, +if invented, should be able to learn unknown symmetry. Researchers +have worked on this, for example [14] proposed learning unknown +symmetries by different principles of family of methods. But it is +equally important to learn by exploiting symmetry more effectively. +For example, if an AGI system can interpret the rules of Gomoku +and realize from the rules that Gomoku is reflection and rotation +invariant, it should directly exploit such symmetry instead of +assuming symmetry is unknown. Ideally, such exploitation should be +switched on easily if one wishes, and hence the term ‘switchable’ in +SLAP, which can be used upon any function or model. If transfer +learning in CNN is analogous to reusing a chair by cutting the legs +and installing new legs to fit another, such ‘switchable learning’ in +SLAP is analogous to turning the switch of an adjustable chair to fit +certain symmetries. Such kind of ‘switch ’ in design can also help AI +be more explainable and transparent, and more easily reused or +transferred, while an AGI system should be able to link and switch +to different sub-systems easily to solve a problem . SLAP can also +reduce memory required. For example, AlphaGo Zero used a +transposition table [4], a cache of previously seen positions and +associated evaluations. Had SLAP been used instead of data +augmentation, such memory size could be reduced by a factor of 8, +or alternatively 8 times more positions or states could be stored. +Indeed memory plays an important role in reinforcement learning as +well by episodic memory, an explicit record of past events to be +taken as reference for making decisions, improving both sample +efficiency and speed in reinforcement learning as experience can be +used immediately for making decisions [15]. It is likely that an AGI +system would, just like human, use memory to solve some problems +rather than always resort to learning from scratch. And in the real +word, a continuous space, there can be much more than 8 equivalent +variants. Recently, [16] suggested that symmetry should be an +important general framework that determines the structure of +universe, constrains the nature of natural tasks and consequently +shape both biological and artificial general intelligence; they argued +that symmetry transformations should be a fundamental principle in +search for a good representation in learning. Perhaps SLAP may +contribute a tiny step towards AGI, by shaping input representations +directly by symmetry transformation. Note that SLAP can be used +upon any function or model and even if some (types) of the outputs +are not invariant but follow the same transformation, these may be +broken down and use the transformation information output from +SLAP to make appropriate transformation back later for these parts +only. A little kid often mistakes b for d at the beginning of learning +alphabets, and it appears that human learning types of objects by +vision might naturally assume symmetry first and then learn non- +symmetry later. If a machine learning problem is to be split into +stages or parts by specified symmetry as a guide, SLAP might help +by wrapping certain parts of a function or neural network model. + + +3 +Methods +3.1 SLAP +SLAP forces the input of different variants (e.g. different rotation +angle) to give the same output variant (and output the transformation +information e.g. angle, though not necessarily used). There can be +multiple ways to achieve this. For rotation and reflection variants of +Gomoku states, one way to implement this is simply flattening the +pixels of 8 variants to 8 lists, compare the lists and always choose +the largest. Below (Fig. 3) was the algorithm used for SLAP in +dealing with rotation and reflection variants of Gomoku states, but +the concept may be applied to other symmetries as well. + +Algorithm SLAP +1: Generate symmetry variants of input, store required transformation +2: Convert each variant to a list +3: Compare each list and find the ‘largest’ list +4: return the ‘largest’ variant & required transformation of the variant + +Fig. 3: SLAP algorithm. Positive large data cluster towards top left. + If the image/state has multiple input channels or planes in one +sample, the first channel/plane is compared first by list comparison. + SLAP was implemented by numpy instead of torch tensor for faster +speed, because numpy uses view for rotation and reflection. The +output variant replaced the input state when SLAP was applied in +training. During inference time, output action probabilities from +neural network would be transformed back using the transformation +information (rotation & reflection) from SLAP. +3.1.1 Invariance +Denote s, t = slap(xi), where slap is SLAP function in pythonic style, +s is the symmetry (of certain group G, with n symmetry variants for +each state) variant and t is corresponding transformation information. +Given property of slap, for all i∈N<=n, + s, t = slap(x1) = slap(x2) = … = slap(xn) + Denote s = slap(xi)[0], t = slap(xi)[1], the pythonic expression to +capture first and second return variables of a function respectively. +Denote h(slap(xi)[0]) as hslap(xi) for any function h. + Given an arbitrary function y = f(x), + y = fslap(xi) ⇒ y = f(slap(xi)[0]) ⇒ y = f(s) for all i +∴ y = fslap(xi) is invariant with respect to i (i.e. symmetry of group G). + When f is the neural network, the composite function resulting +from the neural network, fslap, is invariant to symmetry (of group G). +3.1.2 Differentiability +SLAP was not applied to intermediate layers of neural networks for +Gomoku, so its differentiability was not required in this research. +Approximation would be required to make it differentiable. +3.1.3 Groupoid and SLAP-CC +As Gomoku is only ‘partially’ invariant to translation, it is also +interesting to experiment with translation variants, which are +considered to be groupoid instead of group as they are symmetric +locally but not necessarily symmetric globally. There can be many +more translation variants than rotation and reflection variants, see 2.2. +To save computation, different algorithm (crop and centre) was used +to ‘standardize’ translation variants, denoted as SLAP-CC in the +below, to emphasize that it shared the same general idea as SLAP, +but just different way for implementation. Denoted as cc in the code. +The algorithm of SLAP-CC, shown in Fig. 4, would concentrate +experience around the centre, as input variant was centred to become +output variant. If it could not be exactly centred, the algorithm would +make it slightly lean to top left. + +Algorithm SLAP-CC +1: Find non-empty min & max indices by row & column in input image +2: r_shift = (no. of rows – 1– min row index – max row index) // 2 +3: c_shift=(no. of columns –1–min column index–max column index) // 2 +4: return numpy.roll(image, (r_shift, c_shift), axis=(-2, -1)) + +Fig. 4: SLAP-CC algorithm. Data cluster towards centre. + + Note that since Gomoku is not completely invariant to translation, +SLAP-CC was used to add information as additional planes instead, +as opposed to replacing the input state when SLAP was applied. 2 +planes representing stones of different colours (current and opponent +players respectively) centred together by SLAP-CC, followed by 2 +planes representing original indices for vertical and horizontal +positions respectively (scaled linearly to [1, -1]) were added along +with original 4 planes in Gomoku state representation (see 3.2). The +scaled position indices for whole plane were to give neural network +a sense of original positioning. +3.2 Representation of Gomoku +In this research, the representation of Gomoku followed the style of +AlphaGo Zero / AlphaZero, with simplification and taking [13] into +account for representation design. +For each Gomoku state, there were 4 planes representing current +player stones, opponent stones, last action and current colour +respectively by one-hot-encoding. See Fig. 5 for a typical Gomoku +state in this research, which used simplified board size 8x8 instead. + +Fig. 5: Gomoku state representation example at time t = 4. + For labels, probabilities of a move over all positions were +represented by 8x8 flattened vector. Final outcome (value) of current +player was represented by 1, 0, -1 respectively for win, draw, lose. +3.3 SLAP in Gomoku Reinforcement Learning +SLAP was used to pre-process states for network training and +inference. Transformation information from SLAP was only used in +network inference to convert probabilities (not estimated outcome) +back to corresponding game board positions for MCTS to improve +probabilities of actions, which were used as sampling probabilities +to make a move in self-play (but greedy in evaluation). See Fig. 6. + + +Fig. 6: SLAP used in Gomoku reinforcement learning. + For SLAP-CC, it was applied at the same place as SLAP in the +above flow chart, but data augmentation was kept instead of being +replaced and no transformation information was used to transform +probabilities output of the network. See methods in 3.1.3. +3.4 Testing Benefits for Neural Network Learning +To decouple from reinforcement learning dynamics, synthetic +states of Gomoku were created for testing neural network learning +with SLAP vs with typical data augmentation (by rotation and +reflection), the latter of which had 8 times the number of training +samples. Self-play was not involved in this testing. + Synthetic states were generated by first creating states each with +only 5 stones connected in a straight line (i.e. win status) for all +combinations for current black player, then removing one stone (to +be repeated with another stone 5 times to create 5 different states) +and randomly adding 4 opponent stones to become one about-to- +win state. Together these were one set of 480 about-to-win states. +Different sets could be created since white stones were merely +random. Each set was mixed with 1000 purely random states, also +with 4 stones for each player. 8 mixed sets were created, i.e. 11,840 +samples. 15%, i.e. 1,776 samples, were reserved for validation test. +Labels were assigned as follows: if there were one or more choices +to win immediately (include some purely random states, though the +chance would be very remote), the value of state would be labelled +as 1 and the wining position(s) would be labelled with probability +of move = 1/no. of winning positions, while others were labelled 0; +otherwise the value of state would be labelled as 0 and the +probability of move for each available position would be random +by uniform distribution, normalizing and summing to 1. + Neural networks (see A1) with SLAP vs with data augmentation +would learn from training samples of states and labels to predict +labels of validation data given the input states. Validation loss and +its speed of convergence would be the key metrics. + First, at preliminary stage, for each set of hyperparameters the +neural network ran 1000 iterations each with batch size 512 +sampled from training samples of size 10,064 and 80,512 +respectively for neural networks with SLAP and neural networks +with data augmentation. Sampled with replacement, same as during +reinforcement learning. There were 2400 combinations of +hyperparameters by grid search, shown in Fig. 7: + +Hyperparameter +Tested values +Remarks +use_slap +True, False +False: data augmentation +instead of SLAP +extra_act_fc +True, False +True: add extra layer (size +64) to action policy +L2 +10-3, 10-4, 10-5 +weight decay of optimizer +Num_ResBlock +0, 5, 10, 20 +no. of residual blocks +SGD +True, False +False: Adam optimizer +lr +10-1, 10-2, 10-3, 10-4, 10-5 (learning rate) +dropout +0, 0.1, 0.2, 0.3, 0.4 +Fig. 7: Hyperparameters tested at preliminary stage of CNN learning. + If Num_ResBlock > 0, the residual blocks replaced the common +CNN layers and added a convolutional layer of 256 filters (3x3 +kernel, stride 1, padding 1, no bias, ReLU activation) as the first +layer. No autoclip [17] in optimizer, unlike reinforcement learning. + At stage 2, selected models from previous stage would run for +10,000 iterations instead of 1,000 iterations, with losses recorded +every 10 iterations. +3.5 Testing Benefits for Reinforcement Learning +The baseline algorithm of Gomoku reinforcement learning +followed AlphaGo Zero/AlphaZero (see 2.3). Among their +differences, the baseline algorithm in this research followed the +better version, and thus followed AlphaZero except on symmetry +exploitation. Like AlphaGo Zero, the baseline exploited symmetry +by data augmentation to increase no. of training samples by 8 times, +but random transformation was not done in self-play. Autoclip to +gradients [17] was added in the optimizer for stable learning. + Reinforcement learning required much more computation than +neural network learning, so to save computation, the same neural +network will be used and the testing of hyperparameters would be +based on best models in neural network learning by synthetic +Gomoku states, with some deviations to be tested by grid search. + Stage 1: each of 240 models were trained by self-play of 250 games. +Data buffer size: 1,250 and 10,000 for SLAP and non-SLAP models +respectively, both roughly equivalent to storing latest 60 games. + Stage 2: selected models were trained by self-play of 5000 games. +With more games arranged for training, larger data buffer size could +be used. So data buffer size was increased to 5000 and 4000 +respectively for SLAP and non-SLAP models, roughly equivalent to + +00000000 +00000000 +00000000 +1 +1 +1 +1 +1 +00000000 +00000000 +00000000 +1 +1 +1 +00000000 +00000000 +00000000 +1 +1 +1 +00010000 +00000000 +00000000 +1 +1 +00010000 +00001000 +00000000 +1 +1 +00000000 +00010000 +00010000 +1 +1 +1 +1 +1 +1 +1 +00000000 +00000000 +00000000 +1 +1 +00000000 +00000000 +00000000 +1 +1 +1 +1 +Current player +Opponentplayer +Last action +Current ColourSelf-play: +probs&statevalues→ +Data augmentation +MCTS→actionprobs +gamestates +game→newstates +SLAP +Training samples +probabilities,outcome +game states +SLAP +transform info +states +Est.probabilities +Network Inference +newweights +Network Training +Estimatedoutcomestoring latest 250 games. To align with stage 1 testing initially, the +initial data buffer size was kept same as stage 1 for first 1000 games. +This also got rid of initial poor-quality game state data quickly. +Learning rate multiplier was used to adaptively decrease learning +rate by half if validation loss increased beyond 3-sigma limit, +measured every 100 games. + Evaluation: Independent agent(s), also called evaluation agent or +evaluator, was built by pure Monte Carlo Tree Search (MCTS) +with random policy to play against the trained AI. The strength of +a pure MCTS agent depends on no. of playouts (aka. simulations) +in each move. To facilitate observation of growing strength, multi- +tier evaluation was built by playing 10 games against each of 3 pure +MCTS agents (30 games total), each with 1000, 3000, 5000 +playouts respectively. Overall winning rate (tie counted as half win) +against them would be the key metrics for reinforcement learning. +It was often either a win or loss, and seldom a tie. Assuming that a +tie could be neglected, especially after counting tie as half win, it +simplified as Bernoulli distribution with standard deviation +approximated by √p(1 − p)/30 to calculate confidence interval, +where 30 is the number of trials in each evaluation. +3.6 Code Implementation +The part regarding AlphaZero was upgraded from [18]. Details of +implementation and code repository: https://github.com/chihangs + + +4 +Results +4.1 Impact on Neural Network Learning +4.1.1 SLAP vs Baseline (Data Augmentation) +The best few SLAP and baseline models converged to loss around +2.81 (difference < 0.01), all without residual blocks. 3 SLAP models +(denoted as s0_...) and 3 baseline models (denoted as n0_...) were +selected and their losses were plotted in Fig. 8, where each model +had Adam optimizer, same learning rate 0.001, no dropout , no +residual blocks, but different values of L2 (10-3, 10-4, 10-5). + +Fig. 8: Validation losses of SLAP and baseline models. + Above 6 models were repeated 3 more times to calculate average +time (by no. of iterations) for convergence. SLAP speeded up the +convergence by 95.1% and 71.2% measured by validation loss +reaching 3.0 and 2.9 respectively, 83.2% in average, a conservative +estimate as saving due to smaller sample size was not considered. +4.1.2 Testing Sample Size +Holding validation dataset unchanged, the training data sample size +was reduced by holding out some samples to match required size, +using models with L2=10-4 from Fig. 8. SLAP models converged +when sample size was 5032 or above, but they were more vulnerable +to decreasing no. of training samples and failed to converge when +the sample size decreased to 2516 or below, while their baseline +counterpart models (8 times the sample size) still converged. + +4.1.3 SLAP-CC vs Baseline (Data Augmentation) +SLAP-CC (see 3.1.3) was added to the 3 best baseline models from +Fig. 8. Validation losses of SLAP-CC converged to around 2.8 for +all 3 values of L2, similar to its baseline counterparts. Experiments +were repeated 3 more times to calculate average time (by no. of +iterations) to converge. The time for validation loss to reach 3.0 and +2.9 both worsened by 30.7% in average for SLAP-CC. + +4.2 Impact on Reinforcement Learning +4.2.1 SLAP vs Baseline (Data Augmentation) +The best SLAP model had highest winning rate 86.7%, equivalent to +winning 26 games out of 30. 95% confidence interval was 86.7% +/- +12.2%, i.e. (74.5%, 98.9%). The best baseline model had highest +winning rate 93.3%, equivalent to winning 28 games out of 30; 95% +confidence interval = 93.3% +/- 8.9%, i.e. (84.4%, 100%) + Best SLAP and baseline models had similar winning rates, by +confidence intervals. If winning rate of two thirds (66.6%) is used as +benchmark for this three-tier evaluation, both took 1000 games to +achieve or surpass this. However, non-SLAP took 1250 games only +to first achieve winning rate of 86.6%, while SLAP took 3000 games. +SLAP spent 0.761 second per move in self-play, 10.8% more time +than baseline (only 5% more in a separate speed-optimizing version). +SLAP tended to decrease learning rate multiplier more frequently, +implying more frequent significant increase of validation loss. + +4.2.2 Testing Buffer Size +Best models of SLAP and non-SLAP were repeated but with smaller +data buffer size of only 1,250 and 10,000 respectively throughout +whole reinforcement learning. Similar to stage 2, above models were +trained by 5000 games. With fewer data in buffer, the highest +winning rate achieved for SLAP model was only 73.3%, below the +corresponding confidence interval. The highest winning rate +achieved for non-SLAP model was only 83.3%, below the +corresponding confidence interval. So, it harmed reinforcement +learning when data buffer was too small and it was good decision to +use larger data buffer at stage 2. + +4.2.3 SLAP-CC vs Baseline (Data Augmentation) +SLAP-CC was tested by same configurations as best baseline model +from 4.2.1, but adding information from SLAP-CC and scaled +position indices as extra input feature planes. The new model also +ran for 5000 games. See methods in 3.1.3 and 3.3. The best winning +rate achieved for SLAP-CC model was 96.7%, slightly higher than +the baseline, but within the confidence interval. +NB: Learning rate multiplier did not change throughout training. + + +5 +Discussion +Despite the widely use of data augmentation to increase the variety +of transformation variants in samples to improve machine learning, +this paper proved that using SLAP to decrease the variety could +achieve the same performance of typical data augmentation with +sample size reduced by 87.5% and faster by 83.2% in convolutional +neural network learning, and statistically the same performance for +reinforcement learning with sample size reduced by 87.5%. The +success could be explained by concentrating learning experience to +certain regions when different variants were standardized, implicitly +sharing weights among variants. The proof of invariance (see 3.1.1) +after applying SLAP did not require the network to be CNN and it +could be an arbitrary function, so the applicability of SLAP should +not be restricted to CNN. While SLAP exploited only reflection and + +Slap vs Non-slap, Adam, residual blocks: +4.50 +s0_0.001ir_Adam0dropout_0.001L2 +no_0.001r_Adamodropout_0.001L2 +so_0.001ir_Adamodropout_0.00012 +4.25 +no_0.001ir_Adamodropout_0.000112 +s0_0.001ir_Adamodropout_le-05L2 +no_0.001ir_Adainodropout_1e-05L2 +4.00 +05 +3.25 +300 +2.75 +0 +1000 +2000 +000 +000 +0005 +terationsbybatchrotation symmetries in learning Gomoku, the general concept of +SLAP and the proof of invariance could apply to other symmetries. +As no domain specific features or knowledges (except symmetry) +were used in SLAP, the benefits shown in the experiments should +apply generally for domains that are symmetry invariant. + Shortcomings: in Gomoku reinforcement learning, SLAP tended +to decrease learning rate multiplier more frequently, implying more +frequent significant increase of validation loss. This instability could +be caused by faster neural network learning. Note that AlphaGo Zero +only dropped learning rate twice over 1,000,000 training steps in +their planned schedule [4]. It might imply that SLAP would need +quite different hyperparameters in reinforcement learning (as +opposed to sharing the same hyperparameters of baseline models in +the neural network learning experiment), and more or better searches +of hyperparameters for reinforcement learning would be required, +though it was constrained by computation resources. + Limitations: the results only applied to symmetry-invariant domain, +and SLAP could be more vulnerable if the sample was too small (see +4.1.2). SLAP required 10.8% more time for self-play in 4.2.1, but the +overhead would be insignificant if the simple CNN were replaced by +a deep one. It was not yet proved to speed up reinforcement learning. +Neither was it proved to be able to exploit groupoid patterns. + + +6 +Conclusion and Future Work +SLAP could improve the convergence speed of neural network +(CNN in the experiment) learning synthetic states in Gomoku by +83.2%, with only one eighth of training sample size of baseline +model (data augmentation). Since no domain specific features or +knowledges were used in SLAP, it should also benefit neural +network learning generally for domains that are symmetry invariant, +especially for reflection and rotation symmetry. As SLAP is model- +independent, the benefits should apply to models beyond CNN. But +it was not yet proved to speed up reinforcement learning, though it +could achieve similar performance with smaller training sample size. +Neither was it proved to exploit groupoid variants effectively. + As future work, SLAP may be applied in domains that are not fully +symmetry invariant, by breaking down the neural network layers into +two parts – first learning as if it were fully symmetry invariant. Or +even split into stages by type of symmetries. Although SLAP is not +directly differentiable, one workaround would be similar to that in +transforming Gomoku action probabilities. That is, given the +transformation information as another input, transform the learned +output back to corresponding original position, and then carry out +necessary subsequent computations forward. This helps create more +explainable stages and transfer learning. Another future work might +be differentiable approximation of SLAP. + + +ACKNOWLEDGEMENTS +This research was supervised by Dr Eduardo Alonso. + +REFERENCES +[1] +Hinton G. E., Krizhevsky A. and Wang S. D. 2011, ‘Transforming Auto- +Encoders’, International Conference on Artificial Neural Networks (ICANN), +2011. +[2] +Sabour S., Frosst N. and Hinton G. 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S., Zagoruyko S. and Komodakis N. 2018, ‘Exploring Weight +Symmetry in Deep Neural Networks’, arXiv preprint arXiv:1812.11027, +Dec 2018. +[11] Bergman D. 2019, ‘Symmetry Constrained Machine Learning’, arXiv +preprint arXiv:1811.07051v2, 2019. +[12] Higgins I. et al 2018, ‘Towards a Definition of Disentangled +Representations’, arXiv preprint arXiv:1812.02230, Dec 2018. +[13] Caselles-Dupré H., Garcia-Ortiz M. and Filliat D. 2019, ‘Symmetry-Based +Disentangled +Representation +Learning +requires +Interaction +with +Environments’, 33rd Conference on Neural Information Processing Systems +(NeurIPS 2019), Vancouver, Canada, 2019. +[14] Anselmi F. et al 2017, ‘Symmetry Regularization’, The Centre for Brain, +Mind and Machines (CBMM) Memo No. 63, 26 May 2017. +[15] Botvinick M. et al 2018, ‘Reinforcement Learning, Fast and Slow’, Trends +in Cognitive Sciences, Vol 23, Issue 5, pp. 408-422, 2019. +[16] Higgins I., Racanière S. and Rezende D. 2022, ‘Symmetry-Based +Representations for Artificial and Biological Intelligence’, arXiv preprint +arXiv:2203.09250, 2022. +[17] Seetharaman P. et al 2020, ‘AutoClip: Adaptive Gradient Clipping for +Source Separation Networks’, 2020 IEEE 30th International Workshop on +Machine Learning for Signal Processing (MLSP), 2020. +[18] Song J. 2017, ‘An implementation of the AlphaZero algorithm for Gomoku +(also called Gobang or Five in a Row)’, github.com. Available at: +https://github.com/junxiaosong/AlphaZero_Gomoku (Accessed: Jun-Sep +2022). +[19] K. Maharana, S. Mondal and B. Nemade 2022, ‘A review: Data pre- +processing and data augmentation techniques’, Global Transitions +Proceedings, Vol. 3, Issue 1, pp. 91-99, Jun 2022. + + + +APPENDICES +A1 Neural Network Architecture and Configurations +The architecture and configurations used (unless otherwise stated): +Neural network: consisted of 3 common convolutional layers (32, +64, 128 filters respectively) each with 3x3 kernel of stride 1 and +padding 1 with ReLU activation, followed by 2 action policy +players and in parallel 3 state value layers. The input was 8 x 8 x 4 +image stack comprising of 4 binary feature planes. The action +policy layers had one convolutional layer with 4 filters each with +1x1 kernel of stride 1 with ReLU activation, followed by a fully +connected linear layer to output a vector of size 64 corresponding +to logit probabilities for all intersection points of the board. The +state value layers had one convolutional layer with 2 filters each +with 1x1 kernel of stride 1 with ReLU activation, followed by fully +connected linear layer to a hidden layer of size 64 with ReLU +activation, finally fully connected to a scalar with tanh activation. +Dropout would be applied to all action policy layers and state value +layers except output layers; not applied to common layers. +Optimizer: Adam with autoclip [17] +Batch size per optimisation step: 512 (2048 in AlphaGo Zero) +Data buffer size: 10,000 for data augmentation, 1,250 for SLAP +No. of network optimisation steps per policy iteration: 10 +No. of self-play games per policy iteration: 1 +No. of playouts: 400 (1600 in AlphaGo Zero, 800 in AlphaZero) +Cpuct (constant of upper confidence bound in MCTS) : 5 +Temperature parameter: 1 (same as AlphaZero) +Dirichlet alpha of noise: 0.3 (same as chess in AlphaZero) + Smaller batch size and no. of playouts per move in MCTS were +used because Gomoku is less complex than Go. Dirichlet alpha was +initially set at 0.3 because mini Gomoku (8x8 board) has same board +size as chess and similar no. of available action choices per move. + diff --git a/XtE3T4oBgHgl3EQf1gsc/content/tmp_files/load_file.txt b/XtE3T4oBgHgl3EQf1gsc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52459724cbb5f9b67aede6b6058451f512fdd6bb --- /dev/null +++ b/XtE3T4oBgHgl3EQf1gsc/content/tmp_files/load_file.txt @@ -0,0 +1,579 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf,len=578 +page_content='Switchable Lightweight Anti-symmetric Processing (SLAP) with CNN to Reduce Sample Size and Speed up Learning – Application in Gomoku Reinforcement Learning Chi-Hang Suen1 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' To replace data augmentation, this paper proposed a method called SLAP to intensify experience to speed up machine learning and reduce the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP is a model-independent protocol/function to produce the same output given different transformation variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP improved the convergence speed of convolutional neural network learning by 83% in the experiments with Gomoku game states, with only one eighth of the sample size compared with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' In reinforcement learning for Gomoku, using AlphaGo Zero/AlphaZero algorithm with data augmentation as baseline, SLAP reduced the number of training samples by a factor of 8 and achieved similar winning rate against the same evaluator, but it was not yet evident that it could speed up reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The benefits should at least apply to domains that are invariant to symmetry or certain transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' As future work, SLAP may aid more explainable learning and transfer learning for domains that are not invariant to symmetry, as a small step towards artificial general intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 Problem Convolutional neural network (CNN) is now the mainstream family of models for computer vision, thanks to its weight sharing mechanism to efficiently share learning across the same plane by so- called kernels, achieving local translational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' But CNN is not reflection and rotation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Typically it can be addressed by data augmentation to inputs by reflection and rotation if necessary, but the sample size would increase substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [1] criticised CNN that it could not learn spatial relationships such as orientation, position and hierarchy and advocated their novel capsule to replace CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [2] improved capsule using routing by agreement mechanism and outperformed CNN at recognising overlapping images, but they also admitted that it tended to account for everything in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' This implies capsule is too heavy in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Inspired by the idea of capturing orientation information in capsule network [2], this paper proposed a novel method called Switchable Lightweight Anti- symmetric Process (SLAP), a protocol to produce the same output given different transformation variants, with the main research question: can symmetry variants be exploited directly by SLAP to improve and combine with CNN for machine learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Very often, we know in advance if a certain machine learning task is invariant to certain types of transformation, such as rotation and reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' in Gomoku, the state is rotation (perpendicularly) and reflection (horizontally and vertically) invariant in terms of winning probability, and “partially” translation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Symmetry is often exploited by data augmentation for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' But this greatly increases the dataset size if all symmetry variants are included – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' there are 8 such variants for each Gomoku state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP was invented in this paper to avoid such expansion (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' On the other hand, reinforcement learning is notorious for lengthy training time and large sample size required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Data augmentation may help improve performance in reinforcement learning, but it would increase the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' This research tried to kill two birds by one stone, SLAP, by applying with CNN in reinforcement learning (of 1 City, University of London, Department of Computer Science, email: chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='suen@city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='uk Gomoku), challenging the widely used practice of data augmentation, aiming at reducing the sample size and improving the learning speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Switchable Lightweight Anti-symmetric Process (SLAP) SLAP is a model-independent protocol and function to always produce or choose the same variant regardless of which transformation variant (by specified symmetry) is given, and if required also output the corresponding transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It can be used upon any function or model to produce outputs that are invariant with regard to specified symmetric properties of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If some (type) of the outputs are not invariant but follow the same transformation, the corresponding transformation information from SLAP may be used to transform these outputs back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It can be viewed as standardization of symmetry, as opposed to standardization of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' After processing, symmetric variants are filtered out – that’s why it is named ‘anti-symmetric process’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Ironically, with this anti- symmetric process, the function or model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' CNN) to be fed would look as if it is symmetric with regard to whichever the symmetry variant is the input, and the same output is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It is a novel method to exploit symmetry variants in machine learning without increasing the number of training samples by data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The motivation is to concentrate experience to speed up learning, without enlarging the sample size by data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' See details in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 Gomoku Gomoku, or Five in a Row, is a 2-player board game, traditionally played with Go pieces (black and white stones) on a Go board (19x19), nowadays on 15x15 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For experiments in this research, mini board 8x8 was used instead to save computation, and the rule of freestyle version was adopted: Black and white place stones of his colour alternatively at an unoccupied intersection point of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Black first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Winner: the one who first forms an unbroken chain of 5 stones of his colour in a straight line (horizonal, vertical or diagonal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Draw happens if there is no winner when the board is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Gomoku was chosen to demonstrate the benefit of SLAP because: Gomoku has huge number of state representations (3225 ~= 2x10107), which justify the use of neural network for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Gomoku is rotation and reflection invariant, but only “partially” translation invariant, so ideal to test different transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Gomoku is Markov Decision Process, meeting the basic mathematical assumption of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [4] and [5] showed a general effective reinforcement learning algorithm for board games and Gomoku is simple to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 CNN CNN (convolutional neural network) has been widely used for computer vision but it is known that CNN is weak to deal with changes by rotation/orientation unless with much larger sample size by data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' To address this problem, [1] proposed that neural network should make use of their then novel capsule, learning to recognize an implicitly defined visual entity and output probability of its existence and instantiation parameters such as pose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' they showed that a transforming auto-encoder could be learnt to force the output (which is a vector instead of scalar) of a capsule to represent an image property that one might want to manipulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [2] showed that a discriminatively trained, multi-layer capsule system achieves state-of-the-art performance on MNIST and was considerably better than CNN at recognizing highly overlapping digits, using the so- called routing by agreement mechanism, and yet [2] admitted that one drawback was the tendency of capsule to account for everything in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It implies that the capsule might be too “heavy” for computation and so a lightweight method is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' But the capsule network with routing by agreement algorithm has been proved not to be a universal approximator [3], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' not fit to all kinds of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' As such, this research did not attempt to replace CNN by capsule, but simply created SLAP to combine with CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Instead of forcing the output to represent certain transformation information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' orientation angle), SLAP forces the input of different variants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' different rotation angle) to give the same output variant (and output the transformation information e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' angle, if needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Nevertheless, the invention of SLAP was inspired by [1] & [2] trying to address the weakness of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Groupoid in Gomoku There are different Gomoku states of the same groupoid (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1), which means having local symmetry but not necessarily global symmetry of the whole structure [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Groupoid is more challenging than symmetry or group, as some groupoids may not have the same status, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' But the potential for learning is huge as there are much more variants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 156 variants by translation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1: Gomoku groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Black can stop white win in C, but not in A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 AlphaGo Zero / Alpha Zero For reinforcement learning of Gomoku in this research, the baseline algorithm was chosen to follow that of AlphaGo Zero [4] and Alpha Zero [5] papers because domain knowledge was not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The algorithm was concisely summarized by [7] as follows: Neural network The neural network feature extractor is a type of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It takes state st as input and yields value of state 𝑣𝜃(𝑠𝑡) ∈ [−1, 1] and policy 𝑝𝜃 ⃗⃗⃗⃗ (st) as probability vector over all possible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It has the following loss function (excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' regularization terms): loss = ∑ (𝑣𝜃(𝑠𝑡) − 𝑡 𝑧𝑡)2 – 𝜋⃗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' log(𝑝𝜃 ⃗⃗⃗⃗ (st)) , where 𝑧𝑡, 𝜋⃗ t are final outcome {-1,0,1} and estimate (to be discussed below) of policy from state st respectively, with 1, 0, -1 representing win, draw, lose respectively for current player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Monte Carlo Tree Search (MCTS) as policy improvement operator At each node, action is chosen by maximizing U(s, a), the upper confidence bound of Q-value Q(s, a), calculated by: U(s, a) = Q(s, a) + C * P(s, a) * √∑ 𝑁(𝑠,𝑏) 𝑏 1+𝑁(𝑠,𝑎) where N(s, a) = no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of times taking action a from state s in MCTS simulation, P(s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=') = 𝑝𝜃 ⃗⃗⃗⃗ (s), and the policy estimate of probability is improved by using 𝜋⃗ t = N(s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=') / ∑ 𝑁(𝑠, 𝑏) When a new node (not visited before from parent node) is reached, instead of rollout, the value of new node is obtained from neural network and propagated up the search path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Unless the new node is terminal, the new node is expanded to have child nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Self-play training as policy evaluation operator In each turn, a fixed number of MCTS simulations are conducted from the state st, and action is selected by sampling from the policy estimate of probabilities improved by MCTS, thus generating training sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' At the end of an iteration, the neural network is updated by learning from the training sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The evaluation metric would be based on winning and drawing percentages of the AI against an independent evaluation agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' There are differences among AlphaGo Zero and AlphaZero, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2: AlphaGo Zero [4] AlphaZero [5] Pitting models Yes, model with new weights plays against previous one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' new weights are adopted only if it wins 55% or above No, always use new weights after each iteration of neural network learning Symmetry Data augmentation by rotation and reflection to increase sample size by 8 times for training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' transform to one of 8 variants randomly in self-play for inference Not exploited,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' as it is intended for generalization Action in self-play Sampled proportional to visit count in MCTS in first 30 moves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' then selected greedily by max visit count (asymptotically with highest winning chance) in MCTS Sampled proportional to visit count in MCTS Outcome prediction Assume binary win/loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' estimate & optimise winning probability Also consider draw or other outcomes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' estimate & optimise expected outcome Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2: Differences between AlphaGo Zero and AlphaZero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='4 Other Related Works, Symmetry and AGI On lightweight capsule, DSC-CapsNet was proposed as lightweight capsule network, which focused on computing efficiency and reducing number of parameters [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [9] proposed dense capsule network with fewer parameters – neither had the novel structure proposed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' On symmetric CNN, [10] proposed to impose symmetry in neural network parameters by repeating some parameters and achieved 25% reduction in number of parameters with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2% loss in accuracy using ResNet-101, a type of CNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' but unlike SLAP, symmetry was not imposed in the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [11] incorporated symmetry into neural network by creating symmetry (of specific type) invariant features, but no implementation or idea similar to SLAP was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Studies have shown rotation based augmentation performed better than many other augmentation techniques [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The type of data augmentation used as baseline in this research was rotation and reflection based (also the type used by AlphaGo Zero[4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The novelty lies in the fact that SLAP is opposite to the practice of data augmentation – decreasing the variety of variants in the data instead for machine learning, though also exploiting symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Symmetry is one of the natures of the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Animals can detect the same object or the same prey being moved (translated), or even rotated after being slapped (the novel method was deliberately abbreviated as SLAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Recognising symmetry can also speed up learning patterns, a typical trick used for playing some board games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' To facilitate research exploiting symmetry in machine learning, [12] connected symmetry transformations to vector representations by the formalism of group and representation theory to arrive at the first formal definition of disentangled representations, expected to benefit StateA StateB Stateclearning from separating out (disentangling) the underlying structure of the world into disjoint parts of its representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Upon this work, [13] showed by theory and experiments that Symmetry-Based Disentangled Representation Learning (SBDRL) could not only be based on static observations: agents should interact with the environment to discover its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' They emphasized that the representation should use transitions rather than still observations for SBDRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' This was taken into account for designing the Gomoku representation for reinforcement learning in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' One may expect that an artificial general intelligence (AGI) system, if invented, should be able to learn unknown symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Researchers have worked on this, for example [14] proposed learning unknown symmetries by different principles of family of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' But it is equally important to learn by exploiting symmetry more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For example, if an AGI system can interpret the rules of Gomoku and realize from the rules that Gomoku is reflection and rotation invariant, it should directly exploit such symmetry instead of assuming symmetry is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Ideally, such exploitation should be switched on easily if one wishes, and hence the term ‘switchable’ in SLAP, which can be used upon any function or model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If transfer learning in CNN is analogous to reusing a chair by cutting the legs and installing new legs to fit another, such ‘switchable learning’ in SLAP is analogous to turning the switch of an adjustable chair to fit certain symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Such kind of ‘switch ’ in design can also help AI be more explainable and transparent, and more easily reused or transferred, while an AGI system should be able to link and switch to different sub-systems easily to solve a problem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP can also reduce memory required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For example, AlphaGo Zero used a transposition table [4], a cache of previously seen positions and associated evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Had SLAP been used instead of data augmentation, such memory size could be reduced by a factor of 8, or alternatively 8 times more positions or states could be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Indeed memory plays an important role in reinforcement learning as well by episodic memory, an explicit record of past events to be taken as reference for making decisions, improving both sample efficiency and speed in reinforcement learning as experience can be used immediately for making decisions [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It is likely that an AGI system would, just like human, use memory to solve some problems rather than always resort to learning from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' And in the real word, a continuous space, there can be much more than 8 equivalent variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Recently, [16] suggested that symmetry should be an important general framework that determines the structure of universe, constrains the nature of natural tasks and consequently shape both biological and artificial general intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' they argued that symmetry transformations should be a fundamental principle in search for a good representation in learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Perhaps SLAP may contribute a tiny step towards AGI, by shaping input representations directly by symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Note that SLAP can be used upon any function or model and even if some (types) of the outputs are not invariant but follow the same transformation, these may be broken down and use the transformation information output from SLAP to make appropriate transformation back later for these parts only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' A little kid often mistakes b for d at the beginning of learning alphabets, and it appears that human learning types of objects by vision might naturally assume symmetry first and then learn non- symmetry later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If a machine learning problem is to be split into stages or parts by specified symmetry as a guide, SLAP might help by wrapping certain parts of a function or neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3 Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 SLAP SLAP forces the input of different variants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' different rotation angle) to give the same output variant (and output the transformation information e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' angle, though not necessarily used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' There can be multiple ways to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For rotation and reflection variants of Gomoku states, one way to implement this is simply flattening the pixels of 8 variants to 8 lists, compare the lists and always choose the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Below (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3) was the algorithm used for SLAP in dealing with rotation and reflection variants of Gomoku states, but the concept may be applied to other symmetries as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Algorithm SLAP 1: Generate symmetry variants of input, store required transformation 2: Convert each variant to a list 3: Compare each list and find the ‘largest’ list 4: return the ‘largest’ variant & required transformation of the variant Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3: SLAP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Positive large data cluster towards top left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If the image/state has multiple input channels or planes in one sample, the first channel/plane is compared first by list comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP was implemented by numpy instead of torch tensor for faster speed, because numpy uses view for rotation and reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The output variant replaced the input state when SLAP was applied in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' During inference time, output action probabilities from neural network would be transformed back using the transformation information (rotation & reflection) from SLAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 Invariance Denote s, t = slap(xi), where slap is SLAP function in pythonic style, s is the symmetry (of certain group G, with n symmetry variants for each state) variant and t is corresponding transformation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Given property of slap, for all i∈N<=n, s, t = slap(x1) = slap(x2) = … = slap(xn) Denote s = slap(xi)[0], t = slap(xi)[1], the pythonic expression to capture first and second return variables of a function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Denote h(slap(xi)[0]) as hslap(xi) for any function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Given an arbitrary function y = f(x), y = fslap(xi) ⇒ y = f(slap(xi)[0]) ⇒ y = f(s) for all i ∴ y = fslap(xi) is invariant with respect to i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' symmetry of group G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' When f is the neural network, the composite function resulting from the neural network, fslap, is invariant to symmetry (of group G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Differentiability SLAP was not applied to intermediate layers of neural networks for Gomoku, so its differentiability was not required in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Approximation would be required to make it differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 Groupoid and SLAP-CC As Gomoku is only ‘partially’ invariant to translation, it is also interesting to experiment with translation variants, which are considered to be groupoid instead of group as they are symmetric locally but not necessarily symmetric globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' There can be many more translation variants than rotation and reflection variants, see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' To save computation, different algorithm (crop and centre) was used to ‘standardize’ translation variants, denoted as SLAP-CC in the below, to emphasize that it shared the same general idea as SLAP, but just different way for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Denoted as cc in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The algorithm of SLAP-CC, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4, would concentrate experience around the centre, as input variant was centred to become output variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If it could not be exactly centred, the algorithm would make it slightly lean to top left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Algorithm SLAP-CC 1: Find non-empty min & max indices by row & column in input image 2: r_shift = (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of rows – 1– min row index – max row index) // 2 3: c_shift=(no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of columns –1–min column index–max column index) // 2 4: return numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='roll(image, (r_shift, c_shift), axis=(-2, -1)) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4: SLAP-CC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Data cluster towards centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Note that since Gomoku is not completely invariant to translation, SLAP-CC was used to add information as additional planes instead, as opposed to replacing the input state when SLAP was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 2 planes representing stones of different colours (current and opponent players respectively) centred together by SLAP-CC, followed by 2 planes representing original indices for vertical and horizontal positions respectively (scaled linearly to [1, -1]) were added along with original 4 planes in Gomoku state representation (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The scaled position indices for whole plane were to give neural network a sense of original positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Representation of Gomoku In this research, the representation of Gomoku followed the style of AlphaGo Zero / AlphaZero, with simplification and taking [13] into account for representation design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For each Gomoku state, there were 4 planes representing current player stones, opponent stones, last action and current colour respectively by one-hot-encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 5 for a typical Gomoku state in this research, which used simplified board size 8x8 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 5: Gomoku state representation example at time t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For labels, probabilities of a move over all positions were represented by 8x8 flattened vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Final outcome (value) of current player was represented by 1, 0, -1 respectively for win, draw, lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 SLAP in Gomoku Reinforcement Learning SLAP was used to pre-process states for network training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Transformation information from SLAP was only used in network inference to convert probabilities (not estimated outcome) back to corresponding game board positions for MCTS to improve probabilities of actions, which were used as sampling probabilities to make a move in self-play (but greedy in evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 6: SLAP used in Gomoku reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' For SLAP-CC, it was applied at the same place as SLAP in the above flow chart, but data augmentation was kept instead of being replaced and no transformation information was used to transform probabilities output of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' See methods in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='4 Testing Benefits for Neural Network Learning To decouple from reinforcement learning dynamics, synthetic states of Gomoku were created for testing neural network learning with SLAP vs with typical data augmentation (by rotation and reflection), the latter of which had 8 times the number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Self-play was not involved in this testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Synthetic states were generated by first creating states each with only 5 stones connected in a straight line (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' win status) for all combinations for current black player, then removing one stone (to be repeated with another stone 5 times to create 5 different states) and randomly adding 4 opponent stones to become one about-to- win state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Together these were one set of 480 about-to-win states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Different sets could be created since white stones were merely random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Each set was mixed with 1000 purely random states, also with 4 stones for each player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 8 mixed sets were created, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 11,840 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 15%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 1,776 samples, were reserved for validation test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Labels were assigned as follows: if there were one or more choices to win immediately (include some purely random states, though the chance would be very remote), the value of state would be labelled as 1 and the wining position(s) would be labelled with probability of move = 1/no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of winning positions, while others were labelled 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' otherwise the value of state would be labelled as 0 and the probability of move for each available position would be random by uniform distribution, normalizing and summing to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Neural networks (see A1) with SLAP vs with data augmentation would learn from training samples of states and labels to predict labels of validation data given the input states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Validation loss and its speed of convergence would be the key metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' First, at preliminary stage, for each set of hyperparameters the neural network ran 1000 iterations each with batch size 512 sampled from training samples of size 10,064 and 80,512 respectively for neural networks with SLAP and neural networks with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Sampled with replacement, same as during reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' There were 2400 combinations of hyperparameters by grid search, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 7: Hyperparameter Tested values Remarks use_slap True, False False: data augmentation instead of SLAP extra_act_fc True, False True: add extra layer (size 64) to action policy L2 10-3, 10-4, 10-5 weight decay of optimizer Num_ResBlock 0, 5, 10, 20 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of residual blocks SGD True, False False: Adam optimizer lr 10-1, 10-2, 10-3, 10-4, 10-5 (learning rate) dropout 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 7: Hyperparameters tested at preliminary stage of CNN learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If Num_ResBlock > 0, the residual blocks replaced the common CNN layers and added a convolutional layer of 256 filters (3x3 kernel, stride 1, padding 1, no bias, ReLU activation) as the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' No autoclip [17] in optimizer, unlike reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' At stage 2, selected models from previous stage would run for 10,000 iterations instead of 1,000 iterations, with losses recorded every 10 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='5 Testing Benefits for Reinforcement Learning The baseline algorithm of Gomoku reinforcement learning followed AlphaGo Zero/AlphaZero (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Among their differences, the baseline algorithm in this research followed the better version, and thus followed AlphaZero except on symmetry exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Like AlphaGo Zero, the baseline exploited symmetry by data augmentation to increase no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of training samples by 8 times, but random transformation was not done in self-play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Autoclip to gradients [17] was added in the optimizer for stable learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Reinforcement learning required much more computation than neural network learning, so to save computation, the same neural network will be used and the testing of hyperparameters would be based on best models in neural network learning by synthetic Gomoku states, with some deviations to be tested by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Stage 1: each of 240 models were trained by self-play of 250 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Data buffer size: 1,250 and 10,000 for SLAP and non-SLAP models respectively, both roughly equivalent to storing latest 60 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Stage 2: selected models were trained by self-play of 5000 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' With more games arranged for training, larger data buffer size could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' So data buffer size was increased to 5000 and 4000 respectively for SLAP and non-SLAP models,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='Current ColourSelf-play: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='probs&statevalues→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='Data augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='MCTS→actionprobs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='gamestates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='game→newstates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='SLAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='Training samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='probabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='outcome game states SLAP transform info states Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='probabilities Network Inference newweights Network Training Estimatedoutcomestoring latest 250 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' To align with stage 1 testing initially, the initial data buffer size was kept same as stage 1 for first 1000 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' This also got rid of initial poor-quality game state data quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Learning rate multiplier was used to adaptively decrease learning rate by half if validation loss increased beyond 3-sigma limit, measured every 100 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Evaluation: Independent agent(s), also called evaluation agent or evaluator, was built by pure Monte Carlo Tree Search (MCTS) with random policy to play against the trained AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The strength of a pure MCTS agent depends on no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of playouts (aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' simulations) in each move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' To facilitate observation of growing strength, multi- tier evaluation was built by playing 10 games against each of 3 pure MCTS agents (30 games total), each with 1000, 3000, 5000 playouts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Overall winning rate (tie counted as half win) against them would be the key metrics for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It was often either a win or loss, and seldom a tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Assuming that a tie could be neglected, especially after counting tie as half win, it simplified as Bernoulli distribution with standard deviation approximated by √p(1 − p)/30 to calculate confidence interval, where 30 is the number of trials in each evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='6 Code Implementation The part regarding AlphaZero was upgraded from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Details of implementation and code repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='com/chihangs 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 Impact on Neural Network Learning 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 SLAP vs Baseline (Data Augmentation) The best few SLAP and baseline models converged to loss around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='81 (difference < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='01), all without residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3 SLAP models (denoted as s0_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=') and 3 baseline models (denoted as n0_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=') were selected and their losses were plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 8, where each model had Adam optimizer, same learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001, no dropout , no residual blocks, but different values of L2 (10-3, 10-4, 10-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 8: Validation losses of SLAP and baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Above 6 models were repeated 3 more times to calculate average time (by no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of iterations) for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP speeded up the convergence by 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1% and 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2% measured by validation loss reaching 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='9 respectively, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2% in average, a conservative estimate as saving due to smaller sample size was not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Testing Sample Size Holding validation dataset unchanged, the training data sample size was reduced by holding out some samples to match required size, using models with L2=10-4 from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP models converged when sample size was 5032 or above, but they were more vulnerable to decreasing no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of training samples and failed to converge when the sample size decreased to 2516 or below, while their baseline counterpart models (8 times the sample size) still converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 SLAP-CC vs Baseline (Data Augmentation) SLAP-CC (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3) was added to the 3 best baseline models from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Validation losses of SLAP-CC converged to around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='8 for all 3 values of L2, similar to its baseline counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Experiments were repeated 3 more times to calculate average time (by no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of iterations) to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The time for validation loss to reach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='9 both worsened by 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='7% in average for SLAP-CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Impact on Reinforcement Learning 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1 SLAP vs Baseline (Data Augmentation) The best SLAP model had highest winning rate 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='7%, equivalent to winning 26 games out of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 95% confidence interval was 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='7% +/- 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='5%, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The best baseline model had highest winning rate 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3%, equivalent to winning 28 games out of 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 95% confidence interval = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3% +/- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='9%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='4%, 100%) Best SLAP and baseline models had similar winning rates, by confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' If winning rate of two thirds (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='6%) is used as benchmark for this three-tier evaluation, both took 1000 games to achieve or surpass this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' However, non-SLAP took 1250 games only to first achieve winning rate of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='6%, while SLAP took 3000 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP spent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='761 second per move in self-play, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='8% more time than baseline (only 5% more in a separate speed-optimizing version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP tended to decrease learning rate multiplier more frequently, implying more frequent significant increase of validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2 Testing Buffer Size Best models of SLAP and non-SLAP were repeated but with smaller data buffer size of only 1,250 and 10,000 respectively throughout whole reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Similar to stage 2, above models were trained by 5000 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' With fewer data in buffer, the highest winning rate achieved for SLAP model was only 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3%, below the corresponding confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The highest winning rate achieved for non-SLAP model was only 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3%, below the corresponding confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' So, it harmed reinforcement learning when data buffer was too small and it was good decision to use larger data buffer at stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 SLAP-CC vs Baseline (Data Augmentation) SLAP-CC was tested by same configurations as best baseline model from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1, but adding information from SLAP-CC and scaled position indices as extra input feature planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The new model also ran for 5000 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' See methods in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The best winning rate achieved for SLAP-CC model was 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='7%, slightly higher than the baseline, but within the confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' NB: Learning rate multiplier did not change throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 5 Discussion Despite the widely use of data augmentation to increase the variety of transformation variants in samples to improve machine learning, this paper proved that using SLAP to decrease the variety could achieve the same performance of typical data augmentation with sample size reduced by 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='5% and faster by 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2% in convolutional neural network learning, and statistically the same performance for reinforcement learning with sample size reduced by 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The success could be explained by concentrating learning experience to certain regions when different variants were standardized, implicitly sharing weights among variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The proof of invariance (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1) after applying SLAP did not require the network to be CNN and it could be an arbitrary function, so the applicability of SLAP should not be restricted to CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' While SLAP exploited only reflection and Slap vs Non-slap, Adam, residual blocks: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='50 s0_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001ir_Adam0dropout_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001L2 no_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001r_Adamodropout_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001L2 so_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001ir_Adamodropout_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='00012 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='25 no_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001ir_Adamodropout_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='000112 s0_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001ir_Adamodropout_le-05L2 no_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='001ir_Adainodropout_1e-05L2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='00 05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='25 300 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='75 0 1000 2000 000 000 0005 terationsbybatchrotation symmetries in learning Gomoku, the general concept of SLAP and the proof of invariance could apply to other symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' As no domain specific features or knowledges (except symmetry) were used in SLAP, the benefits shown in the experiments should apply generally for domains that are symmetry invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Shortcomings: in Gomoku reinforcement learning, SLAP tended to decrease learning rate multiplier more frequently, implying more frequent significant increase of validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' This instability could be caused by faster neural network learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Note that AlphaGo Zero only dropped learning rate twice over 1,000,000 training steps in their planned schedule [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It might imply that SLAP would need quite different hyperparameters in reinforcement learning (as opposed to sharing the same hyperparameters of baseline models in the neural network learning experiment), and more or better searches of hyperparameters for reinforcement learning would be required, though it was constrained by computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Limitations: the results only applied to symmetry-invariant domain, and SLAP could be more vulnerable if the sample was too small (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' SLAP required 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='8% more time for self-play in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='1, but the overhead would be insignificant if the simple CNN were replaced by a deep one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' It was not yet proved to speed up reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Neither was it proved to be able to exploit groupoid patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 6 Conclusion and Future Work SLAP could improve the convergence speed of neural network (CNN in the experiment) learning synthetic states in Gomoku by 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='2%, with only one eighth of training sample size of baseline model (data augmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Since no domain specific features or knowledges were used in SLAP, it should also benefit neural network learning generally for domains that are symmetry invariant, especially for reflection and rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' As SLAP is model- independent, the benefits should apply to models beyond CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' But it was not yet proved to speed up reinforcement learning, though it could achieve similar performance with smaller training sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Neither was it proved to exploit groupoid variants effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' As future work, SLAP may be applied in domains that are not fully symmetry invariant, by breaking down the neural network layers into two parts – first learning as if it were fully symmetry invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Or even split into stages by type of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Although SLAP is not directly differentiable, one workaround would be similar to that in transforming Gomoku action probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' That is, given the transformation information as another input, transform the learned output back to corresponding original position, and then carry out necessary subsequent computations forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' This helps create more explainable stages and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Another 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implementation of the AlphaZero algorithm for Gomoku (also called Gobang or Five in a Row)’, github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='com/junxiaosong/AlphaZero_Gomoku (Accessed: Jun-Sep 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Maharana, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Mondal and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Nemade 2022, ‘A review: Data pre- processing and data augmentation techniques’, Global Transitions Proceedings, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 3, Issue 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' 91-99, Jun 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' APPENDICES A1 Neural Network Architecture and Configurations The architecture and configurations used (unless otherwise stated): Neural network: consisted of 3 common convolutional layers (32, 64, 128 filters respectively) each with 3x3 kernel of stride 1 and padding 1 with ReLU activation, followed by 2 action policy players and in parallel 3 state value layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The input was 8 x 8 x 4 image stack comprising of 4 binary feature planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The action policy layers had one convolutional layer with 4 filters each with 1x1 kernel of stride 1 with ReLU activation, followed by a fully connected linear layer to output a vector of size 64 corresponding to logit probabilities for all intersection points of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' The state value layers had one convolutional layer with 2 filters each with 1x1 kernel of stride 1 with ReLU activation, followed by fully connected linear layer to a hidden layer of size 64 with ReLU activation, finally fully connected to a scalar with tanh activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Dropout would be applied to all action policy layers and state value layers except output layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' not applied to common layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Optimizer: Adam with autoclip [17] Batch size per optimisation step: 512 (2048 in AlphaGo Zero) Data buffer size: 10,000 for data augmentation, 1,250 for SLAP No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of network optimisation steps per policy iteration: 10 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of self-play games per policy iteration: 1 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of playouts: 400 (1600 in AlphaGo Zero, 800 in AlphaZero) Cpuct (constant of upper confidence bound in MCTS) : 5 Temperature parameter: 1 (same as AlphaZero) Dirichlet alpha of noise: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 (same as chess in AlphaZero) Smaller batch size and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of playouts per move in MCTS were used because Gomoku is less complex than Go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' Dirichlet alpha was initially set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content='3 because mini Gomoku (8x8 board) has same board size as chess and similar no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} +page_content=' of available action choices per move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQf1gsc/content/2301.04746v1.pdf'} diff --git a/_dAzT4oBgHgl3EQfFvqG/content/tmp_files/2301.01016v1.pdf.txt b/_dAzT4oBgHgl3EQfFvqG/content/tmp_files/2301.01016v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..85c08aa0c4d98c2b1e9fe75d8105f92af9b10a0a --- /dev/null +++ b/_dAzT4oBgHgl3EQfFvqG/content/tmp_files/2301.01016v1.pdf.txt @@ -0,0 +1,1428 @@ +Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities +Mechanical scanning probe lithography of perovskites for fabrication of +high-Q planar polaritonic cavities +N. Glebov,1, a) M. Masharin,1, a) B. Borodin,2 P. Alekseev,2 F. Benimetskiy,3 S. Makarov,1, 4 and A. Samusev*1, 5 +1)ITMO University, School of Physics and Engineering, St. Petersburg, 197101, Russia +2)Ioffe Institute, Saint-Petersburg 194021, Russia +3)Department of Physics and Astronomy, University of Sheffield, S3 7RH, Sheffield, UK +4)Qingdao Innovation and Development Center, Harbin Engineering University, Qingdao 266000, Shandong, +China +5)Experimentelle Physik 2, Technische Universit¨at Dortmund, 44227 Dortmund, Germany +(*Electronic mail: anton.samusev@gmail.com) +(Dated: 4 January 2023) +Exciton-polaritons are unique quasiparticles with hybrid properties of an exciton and a photon, opening ways to realize +ultrafast strongly nonlinear systems and inversion-free lasers based on Bose-Einstein polariton condensation. However, +the real-world applications of the polariton systems are still limited due to the temperature operation and costly fabri- +cation techniques for both exciton materials and photon cavities. 2D perovskites represent one of the most prospective +platforms for the realization of strong light-matter coupling since they possess room-temperature exciton states with +large oscillator strength and can simultaneously provide planar photon cavities with high field localization due to the +huge refractive index of the material. In this work, we demonstrate for the first time the mechanical scanning probe +lithography method for the realization of low-cost room-temperature exciton-polariton systems based on the 2D per- +ovskite (PEA)2PbI4 with exciton binding energy exceeding 200 meV. Precisely controlling the lithography parameters, +we broadly adjust the exciton-polariton dispersion, and radiative losses of polaritonic modes in the range of 0.1 to 0.2 +of total optical losses. Our findings represent a versatile approach to the fabrication of planar high-quality perovskite- +based photonic cavities supporting the strong light-matter coupling regime for the development of on-chip all-optical +active and nonlinear polaritonic devices. +I. +INTRODUCTION +Photonics deals both with fundamental and applied aspects +of operating with optical signals, as well as with prospective +designing energy-efficient optical computing devices. Imple- +menting such devices where light is controlled by light re- +quires systems with strong optical nonlinearity. Optical sys- +tems with the strong coupling of photon cavity mode with an +exciton resonance, resulting in exciton-polariton, demonstrate +a nonlinear response up to 3-4 orders of magnitude higher than +in weakly coupled systems.1 Such systems are realized by +embedding an excitonic material with high exciton oscillator +strength into a photon cavity supporting a mode with strong +field enhancement and long radiative lifetime.2 The search of +excitonic materials as well as the design of photon cavities +suitable for the incorporation with efficient fabrication meth- +ods is therefore of great importance for polaritonics today. +One of the most studied and widely used material plat- +forms for the exciton-polariton systems is the GaAs quantum +well (QW), embedded into the vertical Bragg cavity.3 Due to +the low exciton binding energy, the operation of these po- +lariton systems is limited to cryogenic temperatures.4,5 The +temperature limitations can be overcome with wide-gap semi- +conductor QWs such as ZnO6 or GaN7, but they still re- +quire time-consuming and costly fabrication methods such +as epitaxial growth techniques. Monolayer transition metal +dichalcogenides have become perspective materials for room- +a)These authors contributed equally +temperature polariton systems,8,9 though their potential appli- +cations are still limited by technological scalability. Currently, +halide perovskites represent the promising base for exciton- +polariton systems due to their easy and cost-efficient fabrica- +tion as well as their outstanding excitonic properties making +it possible to implement room-temperature exciton-polariton +systems.10 Moreover, two-dimensional perovskites with enor- +mous exciton binding energy and exceptionally strong ex- +citonic response11 have experimentally demonstrated the +record-high value of Rabi splitting among perovskites exceed- +ing 200 meV at room temperature12 and therefore represent +one of the promising materials for polariton systems. +The most commonly used photon resonator in polaritonic +systems is the vertical Bragg cavity since it provides all nec- +essary requirements such as low optical losses, controllable +lifetimes, and high field enhancement.13 Exciton-polaritons in +perovskite materials and also in 2D-perovskites have been al- +ready demonstrated in the Bragg resonators.12,14,15 Neverthe- +less, such structures have large vertical sizes and also require +sophisticated and costly fabrication methods, which severely +hinder real-world applications.16 Meanwhile, compatible with +on-chip designs planar photon cavities, such as metasur- +faces or photonic crystal slabs (PCSs) can demonstrate com- +parable characteristics and have been recently employed in +exciton-polariton systems with various materials.9,17,18 More- +over, high-Q symmetry-protected bound states in the contin- +uum (BICs), appearing in metasurfaces, when strongly cou- +pled to the exciton resonance,19 allow to even realize polariton +Bose–Einstein condensation.20 Although planar photon cavi- +ties based on perovskites are more suitable for future appli- +cations, there is still a lack of efficient and low-cost cavity +arXiv:2301.01016v1 [physics.optics] 3 Jan 2023 + +Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities +2 +fabrication techniques. +Previously, several methods for perovskite nanostructur- +ing have already been demonstrated, however, all of them +have disadvantageous and limitations. +Thus, some mate- +rial degradation may be caused during focused ion beam +and electron beam lithography21,22. +Direct laser writing +avoids this problem but has a limited lateral resolution above +200 nm.23 Nanoimprinting method maintains the resolution of +ion (or electron) beam lithography and does not cause degra- +dation, but the stamp geometry can not be changed after its +fabrication.18,24 From this point of view, mechanical scanning +probe lithography (m-SPL)25 (Fig. 1a) appears one of the +most versatile and convenient nanostructuring techniques for +the perovskite planar exciton-polariton system since the me- +chanical cutting of perovskites does not cause material degra- +dation, the atomic force microscopy (AFM) tip can be less +than 10 nm in its lateral sizes, and high-precision piezo-stages +of m-SPL allow for the dynamic tuning any of parameters of +the resulting structure. +In this work, we demonstrate a universal and low-cost tech- +nology of 2D-perovskite film nanostructuring for the real- +ization of room-temperature exciton-polariton planar cavities +based on PCS with the precise control of polariton dispersion. +By varying the period and modulation of PCS we change the +exciton-polariton dispersion and its radiative lifetime. The de- +veloped m-SPL method for perovskites opens the way for the +realization of planar polaritonic cavities with on-demand op- +tical properties for nonlinear and active polaritonics. +II. +FABRICATION OF PLANAR PEROVSKITE CAVITIES +A. +(PEA)2PbI4 thin film synthesis +First, a thin film of 2D perovskite (PEA)2PbI4 is synthe- +sized by the solvent engineering method26. The solution for +the synthesis is prepared by dissolving 149.4 mg of PEAI and +138.3 mg of PbI2 in 1 ml of dimethylformamide. It is stirred +with a magnetic stirrer for a little more than 24 hours. The +molarity of the resulting solution is 0.3M. Before the syn- +thesis of 2D-perovskite film, we clean 12×12 mm SiO2 sub- +strates by consistent sonication in soapy water, acetone, and +isopropanol. Then the substrates are dried out and placed in +oxygen plasma cleaner to achieve a hydrophilic surface. The +synthesis of (PEA)2PbI4 films is performed in a glove box +with a dry nitrogen atmosphere with the spin-coating method. +The prepared substrate is placed onto a spin coater, and 20 +µl of the perovskite solution is deposited on top of it. After +the spin coater rotation is launched, it accelerates for 2 sec- +onds and rotates at a speed of 4000 rpm for 60 seconds. The +substrate with the 2D-perovskite thin film in the intermediate +phase is annealed at 70◦ for 10 minutes. The morphology of +the synthesized film is studied with AFM, resulting in 130 nm +film thickness and surface roughness of 15 nm (See Fig. S1 in +Supplementary Information (SI)). +kr +kx +Substrate +(PEA)2PbI4 +kx +ki ++- +Exciton +Exciton +Polariton +Uncoupled photon +2.4 +2.2 +2.0 +1.8 +0.0 +0.2 +-0.2 +-0.4 +0.4 +kx/k0 +Energy, eV +ΩR/2 +(a) +(b) +(c) +hm +d +FIG. 1. +(a) Sketch of mechanical scanning probe lithography of a +thin (PEA)2PbI4 film. AFM cantilever with a single-crystal diamond +tip applies constant pressure on the film and moves with a highly pre- +cise trajectory to create a periodic structure of a photonic crystal slab +(PCS). The inset schematically shows the incident (ki) and reflected +(kr) wavevectors, as well as their in-plane component (kx). (b) Cal- +culated dispersion of the lower polariton branch (red line) resulting +from the strong coupling between the uncoupled exciton resonance +(blue line) and the uncoupled photon cavity mode (orange line). (c) +A sketch of the atomic structure of a 2D-perovskite (PEA)2PbI4. +B. +Mechanical scanning probe lithography +For m-SPL we use an atomic force microscope AIST-NT +SMART SPM and cantilevers with a single-crystal diamond +tip (TipsNano DRP-IN) with a resonant frequency of 500 – +1000 kHz, a normal spring constant of 350 N/m (See SI for +the details), and a tip curvature radius of 25 – 35 nm. Before +the lithography, the film morphology was characterized with +AFM in a semi-contact regime. The use of the piezo-stages +of the atomic force microscope makes it possible to control +the diamond-tipped cantilever position with an accuracy of +nanometers. Thus, fabricating the 1D PCS, we precisely con- +trol the period, the height modulation, and the comb width +(see Fig.1a). +The height modulation is determined by the applied can- +tilever force, which is determined by the shift of the cantilever +from the initial position and its stiffness. The force required +to achieve a modulation of 15 − 50 nm on (PEA)2PbI4 per- +ovskite films is experimentally determined to be in the range +of 5 − 30 µN. Since the tip is conical in shape (See SI for +details), the minimum cavity width depends on the modula- +tion hm. For modulation of 15−50 nm, the width at the half- +height of the cavities is 80 − 130 nm (See SI for the details). +The speed of the cantilever during the lithography process is +limited up to 3 µm/s because at higher values the probe be- + +Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities +3 +hm = 16 nm +(c) +130 +110 +90 +70 +0 +2 +4 +6 +8 +10 +x, μm +h, nm +hm = 24 nm +hm = 40 nm +hm = 49 nm +h, nm +x, μm +5•d = 5•320 nm +5•d = 5•340 nm +5•d = 5•360 nm +5•d = 5•380 nm +130 +100 +0.0 +1.0 +2.0 +3.0 +0 +1 +2 +3 +1 +2 +3 +(a) +y, μm +x, μm +(b) +130 +120 +100 +100 +90 nm +0.5 +1.5 +2.5 +3.5 +1 +3 +5 +7 +9 +130 +100 +130 +100 +130 +100 +FIG. 2. (a) Sketch of mechanical scanning probe lithography of a thin (PEA)2PbI4 film. AFM cantilever with a single-crystal diamond tip +applies constant pressure on the film and moves with a highly precise trajectory to create a periodic structure of a photonic crystal slab (PCS). +The inset schematically shows the incident (ki) and reflected (kr) wavevectors, as well as their in-plane component (kx). (b) Calculated +dispersion of the lower polariton branch (red line) resulting from the strong coupling between the uncoupled exciton resonance (blue line) and +the uncoupled photon cavity mode (orange line). (c) A sketch of the atomic structure of a 2D-perovskite (PEA)2PbI4. +gins to pull out perovskite grains. The optimal speed for the +2D-perovskite film lithography is found to be approximately +1 µm/s. +By choosing the trajectory of the AFM tip with piezo- +stages, the method allows the realization of mostly arbitrary +structures. +Particularly, it is possible to change the period +of the PCS by programming the cantilever movement coor- +dinates with nanometer precision. One of the most impor- +tant advantages of m-SPL is the potential applicability of +this method for the creation of PCSs, bounded waveguides, +or other planar photonic designs on one 2D-perovskite film, +combining them into one photonic on-chip system. +C. +The fabricated photonic crystal slabs +The fabricated PCSs have a lateral size of 15x30 µm2. The +morphology of the structures, studied with AFM is shown +in Fig. 2a. By varying the cantilever displacement coordi- +nate, we fabricated PCSs with the periods of d = 320, 340, +360, 380 nm and modulation of about hm = 20 nm (Fig. +2b). +By changing the pressure force in the range of 9-24 +µN, we also realize structures with different modulations of +hm = 16, 24, 40, 49 nm and a period of d = 340 nm (Fig. +2c). Resulted structures are expected to have different spectral +positions of the resonances and also different optical losses, +which we study further. +III. +OPTICAL SPECTROSCOPY OF THE POLARITONS +In order to study the leaky cavity modes of the fabri- +cated PCSs, we perform angle-resolved spectroscopy mea- +surements based on the back focal plane (BFP) setup. The +BFP of the objective lens (Mitutoyo NIR ×50 with an N.A. +of 0.55) is imaged on a slit spectrometer coupled to a liquid +nitrogen-cooled imaging CCD camera (Princeton Instruments +SP2500+PyLoN) by 4f scheme (see SI for the details). For +the sample illumination as well as for the measurements of +the reflectance spectra, a halogen lamp is used. The plane of +incidence contains both normal to the sample and the direc- +tion of periodicity of the PCS (see Fig. 1a). Before imping- + +Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities +4 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Energy, eV +kX/k0 +Uncoupled exciton +Uncoupled photon +Polariton +2.3 +2.1 +1.9 +0.0 +0.4 +-0.4 +kX/k0 +2.3 +2.1 +1.9 +kX/k0 +2.3 +2.1 +1.9 +kX/k0 +2.3 +2.1 +1.9 +d = 320 nm +d = 340 nm +d = 360 nm +d = 380 nm +kX/k0 +2.3 +2.1 +1.9 +kX/k0 +2.3 +2.1 +1.9 +kX/k0 +2.3 +2.1 +1.9 +kX/k0 +2.3 +2.1 +1.9 +m = 16 nm +m = 24 nm +m = 40 nm +m = 49 nm +Energy, eV +0.0 +0.3 +min +max +PL Intensity +0.0 +0.7 +max +PL Intensity +δR/R +δR/R +0.0 +0.4 +-0.4 +0.0 +0.4 +-0.4 +0.0 +0.4 +-0.4 +0.0 +0.4 +-0.4 +0.0 +0.4 +-0.4 +0.0 +0.4 +-0.4 +0.0 +0.4 +-0.4 +FIG. 3. +(a-d) Angle-resolved reflectance (left) and photoluminescence (right) spectra of the studied PCS based on the (PEA)2PbI4 with the +periods, respectively, d = 320,340,360,380 nm and the modulation hm ≈ 20 nm. (e-h) the same spectra maps of the fabricated PCSs with +the period d = 340 nm and the modulations hm ≈ 16,24,40,49 nm, respectively. Dashed yellow lines correspond to the estimated uncoupled +photon cavity mode dispersions. Dashed green lines correspond to uncoupled exciton resonances. Red solid lines correspond to the polariton +modes fitted with the two-coupled oscillators model. +ing the slit of the imaging spectrometer, the light reflected +from the sample passes through a linear polarizer aligned +such that TE modes are studied. The scheme is also used to +obtain the angle-resolved photoluminescence spectra using a +femtosecond laser (Pharos, Light Conversion) coupled with +a broad-bandwidth optical parametric amplifier (Orpheus-F, +Light Conversion) at the wavelength of 480 nm, 100 kHz rep- +etition rate as a non-resonant excitation source. All measure- +ments were performed at a room temperature of 300K. +We measure angle-resolved reflectance and photolumines- +cence spectra for every of the fabricated PCSs, shown in +Fig. 3. The measured data show the pronounced leaky modes +in the spectral region below the exciton resonance around +2.37 eV. All of the studied samples demonstrate the curving +of the mode dispersion asymptotically approaching the exci- +ton level in the blue spectral region, revealing the signs of the +strong light-matter coupling regime.18 +In order to verify the strong light-matter coupling regime, +we extract the modes from the experimental data by the fol- +lowing procedure: first, we subtract the unbound exciton +photoluminescence signal from the experimental dispersion +at each kx/k0, then we fit the resulting modes by the peak +Lorentz function data. Combining the spectral peak positions +for each of wavenumber kx/k0, we obtain the experimental +mode dispersion. +Since the upper polariton branch (UPB) +above the exciton resonance does not exist due to the strong +non-radiative absorption, the only way to confirm the strong +light-matter coupling regime is to fit the extracted mode with +a lower polariton branch (LPB), estimated by the two-coupled +oscillator model as27: +ELP = +�Ex + �Ec(k) +2 +− 1 +2 +�� +�Ex − �Ec(k) +�2 ++4g2, +(1) +where �Ex = Ex − iγx is complex energy accounting for the +spectral position and the linewidth of the uncoupled exciton +resonance, �Ec(k) = Ec(k)−iγc is a complex dispersion of the +uncoupled cavity photon mode, g - is a light-matter coupling +coefficient. The Rabi splitting ΩR corresponds to the mini- +mal energy distance between UPB and LPB, however, as UPB +does not exist, we can only estimate this value based on the +described model: +ΩR = +� +4g2 −(γc −γx)2; +(2) +The uncoupled photon cavity mode has linear dependence +of the energy on the wavenumber kx/k0 since the refractive + +2.4 +2.4 +2.4 +2.3 +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +0.2 +1.9 +-0.4-0.2 +0.0 +0.2 +0.4 +-0.40.2 +0 +0.2 +0.4 +0.4-0.20.0 +0.4 +0.4-0.2 +00 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 → +2.3 +2.3 - +2.3+ +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +0.2 +0 +0.2 +0.4 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 + +2.3 +2.3 +2.3 - +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 , +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 - +-0.4-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.3 + +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 - +2.1 +2.0 +2.0 +2.0 - +2.0 +1.9 +1.9 / +1.9 +1.9 1 +-0.4-0.20.0 +0.4 +0.4-0.20.0 +0.20.4 +-0.4-0.20.00.2 +0.4 +00-0- +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 + +2.3 +2.3 +2.3 - +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 , +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 - +-0.4-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.3 + +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 - +2.1 +2.0 +2.0 +2.0 - +2.0 +1.9 +1.9 / +1.9 +1.9 1 +-0.4-0.20.0 +0.4 +0.4-0.20.0 +0.20.4 +-0.4-0.20.00.2 +0.4 +00-0- +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 + +2.3 +2.3 +2.3 - +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 , +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 - +-0.4-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.3 + +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 - +2.1 +2.0 +2.0 +2.0 - +2.0 +1.9 +1.9 / +1.9 +1.9 1 +-0.4-0.20.0 +0.4 +0.4-0.20.0 +0.20.4 +-0.4-0.20.00.2 +0.4 +00-0- +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 + +2.3 +2.3 +2.3 - +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 , +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 - +-0.4-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +0.2 +1.9 +-0.4-0.2 +0.0 +0.2 +0.4 +-0.40.2 +0 +0.2 +0.4 +0.4-0.20.0 +0.4 +0.4-0.2 +00 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 → +2.3 +2.3 - +2.3+ +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +0.2 +0 +0.2 +0.4 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +0.2 +1.9 +-0.4-0.2 +0.0 +0.2 +0.4 +-0.40.2 +0 +0.2 +0.4 +0.4-0.20.0 +0.4 +0.4-0.2 +00 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 → +2.3 +2.3 - +2.3+ +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +0.2 +0 +0.2 +0.4 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +0.2 +1.9 +-0.4-0.2 +0.0 +0.2 +0.4 +-0.40.2 +0 +0.2 +0.4 +0.4-0.20.0 +0.4 +0.4-0.2 +00 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.4 +2.4 +2.3 → +2.3 +2.3 - +2.3+ +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 +2.1 +2.0 +2.0 +2.0 +2.0 +1.9 +1.9 +1.9 +1.9 +0.0 +0.2 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +0.2 +0 +0.2 +0.4 +0.0 +0.2 +0.4 +kx/ko +kx/ko +kx/ko +kx/ko2.4 +2.3 + +2.3 +2.3 +2.3 +2.2 +2.2 +2.2 +2.2 +2.1 +2.1 +2.1 - +2.1 +2.0 +2.0 +2.0 - +2.0 +1.9 +1.9 / +1.9 +1.9 1 +-0.4-0.20.0 +0.4 +0.4-0.20.0 +0.20.4 +-0.4-0.20.00.2 +0.4 +00-0- +kx/ko +kx/ko +kx/ko +kx/koMechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities +5 +Energy, eV +1.9 2.0 +2.1 +2.2 2.3 +Energy, eV +1.9 2.0 +2.1 2.2 2.3 +δR/R +1.0 +0.0 +0.0 +1.0 +2.0 +m, nm +d, nm +m, nm +d, nm +(a) +(b) +(c) +(d) +(e) +(f) +δR/R +0.1 +10 +30 +50 +Ecross, eV +2.0 +320 +340 +360 +380 +γrad/γtotal +2.1 +255 +10 +30 +50 +235 +320 +340 +360 +380 +ΩR, meV +Variation of the period +Variation of the modulation +320 nm +360 nm +340 nm +380 nm +16 nm +40 nm +24 nm +49 nm +245 +255 +235 +245 +ΩR, meV +0.2 +0.5 +0.5 +1.5 ++0.2 ++0.6 ++0.8 ++0.2 ++0.5 ++1.1 +FIG. 4. (a), (b) Reflectance spectra at kx/k0 = 0 for the correspond- +ing PCS. They are shifted along the vertical axis at arbitrary values +for better visual perception. (c) The spectral position of the cross- +ing point of counter-propagating polariton modes (at Γ-point) Ecross +depending on PCS period d. (d) The ratio between the radiative and +total losses γrad/γtotal depending on PCS modulation hm. (e), (f) +Rabi splitting ΩR depending on PCS period d and modulation hm, +respectively. +index is considered to have negligible changes in the consid- +ered spectral range without accounting for the exciton reso- +nance. Therefore, we estimate uncoupled photon cavity dis- +persions as Ec(kx) = k · kx + b based on the calculations of +Fourier modal method28 (see SI for details). The coupling co- +efficient g, as well as the half-widths of an unbound photon +γc and exciton γx, are chosen as the optimization parameters +in the fitting of the LPB. The resulting real part of the PL dis- +persion ELP optimized for each of the samples is shown as red +curves in Fig. 3. +The estimated values from the fitting of the uncoupled cav- +ity photon and exciton γc and exciton γx do not exceed 50 +meV and 15 meV, respectively. The resulting values of Rabi +splitting ΩR for each of the PCSs are shown in Figs. 4e and +4f, which are no less than 230 meV. The obtained values fully +satisfy the strong light-matter regime criteria (g > |γC −γX|/2; +ΩR > |γC +γX|/2)29 in all studied samples. +The leaky mode dispersion of the 1D PCS is determined +by the waveguide modes folded towards the first Brillouin +zone with the edges of kBZ +x += ±π/d, where d is a PCS pe- +riod. For a 2D waveguide with the chosen thickness, with +the change of the PCS period, the spectral position of folded +uncoupled leaky modes, and, hence, polariton branches shift +proportionally.30 Actually the difference in the spectral posi- +tion of the polariton modes can be noticed in Figs. 3(a-d). In +order to reveal the dependence of the spectral position of po- +lariton mode as a function of period, we extract the reflec- +tion spectra at normal incidence kx/k0 = 0 (Fig. 4a). +The +frequencies of the modes are estimated by fitting with the +Fano resonance function (See SI for details) and as expected +show a monotonous decrease with the increase of PCS period +(Fig. 4b). +The value of the Rabi splitting ΩR depends on the coupling +coefficient and the linewidths of uncoupled exciton and cavity +photon modes (Eq. 2). Since the coupling coefficient, g de- +pends on the cavity mode localization, oscillator strength, and +the excitonic response,27,31, it should not change strongly with +the PCS period or other geometrical parameters. This was +confirmed by the results of fitting all the experimental data. +In turn, the uncoupled exciton linewidth γX is the property of +the materials and thus should not depend on the PCS design, +which we also confirmed by analyzing the data. Hence, the +only way to tune the Rabi splitting is to vary the radiative part +of leaky mode losses γC, which is dictated by the PCS mod- +ulation and comb width. Thus, with the variation of the PCS +period, we do neither expect nor observe the pronounced de- +pendence of the estimated Rabi splitting ΩR values (Fig. 4e). +The variation of the modulation hm with constant period +provides a different contrast of the experimentally measured +polariton modes, as shown in Figs. 3e-h and in the reflectance +spectra, shown in Fig. 4b. Higher modulation causes higher +coupling of the leaky mode with the free space, or in other +words, increases the radiative losses of the mode. In order +to reveal the dependence, we estimate the ratio γrad/γtotal by +fitting the amplitude and asymmetry parameter of the Fano +resonance (See SI for details) for different modulations hm at +the kx/k0 = 0 and show it in Fig. 4d. The non-radiative losses +are considered to be constant for each of the PCSs because +they are mostly dictated by the material defect states and ex- +citon absorption. Hence, the total optical losses γC rise with +increasing the PCS modulation, which leads to the reduction +of the Rabi splitting values (Fig. 4f). Thus by applying the +different forces on the cantilever during the m-SPL process +it is possible to control the modes contrast and the value of +the Rabi splitting in the planar exciton-polariton PCS leaky +modes. +IV. +CONCLUSION +This research is the first to employ and demonstrate the +method of mechanical scanning probe lithography for the real- +ization of planar room-temperature exciton-polariton systems +based on 2D perovskites. The fabricated PCSs demonstrate +the high-Q polariton modes up to 100. Thanks to the features +of the m-SPL method it is possible to vary the period and mod- +ulation of the structures with nanoscale precision. In this way, +we are able to fully control the dispersion, optical radiative +losses, and the Rabi splitting of the exciton-polariton states in +the planar photon cavity based on (PEA)2PbI4. 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Kivshar, “Halide-perovskite resonant + +Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities +7 +nanophotonics,” Advanced optical materials 7, 1800784 (2019). +36J. A. Steele, H. Yuan, C. Y. Tan, M. Keshavarz, C. Steuwe, M. B. Roeffaers, +and J. Hofkens, “Direct laser writing of δ-to α-phase transformation in +formamidinium lead iodide,” ACS nano 11, 8072–8083 (2017). +37N. Yi, S. Wang, Z. Duan, K. Wang, Q. Song, and S. Xiao, “Tailoring the +performances of lead halide perovskite devices with electron-beam irradia- +tion,” Advanced Materials 29, 1701636 (2017). +38M. S. Alias, Y. Yang, T. K. Ng, I. Dursun, D. Shi, M. I. Saidaminov, D. Pri- +ante, O. M. Bakr, and B. S. Ooi, “Enhanced etching, surface damage re- +covery, and submicron patterning of hybrid perovskites using a chemically +gas-assisted focused-ion beam for subwavelength grating photonic applica- +tions,” The journal of physical chemistry letters 7, 137–142 (2016). +39E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Al- +pha Science Int’l Ltd., 2005). + diff --git a/_dAzT4oBgHgl3EQfFvqG/content/tmp_files/load_file.txt b/_dAzT4oBgHgl3EQfFvqG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2c73465a79b39dabd1553b1159821b485f4a5f3 --- /dev/null +++ b/_dAzT4oBgHgl3EQfFvqG/content/tmp_files/load_file.txt @@ -0,0 +1,1273 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf,len=1272 +page_content='Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities Mechanical scanning probe lithography of perovskites for fabrication of high-Q planar polaritonic cavities N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Glebov,1, a) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Masharin,1, a) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Borodin,2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Alekseev,2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Benimetskiy,3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Makarov,1, 4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Samusev*1, 5 1)ITMO University, School of Physics and Engineering, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Petersburg, 197101, Russia 2)Ioffe Institute, Saint-Petersburg 194021, Russia 3)Department of Physics and Astronomy, University of Sheffield, S3 7RH, Sheffield, UK 4)Qingdao Innovation and Development Center, Harbin Engineering University, Qingdao 266000, Shandong, China 5)Experimentelle Physik 2, Technische Universit¨at Dortmund, 44227 Dortmund, Germany (*Electronic mail: anton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='samusev@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='com) (Dated: 4 January 2023) Exciton-polaritons are unique quasiparticles with hybrid properties of an exciton and a photon, opening ways to realize ultrafast strongly nonlinear systems and inversion-free lasers based on Bose-Einstein polariton condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' However, the real-world applications of the polariton systems are still limited due to the temperature operation and costly fabri- cation techniques for both exciton materials and photon cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2D perovskites represent one of the most prospective platforms for the realization of strong light-matter coupling since they possess room-temperature exciton states with large oscillator strength and can simultaneously provide planar photon cavities with high field localization due to the huge refractive index of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' In this work, we demonstrate for the first time the mechanical scanning probe lithography method for the realization of low-cost room-temperature exciton-polariton systems based on the 2D per- ovskite (PEA)2PbI4 with exciton binding energy exceeding 200 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Precisely controlling the lithography parameters, we broadly adjust the exciton-polariton dispersion, and radiative losses of polaritonic modes in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 of total optical losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Our findings represent a versatile approach to the fabrication of planar high-quality perovskite- based photonic cavities supporting the strong light-matter coupling regime for the development of on-chip all-optical active and nonlinear polaritonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' INTRODUCTION Photonics deals both with fundamental and applied aspects of operating with optical signals, as well as with prospective designing energy-efficient optical computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Imple- menting such devices where light is controlled by light re- quires systems with strong optical nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Optical sys- tems with the strong coupling of photon cavity mode with an exciton resonance, resulting in exciton-polariton, demonstrate a nonlinear response up to 3-4 orders of magnitude higher than in weakly coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 Such systems are realized by embedding an excitonic material with high exciton oscillator strength into a photon cavity supporting a mode with strong field enhancement and long radiative lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 The search of excitonic materials as well as the design of photon cavities suitable for the incorporation with efficient fabrication meth- ods is therefore of great importance for polaritonics today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' One of the most studied and widely used material plat- forms for the exciton-polariton systems is the GaAs quantum well (QW), embedded into the vertical Bragg cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 Due to the low exciton binding energy, the operation of these po- lariton systems is limited to cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4,5 The temperature limitations can be overcome with wide-gap semi- conductor QWs such as ZnO6 or GaN7, but they still re- quire time-consuming and costly fabrication methods such as epitaxial growth techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Monolayer transition metal dichalcogenides have become perspective materials for room- a)These authors contributed equally temperature polariton systems,8,9 though their potential appli- cations are still limited by technological scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Currently, halide perovskites represent the promising base for exciton- polariton systems due to their easy and cost-efficient fabrica- tion as well as their outstanding excitonic properties making it possible to implement room-temperature exciton-polariton systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='10 Moreover, two-dimensional perovskites with enor- mous exciton binding energy and exceptionally strong ex- citonic response11 have experimentally demonstrated the record-high value of Rabi splitting among perovskites exceed- ing 200 meV at room temperature12 and therefore represent one of the promising materials for polariton systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The most commonly used photon resonator in polaritonic systems is the vertical Bragg cavity since it provides all nec- essary requirements such as low optical losses, controllable lifetimes, and high field enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='13 Exciton-polaritons in perovskite materials and also in 2D-perovskites have been al- ready demonstrated in the Bragg resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='12,14,15 Neverthe- less, such structures have large vertical sizes and also require sophisticated and costly fabrication methods, which severely hinder real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='16 Meanwhile, compatible with on-chip designs planar photon cavities, such as metasur- faces or photonic crystal slabs (PCSs) can demonstrate com- parable characteristics and have been recently employed in exciton-polariton systems with various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9,17,18 More- over, high-Q symmetry-protected bound states in the contin- uum (BICs), appearing in metasurfaces, when strongly cou- pled to the exciton resonance,19 allow to even realize polariton Bose–Einstein condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='20 Although planar photon cavi- ties based on perovskites are more suitable for future appli- cations, there is still a lack of efficient and low-cost cavity arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='01016v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='optics] 3 Jan 2023 Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities 2 fabrication techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Previously, several methods for perovskite nanostructur- ing have already been demonstrated, however, all of them have disadvantageous and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Thus, some mate- rial degradation may be caused during focused ion beam and electron beam lithography21,22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Direct laser writing avoids this problem but has a limited lateral resolution above 200 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='23 Nanoimprinting method maintains the resolution of ion (or electron) beam lithography and does not cause degra- dation, but the stamp geometry can not be changed after its fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='18,24 From this point of view, mechanical scanning probe lithography (m-SPL)25 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 1a) appears one of the most versatile and convenient nanostructuring techniques for the perovskite planar exciton-polariton system since the me- chanical cutting of perovskites does not cause material degra- dation, the atomic force microscopy (AFM) tip can be less than 10 nm in its lateral sizes, and high-precision piezo-stages of m-SPL allow for the dynamic tuning any of parameters of the resulting structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' In this work, we demonstrate a universal and low-cost tech- nology of 2D-perovskite film nanostructuring for the real- ization of room-temperature exciton-polariton planar cavities based on PCS with the precise control of polariton dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' By varying the period and modulation of PCS we change the exciton-polariton dispersion and its radiative lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The de- veloped m-SPL method for perovskites opens the way for the realization of planar polaritonic cavities with on-demand op- tical properties for nonlinear and active polaritonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' FABRICATION OF PLANAR PEROVSKITE CAVITIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (PEA)2PbI4 thin film synthesis First, a thin film of 2D perovskite (PEA)2PbI4 is synthe- sized by the solvent engineering method26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The solution for the synthesis is prepared by dissolving 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 mg of PEAI and 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 mg of PbI2 in 1 ml of dimethylformamide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' It is stirred with a magnetic stirrer for a little more than 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The molarity of the resulting solution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Before the syn- thesis of 2D-perovskite film, we clean 12×12 mm SiO2 sub- strates by consistent sonication in soapy water, acetone, and isopropanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Then the substrates are dried out and placed in oxygen plasma cleaner to achieve a hydrophilic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The synthesis of (PEA)2PbI4 films is performed in a glove box with a dry nitrogen atmosphere with the spin-coating method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The prepared substrate is placed onto a spin coater, and 20 µl of the perovskite solution is deposited on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' After the spin coater rotation is launched, it accelerates for 2 sec- onds and rotates at a speed of 4000 rpm for 60 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The substrate with the 2D-perovskite thin film in the intermediate phase is annealed at 70◦ for 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The morphology of the synthesized film is studied with AFM, resulting in 130 nm film thickness and surface roughness of 15 nm (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' S1 in Supplementary Information (SI)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' kr kx Substrate (PEA)2PbI4 kx ki +- Exciton Exciton Polariton Uncoupled photon 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 kx/k0 Energy, eV ΩR/2 (a) (b) (c) hm d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (a) Sketch of mechanical scanning probe lithography of a thin (PEA)2PbI4 film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' AFM cantilever with a single-crystal diamond tip applies constant pressure on the film and moves with a highly pre- cise trajectory to create a periodic structure of a photonic crystal slab (PCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The inset schematically shows the incident (ki) and reflected (kr) wavevectors, as well as their in-plane component (kx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (b) Cal- culated dispersion of the lower polariton branch (red line) resulting from the strong coupling between the uncoupled exciton resonance (blue line) and the uncoupled photon cavity mode (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (c) A sketch of the atomic structure of a 2D-perovskite (PEA)2PbI4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Mechanical scanning probe lithography For m-SPL we use an atomic force microscope AIST-NT SMART SPM and cantilevers with a single-crystal diamond tip (TipsNano DRP-IN) with a resonant frequency of 500 – 1000 kHz, a normal spring constant of 350 N/m (See SI for the details), and a tip curvature radius of 25 – 35 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Before the lithography, the film morphology was characterized with AFM in a semi-contact regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The use of the piezo-stages of the atomic force microscope makes it possible to control the diamond-tipped cantilever position with an accuracy of nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Thus, fabricating the 1D PCS, we precisely con- trol the period, the height modulation, and the comb width (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The height modulation is determined by the applied can- tilever force, which is determined by the shift of the cantilever from the initial position and its stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The force required to achieve a modulation of 15 − 50 nm on (PEA)2PbI4 per- ovskite films is experimentally determined to be in the range of 5 − 30 µN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Since the tip is conical in shape (See SI for details), the minimum cavity width depends on the modula- tion hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' For modulation of 15−50 nm, the width at the half- height of the cavities is 80 − 130 nm (See SI for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The speed of the cantilever during the lithography process is limited up to 3 µm/s because at higher values the probe be- Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities 3 hm = 16 nm (c) 130 110 90 70 0 2 4 6 8 10 x, μm h, nm hm = 24 nm hm = 40 nm hm = 49 nm h, nm x, μm 5•d = 5•320 nm 5•d = 5•340 nm 5•d = 5•360 nm 5•d = 5•380 nm 130 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 0 1 2 3 1 2 3 (a) y, μm x, μm (b) 130 120 100 100 90 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 1 3 5 7 9 130 100 130 100 130 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (a) Sketch of mechanical scanning probe lithography of a thin (PEA)2PbI4 film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' AFM cantilever with a single-crystal diamond tip applies constant pressure on the film and moves with a highly precise trajectory to create a periodic structure of a photonic crystal slab (PCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The inset schematically shows the incident (ki) and reflected (kr) wavevectors, as well as their in-plane component (kx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (b) Calculated dispersion of the lower polariton branch (red line) resulting from the strong coupling between the uncoupled exciton resonance (blue line) and the uncoupled photon cavity mode (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (c) A sketch of the atomic structure of a 2D-perovskite (PEA)2PbI4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' gins to pull out perovskite grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The optimal speed for the 2D-perovskite film lithography is found to be approximately 1 µm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' By choosing the trajectory of the AFM tip with piezo- stages, the method allows the realization of mostly arbitrary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Particularly, it is possible to change the period of the PCS by programming the cantilever movement coor- dinates with nanometer precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' One of the most impor- tant advantages of m-SPL is the potential applicability of this method for the creation of PCSs, bounded waveguides, or other planar photonic designs on one 2D-perovskite film, combining them into one photonic on-chip system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The fabricated photonic crystal slabs The fabricated PCSs have a lateral size of 15x30 µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The morphology of the structures, studied with AFM is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' By varying the cantilever displacement coordi- nate, we fabricated PCSs with the periods of d = 320, 340, 360, 380 nm and modulation of about hm = 20 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' By changing the pressure force in the range of 9-24 µN, we also realize structures with different modulations of hm = 16, 24, 40, 49 nm and a period of d = 340 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Resulted structures are expected to have different spectral positions of the resonances and also different optical losses, which we study further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' OPTICAL SPECTROSCOPY OF THE POLARITONS In order to study the leaky cavity modes of the fabri- cated PCSs, we perform angle-resolved spectroscopy mea- surements based on the back focal plane (BFP) setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The BFP of the objective lens (Mitutoyo NIR ×50 with an N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='55) is imaged on a slit spectrometer coupled to a liquid nitrogen-cooled imaging CCD camera (Princeton Instruments SP2500+PyLoN) by 4f scheme (see SI for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' For the sample illumination as well as for the measurements of the reflectance spectra, a halogen lamp is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The plane of incidence contains both normal to the sample and the direc- tion of periodicity of the PCS (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Before imping- Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities 4 (a) (b) (c) (d) (e) (f) (g) (h) Energy, eV kX/k0 Uncoupled exciton Uncoupled photon Polariton 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 d = 320 nm d = 340 nm d = 360 nm d = 380 nm kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 kX/k0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='9 m = 16 nm m = 24 nm m = 40 nm m = 49 nm Energy, eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 min max PL Intensity 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (a-d) Angle-resolved reflectance (left) and photoluminescence (right) spectra of the studied PCS based on the (PEA)2PbI4 with the periods, respectively, d = 320,340,360,380 nm and the modulation hm ≈ 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (e-h) the same spectra maps of the fabricated PCSs with the period d = 340 nm and the modulations hm ≈ 16,24,40,49 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Dashed yellow lines correspond to the estimated uncoupled photon cavity mode dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Dashed green lines correspond to uncoupled exciton resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Red solid lines correspond to the polariton modes fitted with the two-coupled oscillators model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' ing the slit of the imaging spectrometer, the light reflected from the sample passes through a linear polarizer aligned such that TE modes are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The scheme is also used to obtain the angle-resolved photoluminescence spectra using a femtosecond laser (Pharos, Light Conversion) coupled with a broad-bandwidth optical parametric amplifier (Orpheus-F, Light Conversion) at the wavelength of 480 nm, 100 kHz rep- etition rate as a non-resonant excitation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' All measure- ments were performed at a room temperature of 300K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' We measure angle-resolved reflectance and photolumines- cence spectra for every of the fabricated PCSs, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The measured data show the pronounced leaky modes in the spectral region below the exciton resonance around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='37 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' All of the studied samples demonstrate the curving of the mode dispersion asymptotically approaching the exci- ton level in the blue spectral region, revealing the signs of the strong light-matter coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='18 In order to verify the strong light-matter coupling regime, we extract the modes from the experimental data by the fol- lowing procedure: first, we subtract the unbound exciton photoluminescence signal from the experimental dispersion at each kx/k0, then we fit the resulting modes by the peak Lorentz function data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Combining the spectral peak positions for each of wavenumber kx/k0, we obtain the experimental mode dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Since the upper polariton branch (UPB) above the exciton resonance does not exist due to the strong non-radiative absorption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' the only way to confirm the strong light-matter coupling regime is to fit the extracted mode with a lower polariton branch (LPB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' estimated by the two-coupled oscillator model as27: ELP = �Ex + �Ec(k) 2 − 1 2 �� �Ex − �Ec(k) �2 +4g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (1) where �Ex = Ex − iγx is complex energy accounting for the spectral position and the linewidth of the uncoupled exciton resonance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' �Ec(k) = Ec(k)−iγc is a complex dispersion of the uncoupled cavity photon mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' g - is a light-matter coupling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The Rabi splitting ΩR corresponds to the mini- mal energy distance between UPB and LPB, however, as UPB does not exist, we can only estimate this value based on the described model: ΩR = � 4g2 −(γc −γx)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (2) The uncoupled photon cavity mode has linear dependence of the energy on the wavenumber kx/k0 since the refractive 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='4 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='3 δR/R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 m, nm d, nm m, nm d, nm (a) (b) (c) (d) (e) (f) δR/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 10 30 50 Ecross, eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='0 320 340 360 380 γrad/γtotal 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 255 10 30 50 235 320 340 360 380 ΩR, meV Variation of the period Variation of the modulation 320 nm 360 nm 340 nm 380 nm 16 nm 40 nm 24 nm 49 nm 245 255 235 245 ΩR, meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='8 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='5 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (a), (b) Reflectance spectra at kx/k0 = 0 for the correspond- ing PCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' They are shifted along the vertical axis at arbitrary values for better visual perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (c) The spectral position of the cross- ing point of counter-propagating polariton modes (at Γ-point) Ecross depending on PCS period d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (d) The ratio between the radiative and total losses γrad/γtotal depending on PCS modulation hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' (e), (f) Rabi splitting ΩR depending on PCS period d and modulation hm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' index is considered to have negligible changes in the consid- ered spectral range without accounting for the exciton reso- nance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Therefore, we estimate uncoupled photon cavity dis- persions as Ec(kx) = k · kx + b based on the calculations of Fourier modal method28 (see SI for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The coupling co- efficient g, as well as the half-widths of an unbound photon γc and exciton γx, are chosen as the optimization parameters in the fitting of the LPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The resulting real part of the PL dis- persion ELP optimized for each of the samples is shown as red curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The estimated values from the fitting of the uncoupled cav- ity photon and exciton γc and exciton γx do not exceed 50 meV and 15 meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The resulting values of Rabi splitting ΩR for each of the PCSs are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4e and 4f, which are no less than 230 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The obtained values fully satisfy the strong light-matter regime criteria (g > |γC −γX|/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' ΩR > |γC +γX|/2)29 in all studied samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The leaky mode dispersion of the 1D PCS is determined by the waveguide modes folded towards the first Brillouin zone with the edges of kBZ x = ±π/d, where d is a PCS pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' For a 2D waveguide with the chosen thickness, with the change of the PCS period, the spectral position of folded uncoupled leaky modes, and, hence, polariton branches shift proportionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='30 Actually the difference in the spectral posi- tion of the polariton modes can be noticed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 3(a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' In order to reveal the dependence of the spectral position of po- lariton mode as a function of period, we extract the reflec- tion spectra at normal incidence kx/k0 = 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The frequencies of the modes are estimated by fitting with the Fano resonance function (See SI for details) and as expected show a monotonous decrease with the increase of PCS period (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The value of the Rabi splitting ΩR depends on the coupling coefficient and the linewidths of uncoupled exciton and cavity photon modes (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Since the coupling coefficient, g de- pends on the cavity mode localization, oscillator strength, and the excitonic response,27,31, it should not change strongly with the PCS period or other geometrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' This was confirmed by the results of fitting all the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' In turn, the uncoupled exciton linewidth γX is the property of the materials and thus should not depend on the PCS design, which we also confirmed by analyzing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Hence, the only way to tune the Rabi splitting is to vary the radiative part of leaky mode losses γC, which is dictated by the PCS mod- ulation and comb width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Thus, with the variation of the PCS period, we do neither expect nor observe the pronounced de- pendence of the estimated Rabi splitting ΩR values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The variation of the modulation hm with constant period provides a different contrast of the experimentally measured polariton modes, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 3e-h and in the reflectance spectra, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Higher modulation causes higher coupling of the leaky mode with the free space, or in other words, increases the radiative losses of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' In order to reveal the dependence, we estimate the ratio γrad/γtotal by fitting the amplitude and asymmetry parameter of the Fano resonance (See SI for details) for different modulations hm at the kx/k0 = 0 and show it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The non-radiative losses are considered to be constant for each of the PCSs because they are mostly dictated by the material defect states and ex- citon absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Hence, the total optical losses γC rise with increasing the PCS modulation, which leads to the reduction of the Rabi splitting values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Thus by applying the different forces on the cantilever during the m-SPL process it is possible to control the modes contrast and the value of the Rabi splitting in the planar exciton-polariton PCS leaky modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' CONCLUSION This research is the first to employ and demonstrate the method of mechanical scanning probe lithography for the real- ization of planar room-temperature exciton-polariton systems based on 2D perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' The fabricated PCSs demonstrate the high-Q polariton modes up to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Thanks to the features of the m-SPL method it is possible to vary the period and mod- ulation of the structures with nanoscale precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' In this way, we are able to fully control the dispersion, optical radiative losses, and the Rabi splitting of the exciton-polariton states in the planar photon cavity based on (PEA)2PbI4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Note that the demonstrated method can be used for other halide perovskites Mechanical scanning probe lithography of 2D perovskites for fabrication of planar polaritonic cavities 6 and also can provide any other planar photon cavities, includ- ing 2D metasurfaces and PCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Thus, our work reveals the low-cost and time-efficient method for the fabrication of pla- nar high optical quality exciton-polariton systems based on 2D perovskite film, which is highly demanded for the real- ization of optical nonlinear and active on-chip polaritonic de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work was funded by Russian Science Foundation, grant #21-12-00218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' acknowledges the Deutsche Forschungs- gemeinschaft (Grant SFB TRR142/project A6), the Mercur Foundation (Grant Pe-2019–0022), and TU Dortmund core funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 1N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Bélanger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Villeneuve, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Aitchison, “Solitonlike pulses in self- defocusing algaas waveguides,” JOSA B 14, 3003–3012 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=' Ivchenko, Optical spectroscopy of semiconductor nanostructures (Al- pha Science Int’l Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfFvqG/content/2301.01016v1.pdf'} diff --git a/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf b/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ee65005636924d8999c1f1406a2e318e60c47de5 --- /dev/null +++ b/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:370f800ac75ddf002090de8a6b3ccb398b7595b89c2acab936d8dab854cf35c2 +size 2512265 diff --git a/c9E_T4oBgHgl3EQf0hwq/content/tmp_files/2301.08329v1.pdf.txt b/c9E_T4oBgHgl3EQf0hwq/content/tmp_files/2301.08329v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..abb45089463feafa45dba726ac5e09684b318453 --- /dev/null +++ b/c9E_T4oBgHgl3EQf0hwq/content/tmp_files/2301.08329v1.pdf.txt @@ -0,0 +1,1658 @@ +Viscosity measurements of glycerol in a parallel-plate rheometer exposed to +atmosphere +Jesse T. Ault∗ +Center for Fluid Mechanics, Brown University, Providence, Rhode Island 02912, USA +Sangwoo Shin +Department of Mechanical and Aerospace Engineering, University at Buffalo, +The State University of New York, Buffalo, NY 14260, USA +Allan Garcia, Antonio Perazzo, and Howard A. Stone +Department of Mechanical and Aerospace Engineering, +Princeton University, Princeton, NJ 08544, USA +Glycerol is a hygroscopic fluid that spontaneously absorbs water vapor from the atmosphere. For +applications involving glycerol, care must be taken to avoid exposure to humidity, since its viscosity +decreases quickly as water is absorbed. We report experimental measurements of the viscosity of +glycerol in a parallel-plate rheometer where the outer interface is exposed to atmosphere. The mea- +surements decrease with time as water is absorbed from the atmosphere and transported throughout +the glycerol via diffusion and advection. Measured viscosities drop faster at higher relative humidi- +ties, confirming the role of hygroscopicity on the transient viscosities. The rate of viscosity decrease +shows a non-monotonic relationship with the rheometer gap height. This behavior is explained by +considering the transition from diffusion-dominated transport in the narrow gap regime to the large +gap regime where transport is dominated by inertia-driven secondary flows. Numerical simulations +of the water absorption and transport confirm this non-monotonic behavior. +The experimental +viscosity measurements show unexpectedly fast decreases at very small gap heights, violating the +parallel-plate, axisymmetric model. We propose that this drop-off may be due to misalignment in +the rheometer that becomes non-negligible for small gaps. Theoretical considerations show that sec- +ondary flows in a misaligned rheometer dominate the typical secondary inertial flows in parallel-plate +rheometers at small gaps. Finally, simulations in a misaligned parallel-plate system demonstrate +the same sharp drop-off in viscosity measurements at small gap heights. +This modeling can be +used to estimate the gap height where misalignment effects dominate the transient glycerol viscosity +measurements. +I. +INTRODUCTION +A rotational rheometer is a device in which one component rotates relative to another in order to induce a shear on +the fluid placed in between the two components. Through a characterization of the torques and forces that result as a +function of rotation rate, rheological properties of the fluids can be measured. In the last several decades, rheometry +has emerged as an essential tool for studying the fluid dynamics of complex fluids for measuring properties ranging +from the dynamic viscosity of Newtonian fluids to the viscoelastic responses of non-Newtonian fluids [11, 31]. Owing +to their precision and versatility, rheometers have been utilized to investigate the rheology and mechanics of a wide +range of viscous and viscoelastic fluids such as polymer melts, gels, suspensions, cells, bacterial biofilms, lipid vesicle +solutions, food products, cosmetics, pharmaceuticals, and many others [9, 12, 17, 20, 29, 39, 44]. Rheometry has +also been used for evaluating drag reduction, since these techniques can also be used to characterize slip lengths such +as those generated by nanostructured surfaces [5, 7, 41]. Among various configurations, parallel-plate rheometry is +a system that is commonly used for highly viscous and viscoelastic fluids due to the ability to carefully control the +gap height and the spatially uniform confinement in the system. From a fluid dynamics perspective, a parallel-plate +rheometer generates a shear-driven fluid velocity that is primarily in the azimuthal direction. In the limit of slow +rotation speed or narrow gap heights, the velocity profile in such a system is simply (ur, uθ, uz) = (0, Ωrz/h0, 0), where +Ω is the angular velocity of the upper plate, h0 is the gap height between the plates, and (r, θ, z) are the cylindrical +coordinates with the origin located at the center of the bottom plate (see e.g., Bird et al. [4], Middleman [32] and +others). +In addition to the primary azimuthal flow, secondary recirculating flows exist due to inertial effects for any finite +angular rotation speed [19, 23]. The first experimental evidence of these secondary flows was achieved by Garner et al. +∗ jesse ault@brown.edu +arXiv:2301.08329v1 [physics.flu-dyn] 19 Jan 2023 + +2 +FIG. 1. Problem setup. We measure the viscosity of glycerol in a parallel-plate rheometer exposed to atmosphere at the outer +fluid interface. Due to the strong hygroscopic nature of glycerol, the water vapor present in the atmosphere is absorbed by the +glycerol at the outer boundary of the rheometer at a mass vapor flux of jw. The absorption of water by glycerol leads to a +change in the fluid properties over time including a reduction in the fluid viscosity, which leads to a transient reduction in the +effective viscosity measured by the system. +[21] in a study of the rheological properties of a hydrocarbon-type micellar system. The secondary flow profile in a +parallel-plate rheometer was found by Savins and Metzner more than half a century ago and has been well-described +by various authors (see e.g., Denn [15], Savins and Metzner [38]), where the radial velocity (except near the turning +regions) is given as +ur(r, z) = −ρΩ2h2 +0r +12µ +� +4 +5 +� z +h0 +� +− 9 +5 +� z +h0 +�2 ++ +� z +h0 +�4� +, +(1) +where ρ is the fluid density and µ0 is the fluid viscosity. Notably, the radial flow becomes increasingly important as +the gap height or rotation speed increase (i.e., ur/uθ ∼ Ωh2 +0/ν). +The secondary fluid dynamics can play a surprisingly significant role in a rheometer both by altering the torque +measurements and by triggering instabilities or driving fluid mixing, especially in the case of non-homogeneous fluids. +For example, Jacobi et al. [26] studied the effect of radial flow on the viscosity measurements in parallel-plate and +cone-and-plate settings when the target fluid is stratified with an immiscible fluid. In this case, the authors found +that the radial flow can distort the fluid interface, leading to drastically different torque measurements and even +fluid dewetting. Another situation in which the secondary recirculation plays a key role is in the case of an initially +homogeneous fluid that becomes non-homogeneous during the measurement procedure due to mass transfer that +occurs at the exposed outer edge, such as by solute sorption or solvent evaporation/condensation. This is particularly +important for parallel-plate rheometry since inhomogeneities alter the stress profiles, such that small changes in the +fluid composition at the outer edge can have significant impacts on the torque measurement. This secondary flow +is expected to play a significant role in the viscosity measurements of glycerol over time, since any water absorbed +at the outer edge of the rheometer can be transported radially inwards by the secondary flow, redistributing the +relatively lower viscosity glycerol (where the water fraction is higher). Since the torque in a parallel-plate rheometer +is primarily generated near the outer edge of the system, this redistribution of the lower-viscosity fluid must have +direct consequences on the measured torque value. +Previously, we reported that the strong hygroscopic nature of glycerol can cause unreliable viscosity measurements +in a cone-and-plate rheometer due to the continuous vapor absorption from the outer edge [40]. Motivated by this +observation, here we present a systematic study on the transient measurement of the viscosity of glycerol using a +parallel-plate rheometer (see Figure 1). As the secondary recirculating flow effectively disperses the absorbed water +throughout the glycerol layer, we find that the rate of decrease of the measured viscosity is a complex function of +the rheometer gap height, angular velocity, and the relative humidity. While the viscosity generally decreases over +time as water is absorbed, we find that the rate of decrease is a non-monotonic function of the gap height. Using the +theoretical solutions for the flow profile in a parallel-plate rheometer (i.e., Eq. (1)) along with numerical simulations, +we show that this non-monotonic behavior is consistent with existing theory provided the gap height is not too small. +In the limit of very small gap heights, we find that the behavior of the transient viscosity measurements is inconsistent +with the existing theory for a parallel-plate rheometer. We hypothesize that misalignment effects in the rheometer +result in non-negligible secondary flows at small gap heights that are responsible for this discrepancy in the measured +viscosity data. By developing new theoretical solutions and computational simulations for the misaligned parallel- +plate geometry, we show that this hypothesis is consistent with the measured viscosity data and that the misaligned +rheometer model can predict the viscosity measurements across the full range of gap heights. + +3 +II. +EXPERIMENTAL VISCOSITY MEASUREMENTS +Here, we consider the transient viscosity measurements of glycerol in a parallel-plate rheometer where the outer +fluid interface is exposed to the atmosphere. Due to the hygroscopic nature of glycerol, it will absorb water vapor from +the atmosphere at the outer boundary as shown in Figure 1, which will subsequently lead to a local reduction in the +viscosity of the fluid and a net reduction of the torque measured by the rheometer. Thus, the measured viscosity of +glycerol in a parallel-plate rheometer is expected to decrease with time when exposed to atmosphere. Intuitively, the +rate of decrease should depend on the water concentration/flux experienced at the outer boundary, which is influenced +by the relative humidity in the atmosphere. The rate of decrease should also depend on the secondary recirculating +flows in the rheometer, since these redistribute the relatively less viscous fluid where the water concentration is higher, +thereby altering the stress distribution and total torque experienced by the upper plate of the rheometer. +A. +Experimental methods +Glycerol was purchased from Sigma-Aldrich. +Viscosity measurements were performed using a stress-controlled +rheometer (Physica MCR 301, Anton Paar) with a parallel-plate configuration (plate diameter = 50 mm). +The +rheometer was placed inside an acrylic chamber in which the relative humidity (RH) was controlled using multiple +vapor sources and a stream of dehumidified air. RH was constantly monitored using a digital hygrometer (VWR). +Experiments were performed at 23◦C. +B. +Experimental results +The experimental results for the transient viscosity measurements are presented in Figure 2. First, the transient +viscosity measurements of glycerol in a relative humidity environment of 53% are presented in Figure 2a. +The +experiments were performed with a rheometer with plate radius R = 2.5 cm at an angular velocity of Ω = 0.4 rad/s. +Results are normalized by the initial viscosity µi to account for any small amount of water absorption by the glycerol +while setting up the experiment. As can be seen, the measured viscosities decrease at varying rates depending on the +gap thickness. These trends are all monotonic. However, the rate of viscosity decrease is seen to be a non-monotonic +function of the gap thickness. That is, decreasing the gap height from h0 = 1.0 mm to h0 = 0.5 mm results in the +viscosity dropping at a slower rate. However, subsequently decreasing the gap height results in the viscosity dropping +at a faster and faster rate, until the viscosity drops sharply for the smallest gap height. Furthermore, experimental +results for the final measured viscosity µf/µi after t = 60 minutes are shown in Figure 2b as a function of gap height +h0 for two different relative humidities. +Here, with RH = 72% the viscosities are seen to drop faster than with +RH = 45% due to the increased mass flux of water vapor to the glycerol from the atmosphere. In addition, the final +measured viscosity values also show the same non-monotonic behavior as in Figure 2a. That is, decreasing h0 first +leads to an increase in µf/µi, followed by a sharp decrease for h0 < 0.5 mm. +Next, the transient decrease in measured viscosity values are shown in Figure 2c as functions of relative humidity +at a rotation speed of 0.4 rad/s and a gap height of 0.1 mm. Here the results show a monotonic relationship in which +increasing the relative humidity leads to a faster decrease in viscosity measurements due to the increased flux of water +into the glycerol. Finally, the transient measured viscosities are shown in Figure 2d for varying rotation speeds at a gap +height of 1 mm and a relative humidity of 54%. Here again a non-monotonic relationship with Ω is observed. At first, +increasing the rotation speed from 0.4 rad/s to 4.0 rad/s leads to a slower decrease in viscosity, while subsequently +increasing the rotation speed to 40.0 rad/s leads to a much faster decrease in the measured viscosity values. The key +observations from the experimental measurements are: +1. Increased relative humidity leads to faster viscosity decrease for all measured cases. +2. The rate of viscosity decrease varies non-monotonically with the gap height. That is to say, for different gap +heights, the measured viscosity decreases at a different rate over time. +3. There is a sharp decrease in the measured final viscosity for very small gap heights. +Observation 1 above is a natural and intuitive result, since the hygroscopic nature of glycerol leads it to absorb more +water from the atmosphere at higher humidities. Since the viscosity of a glycerol–water mixture varies monotonically +with the water mass fraction, the rate of viscosity decrease can be expected to vary monotonically with the relative +humidity. However, an intuitive explanation for observations 2 and 3 is not immediately obvious without a more +careful consideration of the fluid dynamics and the coupled transport of dissolved water in the rheometer. In the + +4 +FIG. 2. Experimental viscosity measurements. (a) Transient decrease in the measured viscosity as a function of time, normalized +by the initial viscosity µi with Ω = 0.4 rad/s and R = 2.5 cm. Experiments were performed at a relative humidity of RH = 53%. +(b) Measured final viscosities µf of glycerol at t = 3600 s normalized by µi with Ω = 0.4 rad/s and R = 2.5 cm. Experiments +were performed at relative humidities of RH = 45% and RH = 72%. (c) Normalized transient viscosity measurements over time +for varying relative humidities at a gap height of 0.1 mm and a rotation rate of 0.4 rad/s. (d) Normalized transient viscosity +measurements over time for varying rotation rates at a gap height of 1 mm and a relative humidity of 54%. +following sections, we will seek a physical explanation for these two experimentally observed behaviors using theory +and numerical simulations. +III. +FLUID DYNAMICS OF A GLYCEROL–WATER MIXTURE +Before proceeding to analyze the specific fluid dynamics and transient viscosity evolution in a parallel-plate rheome- +ter, in this section we first introduce all of the relevant governing physics that applies to such systems. This includes +the governing Navier-Stokes equations for variable viscosity fluids, the advection-diffusion equation for the absorbed +water concentration with variable diffusivity, the empirical relationships for the coefficients of viscosity and diffusivity +of glycerol-water mixtures, as well as the saturation concentration of absorbed water in glycerol as a function of the +relative humidity that will be used in the modeling. + +5 +A. +Navier-Stokes equations for a variable viscosity fluid +In a glycerol-water mixture, the viscosity is strongly dependent on the local mass fraction of water. In systems +with a homogeneous concentration of absorbed water, the constant viscosity form of the Navier-Stokes equations is +appropriate. However, when gradients in water concentration exist in the system, due to circumstances such as mixing +streams or the absorption of water at a gas-liquid interface, the viscosity of the fluid must be treated as a function of +time and position. Note that for glycerol-water mixtures, the density is a weak function of the water concentration, +ranging from approximately 1260 kg/m3 for pure glycerol down to 1000 kg/m3 for pure water. Here, we neglect +this variation and use the density of pure glycerol, assuming the water mass fraction does not get too large. The +corresponding continuity equation for an incompressible flow is simply ∇∗ · u∗ = 0. In such a system, extra stresses +arise in the fluid that are related to the gradients of viscosity, and the appropriate form of the Navier-Stokes equations +(for a Newtonian constitutive equation) is: +∇∗ · u∗ = 0 +and +ρDu∗ +Dt∗ = −∇∗p∗ + µ∗∇∗2u∗ + ∇∗µ∗ · ∇∗u∗ + ∇∗µ∗ · (∇∗u∗)T , +(2) +where ρ is the fluid density, µ is the fluid viscosity, and T denotes the transpose. Here, ∗’s denote dimensional variable +quantities. For the case of the flow in a rheometer system, we will represent the flow using cylindrical coordinates, +and we nondimensionalize the governing equations with +r = r∗ +R , +z = z∗ +h0 +, +ur = u∗ +r +ΩR, +uθ = u∗ +θ +ΩR, +uz = u∗ +z +Ωh0 +, +p = +p∗ +µ0ΩR2/h2 +0 +, +µ = µ∗ +µ0 +, +and +t = Ωt∗, +(3) +where µ0 is a reference viscosity value. +With these nondimensionalizations, the component form of Eq. +(2) in +cylindrical coordinates is given by +Re +�∂ur +∂t + ur +∂ur +∂r + uθ +r +∂ur +∂θ − uθ2 +r ++ uz +∂ur +∂z +� += −∂p +∂r + µ +�ϵ2 +r +∂ +∂r +� +r∂ur +∂r +� ++ ϵ2 +r2 +∂2ur +∂θ2 ++∂2ur +∂z2 − ϵ2 ur +r2 − 2ϵ2 +r2 +∂uθ +∂θ +� ++ 2ϵ2 ∂µ +∂r +∂ur +∂r + ϵ2 +r +∂µ +∂θ +�1 +r +∂ur +∂θ + ∂uθ +∂r − uθ +r +� ++ ∂µ +∂z +�∂ur +∂z + ϵ2 ∂uz +∂r +� +, +(4a) +Re +�∂uθ +∂t + ur +∂uθ +∂r + uθ +r +∂uθ +∂θ + uθur +r ++ uz +∂uθ +∂z +� += −1 +r +∂p +∂θ + µ +�ϵ2 +r +∂ +∂r +� +r∂uθ +∂r +� ++ ϵ2 +r2 +∂2uθ +∂θ2 ++∂2uθ +∂z2 − ϵ2 uθ +r2 + 2ϵ2 +r2 +∂ur +∂θ +� ++ ϵ2 ∂µ +∂r +�∂uθ +∂r + 1 +r +∂ur +∂θ − uθ +r +� ++ 2ϵ2 +r +∂µ +∂θ +�1 +r +∂uθ +∂θ + ur +r +� ++ ∂µ +∂z +�∂uθ +∂z + ϵ2 +r +∂uz +∂θ +� +, +(4b) +Re ϵ2 +�∂uz +∂t + ur +∂uz +∂r + uθ +r +∂uz +∂θ + uz +∂uz +∂z +� += −∂p +∂z + µ +�ϵ4 +r +∂ +∂r +� +r∂uz +∂r +� ++ ϵ4 +r2 +∂2uz +∂θ2 + ϵ2 ∂2uz +∂z2 +� ++ ∂µ +∂r +� +ϵ4 ∂uz +∂r + ϵ2 ∂ur +∂z +� ++ 1 +r +∂µ +∂θ +�ϵ4 +r +∂uz +∂θ + ϵ2 ∂uθ +∂z +� ++ 2ϵ2 ∂µ +∂z +∂uz +∂z , +(4c) +where the Reynolds number is defined as Re = ρΩh2 +0/µ0, and the gap aspect ratio is ϵ = h0/R, which is typically +small in a parallel-plate rheometer. The corresponding continuity equation is given by +1 +r +∂ +∂r (rur) + 1 +r +∂uθ +∂θ + ∂uz +∂z = 0. +(5) +Along with boundary conditions, these equations govern the motion of variable viscosity fluids in a parallel-plate +rheometer. The typical boundary conditions for such a system include no-slip conditions at the lower (stationary) +and upper (rotating) plates, as well as a stress-free condition at r = 1 at the gas-liquid interface. + +6 +B. +Water absorption and transport +Along with the equations governing the fluid dynamics in the previous section, the system further requires a +transport equation to model the absorption and transport of water in the glycerol. In particular, this transport can +be modeled with an advection-diffusion equation that is given by +∂c +∂t∗ = ∇∗ · (D∗∇∗c) − u∗ · ∇∗c, +(6) +where c is the mass fraction of water in the glycerol, D∗ is the diffusivity of water in glycerol, and u∗ is the dimensional +fluid velocity vector. Here, D∗ is a spatially/temporally varying function of the local water concentration c. Using +the same nondimensionalizations as above, Eq. (6) in cylindrical coordinates becomes +Pe ϵ2 ∂c +∂t = ϵ2 +r +∂ +∂r +� +rD ∂c +∂r +� ++ ϵ2 +r2 +∂ +∂θ +� +D ∂c +∂θ +� ++ ∂ +∂z +� +D ∂c +∂z +� +− Pe ϵ2 +� +ur +∂c +∂r + uθ +r +∂c +∂θ + uz +∂c +∂z +� +, +(7) +where D = D∗/D0 is the nondimensional diffusivity with reference value D0 and Pe = ΩR2/D0 is the Peclet number +representing the ratio of the timescale for diffusion of water in the radial direction to the convective timescale. +Examining Eq. (7), the diffusion in the z-direction is O(ϵ−2) larger than the radial diffusion due to the separation +in length scales: in the low-inertia, thin-gap limit, the water concentration will be approximately uniform in the +z-direction. +Along with Eq. (7), the absorbed water concentration must satisfy certain boundary conditions in the system. In +particular, the concentration satisfies a no-flux condition at both the upper and lower plates of the rheometer (i.e. +∂c +∂z = 0 in the parallel-plate case). In addition, a boundary condition for the water concentration is needed at the +outer glycerol/air interface. In general, this condition could be represented by a water flux condition such as by +−D ∂c +∂r +�� +r=1 = jw, where the flux of water jw could be a function of the local water concentration in the glycerol at the +interface as well as the water vapor concentration and distribution in the air near the interface. The solution of such +a flux will typically require also solving the water vapor transport problem in the surrounding environment, since +these transport processes are coupled at the interface. For example, the transport of water vapor in the surrounding +atmosphere may be affected by the rotation speed of the rheometer, which can drive flow in the surrounding air +that may alter the vapor transport at the interface. Here, we neglect these effects and assume that the water vapor +transport in the atmosphere is fast relative to the water concentration transport in the glycerol. This assumption +is valid due to the substantially higher diffusivity of water vapor in air than absorbed water in glycerol, provided +the recirculation in the rheometer is not significantly fast. Thus, we assume that the water mass fraction at r = 1 +instantaneously reaches its saturation value based on the local relative humidity in the surrounding air. +Considering both Equations (4) and (7), we see how the dynamics of the problem are fully coupled. In the fluid +problem, the Navier-Stokes equations are coupled to the water transport problem via the dependence of viscosity +on water mass fraction. The water transport problem is further coupled to the Navier-Stokes equations through the +dependence on the flow velocity. Finally, the transport is further complicated by the dependence of diffusivity on water +mass fraction. Due to this two-way coupling and the empirical nature of both the µ(c) and D(c) relationships, it is +difficult to seek theoretical solutions to the coupled dynamics. Thus, here we primarily rely on numerical simulations +to solve the coupled transport problem. +C. +Physical properties of glycerol–water mixtures +Detailed empirical formulas for the viscosity of a glycerol–water mixture have been proposed by Cheng [6] which +are valid for water mass concentrations in the range of 0–100% and for temperatures ranging from 0 to 100◦C. The +viscosity of a glycerol–water mixture at 22◦C varies from around µg = 1.1 Pa·s for pure glycerol down to µw = 0.96 +mPa·s for pure water, spanning a range of approximately three orders of magnitude. Here, we choose a reference +viscosity value corresponding to that of pure glycerol µ0 = µg, such that the nondimensional viscosity varies from an +initial condition of 1 down to as low as ∼ 8.73 × 10−4 if the water mass fraction were to approach 1. The empirical +relationship determined between nondimensional viscosity and water concentration used here is shown in Figure 3a. +The diffusivity of water in glycerol is also a function of the water mass fraction, and so D∗ is expected to evolve as +a function of both position and time as more water is absorbed and transported throughout the system. An empirical +relationship for the diffusivity of water in glycerol has also been developed for mixtures at 25 ◦C by D’Errico et al. + +7 +FIG. 3. (a) Empirical nondimensional viscosity and diffusivity relationships used in this study. (b) Saturation concentration +of water in glycerol and corresponding specific gravity as functions of relative humidity. +[16], and is given by +D∗ = 1.024 − 0.91xg +1 + 7.5xg +· 10−9 m2 +s , +(8) +where xg is the mole fraction of glycerol that is related to c via +xg = +Mw(1 − c) +Mw(1 − c) + Mgc, +(9) +where Mw is the molar mass of water and Mg is the molar mass of glycerol. For reference, the diffusivity of water +in pure glycerol (c = 0) is 1.341 × 10−11 m2/s, which increases up to approximately 1.024 × 10−9 m2/s as the water +mass fraction approaches 1. Here we again choose a reference value equal to the diffusivity in pure glycerol D0 = Dg, +such that D varies from 1 at c = 0 up to approximately 76.4. This empirical relationship from D’Errico et al. [16] is +also shown in Figure 3a. Finally, the saturation concentration of water in glycerol as a function of relative humidity +is needed for the absorption boundary condition at r = 1. These values were measured by the Glycerine Producers’ +Association and are given in Table 15 of Glycerine Producers’ Association and others [22]. For reference, these values +are plotted in Figure 3b along with the specific gravity as functions of relative humidity. With the full governing +equations and empirical relationships for the physical parameters described above, we can now move on to consider +the coupled fluid dynamics and water concentration transport in a rheometer. In the following section we first review +the classical result for the axisymmetric parallel-plate rheometer with constant viscosity before moving on to cases +with variable viscosity. +D. +Note on the assumption of constant density +Before moving on, we briefly comment on the assumption of constant density. As water is absorbed into the glycerol, +the resulting density gradients introduce the possibility for buoyancy-driven flows. An estimate for the magnitude +of such effects is given by considering that a vertical change in density ∆ρ implies a radial pressure gradient on the +order of ∆ρgh0/R. We can balance this radial pressure gradient with a radial viscous stress gradient µub/h2 +0, where +ub is the characteristic magnitude of the buoyancy-driven flow. Thus, we have ub ∼ ∆ρgh3 +0 +µR . For buoyancy effects to +be negligible, we need ub to be small relative to the magnitude of the inertial secondary-velocity components which +are O +� +ρΩ2h2 +0R +µ +� +, as shown in Equation (1). Thus, we need +∆ρgh3 +0 +µR +≪ ρΩ2h2 +0R +µ +−→ +∆ρ +ρ +≪ Ω2R2 +gh0 +. +(10) + +8 +The left-hand side of this inequality has a maximum value of around 0.2 when the saturation water concentration +approaches 100%, and so it will typically take a smaller value depending on the relative humidity. The right-hand +side of the inequality depends on the system parameters, but a typical value with Ω = 4.0 rad/s, R = 2.5 cm, and +h0 = 0.5 mm is approximately 2.0. Thus, the buoyancy-driven flow can be expected to be less than 10% of the inertial +secondary flow. At larger gap heights and smaller rotation speeds, this inequality suggests that buoyancy effects +become relatively more important, and could even dominate the dynamics. However, in the very slow rotation case, +the dynamics are nearly 1D, such that the density variation is almost entirely in the radial direction and the negligible +vertical density gradient should not affect the radial pressure gradient. So, Equation (10) should be considered only +as an estimate of the importance of buoyancy effects. +IV. +CLASSICAL RESULT: AXISYMMETRIC FLOW WITH CONSTANT VISCOSITY +Before moving on to consider the full coupled dynamics of glycerol absorbing water in a parallel-plate rheometer, +we first review the classical Newtonian, constant viscosity flow solution in a parallel-plate rheometer. Understanding +this flow is important because it illustrates the types of secondary flows we should expect in the rheometer, and also +provides some initial insights into the non-monotonic relationship between measured viscosities and rotation speed +and gap height in the full system. The axisymmetric parallel-plate rheometer has a well-known solution that has +been previously described by multiple authors (see, for example Bird et al. [4], Middleman [32]). Here, we briefly +reproduce this calculation in our notation for consistency with later sections. We consider a parallel-plate rheometer +with radius R and gap thickness h0. The lower plate is stationary and the upper plate rotates at a rotation speed of +Ω. We model the system using cylindrical coordinates with the origin located at the center of the bottom plate. Thus, +the governing equations are the axisymmetric and constant viscosity forms of Eqs. (4) and (5). The corresponding +boundary conditions are no-slip at both the upper and lower plates, i.e. (ur,axi, uθ,axi, uz,axi) = (0, 0, 0) at z = 0 and +(ur,axi, uθ,axi, uz,axi) = (0, r, 0) at z = 1. For small gap heights ϵ ≪ 1 a solution for the velocity and pressure can be +sought in the form of an expansion in powers of ϵ2. Up to O(ϵ4), this axisymmetric solution is given by +ur,axi(r, z) = − 1 +12rRe z(z − 1) +� +−4 +5 + z + z2 +� ++ O(ϵ4), +(11a) +uθ,axi(r, z) = rz − rRe2z +6300 (8 + z3(35 − 63z + 20z3)) + O(ϵ4), +(11b) +uz,axi(z) = 1 +30Re z2(z − 1)2(2 + z) + O(ϵ4), +(11c) +paxi(r) = 3r2 +20 Re + 1 +30Re ϵ2z(4 − 9z + 5z3) + O(ϵ4). +(11d) +Here, the subscript ‘axi’ denotes the axisymmetric case, and the expression for ur,axi is equivalent to the result +first presented by Savins and Metzner [38], which was given above in dimensional form as equation (1). The primary +flow is the uθ,axi = rz component with an O(Re2) correction, while the leading-order secondary flows in the r- and +z-directions are both O(Re). Since the flow of interest is axisymmetric, the primary velocity components of interest for +redistributing absorbed species at the outer edge of the rheometer are the secondary velocity components, especially +the radial component, since this will transport absorbed water from the outer edge inwards through the gap. +A visualization of the secondary velocity components is given in Figure 4. Here, the uz,axi component shows that +there is an upward drift that is independent of r throughout the rheometer. The radial component shows that in +the upper-half of the gap the flow is directed radially outwards, and in the lower-half of the gap the flow is directed +radially inwards. Keep in mind that the theoretical solution presented in Eq. (11) must break down near the outer +edge of the rheometer where r → 1, since the lubrication approximation fails in that region. In the true system, near +the outer edge the outward-traveling flow in the upper-half of the rheometer must turn downwards for continuity and +turn around to then travel radially inwards. The width of this turning region should be O(ϵ), and thus is progressively +more confined at the outer edge as the gap height decreases. This effect is not captured by Eq. (11), although it may +have an important role in the transport of absorbed water in the glycerol/water system. +With this picture of the secondary flows, we can hypothesize an explanation for the non-monotonic behavior of the +measured viscosity with gap height and rotation speed in the experimental results. First, in the slow rotation speed +or small gap-height limit, absorbed water can only transport radially inwards via diffusion which is quite slow because +the radial fluid velocity is O(Re). In reality, the diffusive transport will proceed faster than expected based on the +diffusivity of water in pure glycerol, because the absorbed water increases the diffusion coefficient as it is absorbed. +However, even using the diffusivity value that results for c → 1, the radial diffusion remains a slow and inefficient +process. Thus, in the low-Re limit, the absorbed water remains highly confined near the outer edge of the rheometer, + +9 +FIG. 4. Secondary velocity components ur,axi and uz,axi in a parallel-plate rheometer in the small-gap limit (ϵ ≪ 1). The radial +velocity proceeds outward along the upper-half of the gap, reverses at the outer edge, and proceeds radially inward along the +lower-half of the gap. The z-component shows an upward drift along the middle of the gap that is independent of r. Both +secondary velocities are O(Re). Results correspond to Eq. (11). +except over impractically long experimental timescales. On the one hand, this confines the relatively low viscosity +fluid at the outer edge of the rheometer, where the torque is primarily generated, but it limits the amount of water +that is absorbed since the concentration gradient is relatively diffuse at r = 1. As Re increases, the secondary flow +begins to pull some of the absorbed water radially inward, steepening the gradient at r = 1 and increasing the total +flux of water into the system, while still leaving the absorbed water relatively confined at the outer edge. This results +in a greater amount of absorbed water near the outer edge of the rheometer and a faster decrease in the measured +viscosity. As Re continues to increase, the secondary flows continue to more strongly pull the absorbed water away +from the outer edge, further increasing the amount of absorbed water via the steeper gradient at r = 1. However, it +is plausible that above a certain Re the secondary flows become sufficient enough to pull the absorbed water radially +inwards a distance that is sufficiently far from the outer edge such that the net effect on the measured torque begins +to lessen. That is, despite the fact that more water is absorbed, this low viscosity fluid is redistributed towards the +inner part of the rheometer where the effect on the torque and measured viscosity is less. Thus, the constant viscosity, +axisymmetric results can provide one possible explanation for the non-monotonic behavior seen between the measured +viscosity and the gap height and rotation speed. Furthermore, in the high-Re limit, the absorbed water concentration +can be expected to be well-mixed, promoting a rapid flux of water into the glycerol by maintaining a strong gradient +at r = 1 and rapidly redistributing the low viscosity fluid throughout the system. In such a regime, more care should +be taken with determining the flux boundary condition, since the rate of water transport in the glycerol may approach +the rate of water vapor transport in the outer flow problem where depletion of water vapor near the interface may +limit the available flux into the glycerol. As a quick point of reference, with the definition of Reynolds number given +by Re = ρgΩh2 +0/µg, with the characteristic density and viscosity based on values for pure glycerol, the experimental +results presented above in Figure 2 have Reynolds numbers ranging from ∼ 1×10−6 up to ∼ 0.05. While these values +seem small, recall that the dimensional viscosity can vary by over three orders-of-magnitude, such that a locally +defined Reynolds number could be significantly larger. +Finally, we note that the constant viscosity parallel-plate model is apparently inconsistent with the experimental +observation of rapidly decreasing viscosity measurements at very small gap heights. In the limit of Re ≪ 1, the +secondary flows in a parallel-plate rheometer are negligible. In this case, the transport equation simplifies to a purely +1D radial diffusion problem, and the evolution of water concentration becomes independent of both the gap height +and rotation speed. Furthermore, the viscosity distribution likewise is independent of h0 and Ω, which is inconsistent +with the sharp decrease in µf/µi seen in Figure 2b at very small gap heights. This will motivate us later in the paper +to consider the potential role of misalignment. First, we examine in more detail the one-dimensional diffusive limit +with variable viscosity. + +10 +V. +ONE-DIMENSIONAL DIFFUSIVE LIMIT WITH VARIABLE VISCOSITY +First, we consider the evolution of the viscosity distribution and measured effective viscosity of glycerol absorbing +water in the inertialess, one-dimensional diffusive limit. For small Reynolds numbers and gap heights, the axisymmetric +form of Eq. (7) becomes +Pe∂c +∂t = 1 +r +∂ +∂r +� +rD ∂c +∂r +� ++ 1 +ϵ2 +∂ +∂z +� +D ∂c +∂z +� +. +(12) +Considering that the water concentration boundary condition at r = 1 is independent of z, along with the no-flux +conditions at the upper and lower plates, when ϵ ≪ 1 it must be the case that c is approximately independent of z, +so that Eq. (12) further simplifies to +Pe∂c +∂t = 1 +r +∂ +∂r +� +rD ∂c +∂r +� +, +(13) +which is simply a 1D radial diffusion equation with variable diffusivity D(c). In this regime, a better choice for the +characteristic time scale would be the characteristic radial diffusion time R2/D0, the use of which would yield the +same equation without the Pe factor on the left-hand side. For consistency, we continue to use the convective 1/Ω +timescale as the characteristic timescale. Here, the only boundary conditions that are needed are symmetry at r = 0 +and the saturation water mass fraction at r = 1, i.e., c(r = 1) = csat. +As the water concentration evolves, the anticipated viscosity measurement from the rheometer can be predicted +through the use of the viscosity distribution as follows. +A parallel-plate rheometer cannot measure the viscosity +distribution throughout the fluid layer, but rather simply infers an effective viscosity µ∗ +eff by measuring the total +torque exerted on the upper plate as it spins. In dimensional form, the azimuthal velocity at small gap heights is +u∗ +θ = Ωr∗z∗/h0. This velocity profile is valid regardless of the viscosity distribution since c is a function of r and t +only. The total torque experienced by the upper plate is then given by +T = +2π +� +0 +R +� +0 +µ∗Ω +h r∗3 dr∗ dθ. +(14) +If the viscosity is constant and uniform, the total torque on the upper plate is then +T = πµ∗ΩR4 +2h +−→ +µ∗ = 2hT +πΩR4 . +(15) +The rheometer assumes a constant viscosity fluid and reports the “effective” viscosity of the fluid that is calculated +from Eq. (15) based on the measured torque. In the experimental system, the initial condition is assumed to be pure +glycerol, such that Tinit = πµgΩR4/(2h), and we have +µ∗ +eff +µg += T(t) +Tinit += +2h +πµgΩR4 +2π +� +0 +R +� +0 +µ∗Ω +h r∗3 dr∗ dθ +−→ +µeff = 2 +π +2π +� +0 +1 +� +0 +µr3 dr dθ, +(16) +which simplifies to +µeff = 4 +1 +� +0 +µr3 dr +(17) +for axisymmetric flow. +Here, we perform 1D transient simulations of Eq. (12) using the finite-difference method with second-order accuracy +in space and first-order accuracy in time. Convergence studies were performed in space and time to verify the results. +Using these simulations, we compute the effective nondimensional viscosity over time as water is absorbed at the outer +edge and diffuses radially inwards. We perform these simulations over a range of csat values which reproduces the +effect of varying relative humidities. First, for comparison with experiments, the results are simulated for one hour +to determine the degree of viscosity decrease that can be achieved via pure diffusion over the experimental timescale. +These results are shown in Figure 5a. As can be seen, diffusion alone is sufficient to generate a significant decrease in + +11 +FIG. 5. Transient nondimensional viscosity measurements in the inertialess, 1D, axisymmetric regime. (a) Results simulated +over 3600 seconds for comparison with experimental measurements. Results show substantial decreases in measured viscosities +at large csat values, but not as significant as those seen in the experiments. Figure inset shows the final nondimensional viscosity +µf versus csat. (b) Results extended to much longer times to show the final saturation of the glycerol, which corresponds to +the curves leveling off. Clearly, higher values of csat reach saturation more quickly. +measured viscosities over this timescale, although not to the degree seen in the experiments. For example, consider +the experimental results in Figure 2c, which were performed at a Reynolds number of Re = ρΩh2 +0/µ0 = 4.6×10−6 and +aspect ratio of ϵ = h0/R = 4 × 10−3. Clearly, in such a regime the inertialess, 1D model would be expected to apply. +However, the experimental results show a much larger decrease in viscosity over this timescale. The RH = 72% results +(corresponding to approximately csat = 0.386) drop to around µ = 0.38, and the RH = 45% results (corresponding to +csat = 0.185) drop to around µ = 0.6. However, in the 1D limit, the corresponding numerical predictions for csat = 0.4 +and csat = 0.2 only decrease to around µ = 0.78 and µ = 0.86, respectively. +Thus, the experiments show a much larger decrease in viscosity over this timescale than the 1D model with pure +diffusion. Furthermore, the results shown in Figure 5a are clearly still evolving over this timescale, whereas in the +long-time limit we expect all of the glycerol to homogenize at the saturation concentration based on the relative +humidity. Therefore, we extend these results to much longer times in Figure 5b, which shows the measured effective +viscosities level off as the water concentration saturates. Here we see the influence of the variable diffusivity on the +timescale for the diffusive process. With the characteristic diffusivity D0, the timescale for the process would be +expected to be t∗ = O(R2/D0). However, as can be seen, most of the cases have fully saturated well before this +timescale, especially at larger csat values. This is due to the enhanced diffusion at larger water concentrations. In +fact, a much better prediction for the timescale of this 1D diffusive process is to use the diffusivity based on csat, +which we call Dsat. The rescaled results are shown in Figure 6, which shows that for each case the water concentration +in the glycerol has fully saturated over the timescale t∗ = O(R2/Dsat). For each case, two regimes can be seen. In +the early times, the water concentration in the glycerol is non-uniform, and so the diffusive transport in the domain +proceeds with a spatially varying diffusivity coefficient. At late times, the water concentration throughout the system +has nearly equilibrated at around the saturation concentration, such that the diffusion coefficient is nearly uniform +and the results all decay exponentially with the same rate constant. +This constant can be simply calculated by +considering a 1D radial diffusion problem with constant diffusivity (since this is nearly the case at long times), where +the transport in dimensional form is governed by +∂c +∂t∗ = Dsat +1 +r∗ +∂ +∂r∗ +� +r∗ ∂c +∂r∗ +� +with +∂c +∂r∗ +���� +r∗=0 += 0 +and +c(r∗ = R) = csat. +(18) +The solution to this is given by +c(r∗, t∗) = csat + +∞ +� +n=1 +ane−Dsatt∗λ2 +nJ0(λnr), +(19) + +12 +FIG. 6. Rescaled effective viscosities in the 1D, inertialess, axisymmetric limit. For each case the water concentration fully +saturates approximately over the timescale t∗ = O(R2/Dsat), which is consistent with diffusion primarily occurring at the +saturation concentration diffusivity. At late times, the rescaled viscosities all approach the saturation values exponentially with +a rate constant of 5.78, consistent with the 1D theory. +where the an are coefficients that depend on the initial condition, and J0 is the zeroth-order Bessel function of the +first kind. The λn eigenvalues here are the roots of J0 divided by the radius R. Thus, at late times we see that +c − csat ∼ exp +� +−χ2 +1t∗Dsat/R2� +, where χ1 = 2.40483 is the first root of the J0 function. Thus we see the χ2 +1 = 5.7832 +exponential decay seen in Figure 6. +The previous results are strictly valid in the inertialess (Re ≪ 1), small gap (ϵ ≪ 1), and axisymmetric limits. +These calculations are significantly simplified compared to the solution for the inertial regime since (1) the water +concentration profile can no longer be assumed to be independent of z due to the secondary velocity components and +(2) the fluid velocity profiles must be recalculated continuously as the concentration profile evolves while taking into +account the spatial variations in viscosity. Nevertheless, it is clear that we must extend our results to the inertial +regime, since the 1D diffusion-dominated results cannot reproduce the same degree of viscosity decrease over the +timescale of the experiments. These simulations are pursued in the following section. +VI. +INERTIAL REGIME WITH VARIABLE VISCOSITY +Having explored the purely diffusion-dominated 1D axisymmetric regime in the previous section, we now extend +our results to the inertial, axisymmetric regime. Recall that in the inertial regime, the coupled dynamics are governed +by four dimensionless parameters, which are: +Pe = ΩR2 +D0 +, +Re = ρΩh2 +0 +µ0 +, +ϵ = h0 +R , +and +csat, +(20) +whereas in the diffusion-dominated case the dynamics are governed only by csat. Thus, the system is governed by a +relatively large parameter space. However, note that the ratio µ0/(ρD0) = µg/(ρDg) is fixed for glycerol, and the +Peclet number can be written as +Pe = ΩR2 +D0 += +�ρΩh2 +0 +µ0 +� �R2 +h2 +0 +� � µ0 +ρD0 +� += Re +ϵ2 +� µ0 +ρD0 +� +, +(21) +so that the Peclet number is uniquely determined by the choice of Re and ϵ. + +13 +FIG. 7. Sample computational mesh design for the inertial, axisymmetric simulations. The grid has been coarsened by a +factor of 3 in the r- and z-directions for visualization purposes. Local mesh refinement is used near r = 1 to resolve the water +concentration boundary layer. (a) Top-down view of the axisymmetric wedge mesh geometry. (b) Side view of the wedge mesh. +(c) Zoom in of the local refinement near r = 1. Several extra layers of very thin cells exist on the right-hand side which are +difficult to see in order to resolve sharp concentration gradients that can occur at the boundary when inertial effects come into +play. +A. +Numerical methods +Numerical simulations were performed using OpenFOAM [43] with an axisymmetric wedge-shaped mesh geometry +with a wedge angle of 1◦. Local mesh refinement was used near r = 1 to resolve the water concentration boundary +layer. +A sample mesh design is shown in Figure 7. +Simulations were performed using a custom in-house solver +that iteratively updates the water concentration profile for 100 timesteps using a timestep of 0.01 seconds using +second-order backward time-stepping and then recalculates the new steady-state velocity/pressure profiles using the +SIMPLE algorithm [3, 27]. Thus, the solver assumes that the fluid velocity does not change much during one timestep. +Convergence tests were performed to confirm that recalculating the velocity every 100 timesteps had a negligible impact +on the calculated results compared to re-solving every timestep. The solver assumes that the velocity/pressure profiles +are quasi-steady and only evolve when the water-concentration profile changes. The SIMPLE algorithm was used with +relative pressure and velocity tolerances of 1 × 10−5. Convergence tests also confirmed the results were insensitive to +these tolerances. Finally, grid resolution convergence tests were performed, and a final base grid of 375 × 30 cells in +the r × z directions was chosen. The cells within the region from r = 1 − 2ϵ to 1 were all further refined by one level. +Finally, the final layer of cells at r = 1 was further refined by halving three times. Using this grid, convergence tests +indicate that the errors due to spatial discretization should be less than 1%. Torque measurements were calculated +by integrating the wall shear stress over the upper plate. +B. +Results +Using the numerical methods described in the previous section, simulations were performed across a range of gap +heights, Re, and relative humidities (through their proxy csat). Before introducing the final measured viscosity values +for comparison with the experiments, we first present results illustrating the evolution and dynamics of the water +concentration field in the glycerol over a range of gap heights and rotation speeds. A comparison of the evolving water +concentration profiles at various rotation speeds is shown in Figure 8, and we give the results in dimensional form to +make more clear the relationship of the changes to the experimental results presented earlier. These simulations were +performed at csat = 0.2, and ϵ = (1.0 × 10−3 m)/(2.5 × 10−2 m) = 0.04 with rotation speeds of (a) 0.4 rad/s, (b) 1.0 +rad/s, (c) 4.0 rad/s, and (d) 10.0 rad/s. The corresponding nondimensional parameters for these cases are summarized +in Table I. Here, the Reynolds number ranges from Re = 4.58 × 10−4 up to 1.15 × 10−2. This seems counter-intuitive, +since even the smallest rotation speed case shows some deviation from a purely 1D, diffusive transport, as can be +seen by the concentration variation in the z-direction, whereas the relatively small Reynolds numbers suggest inertial +effects should be small for all of these cases. However, consider that u∗ +r,axi ∼ ΩRRe. Then the characteristic time +for convection in the radial direction is τconv,rad = (Ω Re)−1, while the characteristic time for diffusion in the radial +direction is τdiff,rad = R2/D0. Thus, an appropriate radial Peclet number is Perad = +τdiff,rad +τconv,rad = ΩR2Re +D0 +. These values +are also tabulated in Table I. As can be seen, even for the smallest angular velocity case with Ω = 0.4 rad/s, the +radial Peclet number is still greater than O(103), increasing up to O(106) at 10 rad/s. +Thus, even at relatively +small Reynolds numbers, the radial transport will be dominated by convection due to the relatively low diffusivity +coefficients. For reference, Table I also tabulates the nondimensional parameters based on the viscosity and diffusivity +values associated with the saturation concentration csat rather than reference values based on pure glycerol. Here, the +Reynolds numbers are increased while both the convective and diffusive timescales are decreased due to the reduced +viscosity and increased diffusivity at increased water concentrations. + +14 +FIG. 8. Numerical results for the evolving water concentration profile c over time for different rotation speeds Ω at csat = 0.2, +and ϵ = (1.0 × 10−3 m)/(2.5 × 10−2 m) = 0.04. Here, the angular speeds are (a) Ω = 0.4 rad/s, (b) Ω = 1.0 rad/s, (c) Ω = 4.0 +rad/s, and (d) Ω = 10.0 rad/s. The corresponding nondimensional parameters are summarized in Table I. Here, the Reynolds +number (based on the saturation viscosity rather than µg) ranges from 9.71×10−3 up to 0.243 as the role of secondary (inertial) +flows clearly grows with Ω. +TABLE I. Summary of the simulation parameters used in Figures 8 and 10. The timescale values τconv and τdiff have units of +seconds. Here, the parameters with ‘sat’ subscripts are calculated based on the fluid properties at the appropriate saturation +mass fraction of water, and parameters without this subscript are calculated based on the fluid properties of pure glycerol. +Examining the transport dynamics in Figure 8, we see that the transport of water concentration is consistent with +the axisymmetric, constant viscosity flow picture described above. In particular, in the constant viscosity case, flow +proceeds radially outward along the upper plate, turns downward, and then flows radially inward along the lower +plate. This is shown in more detail in Figure 9a. Here, the arrows are color-coded and scaled by the magnitude of the +secondary velocity components |u∗ +sec|, and the background is color-coded by the water concentration profile. Here, +u∗ +sec is the velocity field on the slice in the r- and z-direction. As can be seen, the water begins to diffuse inwards +from the outer edge, but then the secondary velocity pulls the absorbed water down along the outer edge and then + +15 +FIG. 9. Detailed look at the glycerol/water dynamics for the case corresponding to Figure 8d taken at t = 1 × 104 (see Table +I for all relevant parameters. (a) Secondary velocity vectors colored and scaled by the magnitude of the secondary velocity +superimposed on a colormap of the water concentration profile. As can be seen, the water begins to diffuse inward from the +outer boundary, where the secondary flow pulls the absorbed water downward and then radially inward along the bottom plate, +leading to a steep concentration gradient at the outer edge as the rotation speed is increased. (b) Full water concentration +profile over a full axisymmetric cross-section. (c-f) Nondimensional diffusivity coefficient, viscosity, radial velocity component, +and z velocity component. +radially inwards along the bottom plate. This creates a very thin boundary layer region near the outer edge of the +rheometer, which gets thinner as Perad increases. The full solute concentration profile corresponding to Figure 9a is +shown in 9b, and the corresponding dimensionless diffusivity, viscosity, radial velocity, and z velocity are shown in +9c-f, respectively. This case corresponds to the parameters previously shown in Figure 8d with the parameters shown +in Table I, and all results are at the nondimensional time t = Ωt∗ = (10 rad/s)(1000 s) = 1 × 104. +As can be seen in Figure 9, the regions of high water concentration correspond to the regions of increased diffusivity +and decreased viscosity. The radial velocity component resembles the flow for the axisymmetric constant viscosity +case with outward radial flow along the upper half of the domain and inward radial flow along the lower half, except +that the magnitude of both is increased throughout the extent of the low viscosity region. Furthermore, at the front +of the propagating front of water concentration, there is a steep gradient in viscosity that corresponds to an upward +secondary velocity due to the viscosity gradients as seen in Figure 9d,f. +Finally, a last illustration of the water concentration dynamics at higher csat is presented in Figure 10 as a function +of gap height. These results show the evolution of the water concentration profile over time at csat = 0.5 and Ω = 4.0 +rad/s for gap heights ranging from 0.05 mm to 2.0 mm. Again, all of the relevant nondimensional parameters are +given in Table I based on both the pure glycerol reference values and the saturation values. Here, the Reynolds + +16 +FIG. 10. Numerical results for the evolving water concentration profile c over time for different gap heights h0 at csat = 0.5, and +Ω = 4.0 rad/s. Here, the gap heights are (a) 0.05 mm, (b) 0.1 mm, (c) 0.2 mm, (d) 0.5 mm, (e) 0.75 mm, (f) 1.0 mm, (g) 1.25 +mm, (h) 1.5 mm, and (i) 2.0 mm. The corresponding nondimensional numbers are summarized in Table I. Over this parameter +range, the Reynolds numbers (based on the saturation viscosity) range from 2.30 × 10−3 up to 3.68 and the gap aspect ratio +ranges from 0.002 to 0.08. Thus, these cases capture the full transition from the 1D, diffusive limit, up to the inertial regime. +number based on the saturation parameters ranges from 2.30 × 10−3 up to 3.68, representing a transition from the +inertialess regime into the moderate inertial regime. The radial Peclet numbers based on the saturation properties +remain relatively high, increasing from 1.46 × 104 up to 2.34 × 107 as the gap height increases, suggesting that the +radial transport of water is dominated by convection in these regimes. Nevertheless, the transport at the smallest gap +heights is approximately 1D, suggesting that the Perad threshold for this transition is at a relatively large magnitude +in this particular system. +Note that the qualitative picture of the water concentration evolution is different in Figures 8 and 10. In particular, +Figure 10f corresponds to the same gap height and rotation speed as Figure 8c, except with an increased csat value +of 0.5 versus 0.2, respectively. While this seems like a relatively minor change, the corresponding µsat value is an +order of magnitude smaller at csat = 0.5, which leads to an order of magnitude stronger secondary flows in the +region of locally low viscosity. This generates an enhanced mixing that leads to a more homogeneously propagating +front of water concentration. This can be visualized in Figure 11 for the case corresponding to Figure 10f at early +times. As can be seen, the local secondary recirculation in the low-viscosity region completely dominates the expected +global secondary recirculation for the axisymmetric, constant viscosity case. In fact, that global velocity field (the +axisymmetric constant viscosity solution) is negligible on the figure. This enhanced local recirculation at larger csat +values explains why the water propagates as a more uniform front in that regime, as opposed to being pulled down +and inward along the lower plate as was illustrated in Figure 9a for a smaller csat value. +Finally, having characterized and visualized the coupled transport dynamics in the parallel-plate, axisymmetric, +inertial regime, we now calculate the measured effective viscosities in these simulations to see if the proposed model +can fully capture the trends seen in the experimental results. The final measured dimensionless viscosities µf at +t∗ = 3600 s are presented in Figure 12 as functions of gap aspect ratio ϵ, saturation concentration csat, and angular +rotation speed Ω. The corresponding angular rotation speeds in the figure are (a) 0.4 rad/s, (b) 1.0 rad/s, (c) 2.0 +rad/s, and (d) 4.0 rad/s. In the figure, the dashed lines correspond to the predictions of the 1D, axisymmetric, +diffusion-dominated results previously described in Figures 5 and 6. +Note that in the experiments, the reported +viscosities were nondimensionalized by µ∗ +i , the initial measured viscosity at t = 0, and in the simulations the reported +viscosities have been nondimensionalized by µg. Here, several key relationships and trends emerge from the results. +First, we see clearly that in every case, the results approach the 1D diffusion-dominated limit as ϵ → 0 for constant +Ω, and they appear to also approach this limit as Ω → 0 for constant ϵ. For a given Ω, deviations from this limit +increase as ϵ increases, due to enhanced inertial effects, as well as for increased csat. This latter trend is also due to + +17 +FIG. 11. +Enhanced secondary recirculation in the low-viscosity region corresponding to larger csat values. +These results +illustrate the enhanced mixing effect that is seen at early times with csat = 0.5 for the parameters shown in Figure 10f. At +large values of csat, the local viscosity drops in regions of large c to such a degree that the local recirculation dominates the +expected secondary motions for constant viscosity, axisymmetric flow. +an increase in inertial effects, although indirectly through a decreasing in the local viscosity. Furthermore, increasing +Ω clearly leads to more significant deviations from the 1D limit due to increasing secondary inertial flows. +Comparing these results with the experimental results, the axisymmetric inertial simulations do seem to capture +many features of the experimental results. In particular, we generally see decreased µf values at larger gap heights +and larger csat values (i.e., RH values), which are consistent with Figure 2. Specifically, the numerical results in +Figure 12a correspond to the same rotation speed (Ω = 0.4 rad/s) as Figure 2b. Similar trends are seen (except at +small gap heights, which will be discussed below), with slightly less significant decreases in viscosity in the simulations +compared to the experiments. Increasing the rotation speed to 1.0 rad/s in the simulations shows more significant +viscosity decreases than the experimental results at 0.4 rad/s. So the experimental results at 0.4 rad/s agree well +quantitatively with numerical predictions slighty above 0.4 rad/s. One trend seen in the experiments that we do +not see in the simulations is the non-monotonic relationship between viscosity decrease and angular rotation rate +seen in Figure 2d. However, we do see evidence of a non-monotonic relationship with increasing inertial effects in +the simulations. In particular, at Ω = 4.0 rad/s with csat = 0.1 (orange curve in Figure 12d), the final measured +viscosity first decreases and then increases with increasing gap height, demonstrating that the axisymmetric case can +demonstrate such trends. +The most significant experimental result that these simulations cannot explain is the large decrease in measured +viscosity values at small gap heights. In fact, one of the most consistent results of the axisymmetric, inertial simulations +is the approach to the 1D, diffusion-dominated regime at small gap heights for any angular rotation speed. Thus, these +axisymmetric simulations apparently fail to account for some effect that becomes dominant at small gap heights. We +hypothesize that this is due to misalignment effects that only become significant at very small gap heights in practical +parallel-plate rheometers. In the next section, we perform additional simulations based on a misaligned geometry in +an attempt to validate this hypothesis. +VII. +ROLE OF MISALIGNMENT +In the previous section, we clearly saw that the axisymmetric, inertial, variable viscosity model fails to account for +the sharp decrease in measured viscosity at small gap heights. Thus, we must consider what possible sources of error +could account for these effects. A variety of experimental challenges exist for performing accurate measurements with +a rheometer, such as underfilling of the parallel-plate gap, instrument inertia, and surface tension effects [19, 24]. In +addition to these, there are practical sources of error associated with the mechanical uncertainties in the rheometer +itself. A key source of these errors comes from deviations in the geometry of the gap containing the fluid. These +errors in the gap geometry could arise from non-parallelism, non-concentricity, non-flatness of the plates, non-zero + +18 +FIG. 12. Compilation of all measured final dimensionless viscosities µf at t∗ = 3600 s from the axisymmetric, inertial, variable +viscosity simulations for comparison with the experimental results. Dashed lines indicate the 1D, inertialess, diffusion-dominated +results described in the previous section. Results are plotted separately by rotation speed with values of (a) 0.4 rad/s, (b) 1.0 +rad/s, (c) 2.0 rad/s, and (d) 4.0 rad/s. Clearly, deviations from the diffusion-dominated limit increase with gap height and +angular rotation speed due to the increase of inertial secondary flows, as well as with increasing csat due to local reductions in +viscosity (and consequent increases in inertial effects). +slip lengths at the upper or lower plates, edge effects at the outer edge of the rheometer, or errors in the gap- +zeroing procedure [1, 2, 10, 13, 25, 28, 30]. One reason the discussion of these sources of error arose was due to the +experimental observation that as gaps decreased below several hundred microns, measured viscosities began to have +systematic errors, typically decreasing with the gap height as also shown in Figure 2b [33, 42]. +Based on these observations, a variety of studies suggest that a key factor in this discrepancy in our measurements +and simulations could be the misalignment of the rotating plate. Although it is commonly assumed that the plates +are perfectly aligned, a number of reports indicate that a small but finite misalignment is prevalent in parallel-plate +and cone-and-plate rheometers [2]. In fact, the gap height can vary over 50 µm across a few centimeters in a parallel- +plate rheometer due to the non-parallelism in the gap, causing a significant error in the viscosity measurements in +narrow-gap, high-shear-rate experiments [14, 34]. This is due to the fact that the misalignment introduces additional +lubrication forces in the fluid layer. A variety of semi-empirical techniques have been developed to account for these +systematic errors at small gap heights. For example, a simple linear approximation has been proposed in which a +simple gap error is defined to correct the measured values [10, 14, 18]. Another technique involves using ultrasound +time-of-flight measurements to detect the varying thickness of the fluid layer in the case of misalignment, which can + +19 +FIG. 13. +Misaligned parallel-plate rheometer geometry and coordinate system. +The upper plate is misaligned by a small +deflection angle φ and rotates at angular speed Ω. With z nondimensionalized by h0 and r nondimensionalized by R, the +z-coordinate defining the upper plate is h(r, θ, φ) = 1 + φϵ−1r cos θ. Note that for small angles, the angle φ can range from 0 +to a maximum of ϵ. +be used to calculate the degree of misalignment [35]. A numerical solution of the flow in a misaligned parallel-plate +rheometer was also presented by Andablo-Reyes et al. [1]. Also, Clasen [8] introduced a system that can self correct +non-parallelism to a degree using hydrodynamic lubrication forces. Finally, a theoretical description of the velocity +and stress profiles in a slightly misaligned cone-and-plate rheometer was achieved by Dudgeon and Wedgewood [18] +using a domain perturbation study in the limit of zero Reynolds number. +In this section, we consider the role of misalignment on the transport of absorbed water throughout the glycerol, +and the effects of this misalignment on the measured viscosity values. In general, such an analysis would need fully +three-dimensional simulations in a misaligned rheometer geometry. We considered performing such simulations, but +found them to be intractable due to the extremely high computational cost of performing them. In particular, they +require 2-3 orders of magnitude more grid cells than the axisymmetric simulations in order to resolve the water +concentration boundary layer at the outer edge of the rheometer. Furthermore, all of the meshing techniques we tried +that would maintain this resolution at the outer edge ultimately resulted in very high aspect ratio cells at some point +in the domain that affect resolution and greatly increase the number of iterations needed to solve the velocity/pressure +profile with the SIMPLE algorithm, which must be repeated continuously as the water concentration field evolves. +For these reasons, we consider a simplified, depth-averaged case that is valid in the limit of small gap heights. This +model and the corresponding simulations and results will be described in the following sections. +A. +Theory +Here, we consider a misaligned parallel-plate rheometer with radius R and gap thickness h(r, θ, φ), where the upper +plate is slightly tilted by the small angle φ. Once again, the lower plate is stationary, and the upper plate rotates at +a rotation speed of Ω. The coordinate system and problem setup are shown in Figure 13. The boundary conditions +for the system are u = 0 at z = 0 (on the lower plate), and +ur = r cos θ sin θ sin φ tan φ, +(22a) +uθ = r +� +cos2 θ sec φ + cos φ sin2 θ +� +, +(22b) +uz = −rϵ−1 sin θ sin φ, +(22c) +at z = h(r, θ, φ) = 1 + φϵ−1r cos θ (the upper plate). Note that these simplify to ur = uz = 0 and uθ = r at z = 1 as +in the axisymmetric case when φ = 0. For small angles, the angle φ can range from 0 to a maximum of ϵ. Thus, φ/ϵ + +20 +ranges from 0 to 1, and small values of φ/ϵ correspond to small plate deflections. Here, φ/ϵ = 0 corresponds to the +case of no misalignment, and φ/ϵ = 1 corresponds to the case where the plates come in contact at one edge. Further, +recall that as before we generally also need to apply boundary conditions at r = 1. In a practical experiment, this +boundary condition represents a fluid-air interface that is typically not flat and experiences surface tension effects. +However, in the small gap limit, we lose the ability to impose such a boundary condition, and we note that this +contributes to the error in velocity/pressure profiles in the O(ϵ) region near r = 1. +The governing equations are again the Navier-Stokes equations with variable viscosity and the continuity equation, +which are given by Eqs. (4) and (5), respectively, as well as the water concentration advection–diffusion equation given +by Eq. (7). As mentioned above, the numerical simulation of the full system of coupled equations in a well-resolved +3D geometry is computationally expensive. In the limit of narrow gap heights ϵ ≪ 1 and negligible inertia Re ≪ 1, +the Navier-Stokes equations simplify to +0 = −∂p +∂r + µ∂2ur +∂z2 + ∂µ +∂z +∂ur +∂z , +(23a) +0 = −1 +r +∂p +∂θ + µ∂2uθ +∂z2 + ∂µ +∂z +∂uθ +∂z , +(23b) +0 = ∂p +∂z . +(23c) +Furthermore, in the thin gap limit, the water concentration can be assumed to be approximately uniform in the depth +direction, which gives ∂c +∂z ≈ 0 and ∂µ +∂z ≈ 0. This gives +0 = −∂p +∂r + µ∂2ur +∂z2 , +0 = −1 +r +∂p +∂θ + µ∂2uθ +∂z2 , +0 = ∂p +∂z , +and +1 +r +∂ +∂r (rur) + 1 +r +∂uθ +∂θ + ∂uz +∂z = 0. +(24) +Note here that c(r, θ, t), µ(r, θ, t) and p(r, θ, t) in this limit. With the fact that ∂p/∂z = 0 in this limit, the next +leading-order form of the z-component of the Navier-Stokes equations becomes +0 = µ∂2uz +∂z2 + ∂µ +∂r +∂ur +∂z + 1 +r +∂µ +∂θ +∂uθ +∂z . +(25) +By examining the form of the gap height distribution h(r, θ, φ) = 1 + φϵ−1r cos θ we see that the magnitude of the +perturbation is O(φ/ϵ). This suggests the use of a solution given by +ur(r, θ, z, t) = +�φ +ϵ +� +ur,1(r, θ, z, t) + +�φ +ϵ +�2 +ur,2(r, θ, z, t) + . . . , +(26a) +uθ(r, θ, z, t) = rz + +�φ +ϵ +� +uθ,1(r, θ, z, t) + +�φ +ϵ +�2 +uθ,2(r, θ, z, t) + . . . , +(26b) +uz(r, θ, z, t) = +�φ +ϵ +� +uz,1(r, θ, z, t) + +�φ +ϵ +�2 +uz,2(r, θ, z, t) + . . . , +(26c) +p(r, θ, t) = +�φ +ϵ +� +p1(r, θ, t) + +�φ +ϵ +�2 +p2(r, θ, t) + . . . , +(26d) +which is valid in the limit φ/ϵ ≪ 1. Substituting this expansion into Eqs. (24) and (25) and applying the boundary +conditions gives +ur,1(r, θ, z, t) =z(z − 1) +2µ +∂p1 +∂r , +(27a) +uθ,1(r, θ, z, t) = − r2z cos θ + z(z − 1) +2rµ +∂p1 +∂θ , +(27b) +uz,1(r, θ, z, t) = − +z2 +12r2µ +� +6r3µ2 sin θ − (2z − 3) +�∂µ +∂θ +∂p1 +∂θ + r2 ∂µ +∂r +∂p1 +∂r +� ++(2z − 3)µ +�∂2p1 +∂θ2 + r +�∂p1 +∂r + r∂2p1 +∂r2 +��� +. +(27c) + +21 +This procedure also yields a PDE governing the pressure distribution that is given by +µ +�∂2p1 +∂θ2 + r +�∂p1 +∂r + r +� +6rµ sin θ + ∂2p1 +∂r2 +��� += ∂µ +∂θ +∂p1 +∂θ + r2 ∂µ +∂r +∂p1 +∂r . +(28) +With some known distribution of viscosity in the system, a numerical solution of Eq. +(28) yields the pressure +distribution in the gap. This in turn can be used to calculate the velocity profiles from Eqs. (26) and (27). The +velocity profiles can then be used to update the water concentration distribution via the advection–diffusion equation. +With the assumption that c is independent of z (valid in the small gap limit), the solute transport equation becomes +Pe∂c +∂t = 1 +r +∂ +∂r +� +rD ∂c +∂r +� ++ 1 +r2 +∂ +∂θ +� +D ∂c +∂θ +� +− Pe +� +ur +∂c +∂r + uθ +r +∂c +∂θ +� +. +(29) +Since c is independent of z, we consider solving the depth-averaged version of this equation instead, which is simply +Pe∂c +∂t = 1 +r +∂ +∂r +� +rD ∂c +∂r +� ++ 1 +r2 +∂ +∂θ +� +D ∂c +∂θ +� +− Pe +� +¯ur +∂c +∂r + ¯uθ +r +∂c +∂θ +� +, +(30) +where bars denote depth-averaged quantities, ¯c = c, ¯D = D, and +¯ur = +1 +1 + φ +ϵ r cos θ +1+ φ +ϵ r cos θ +� +z=0 +ur(r, θ, z)dz = − 1 +12µ +φ +ϵ +∂p1 +∂r + O +�φ +ϵ +�2 +, +(31a) +¯uθ = +1 +1 + φ +ϵ r cos θ +1+ φ +ϵ r cos θ +� +z=0 +uθ(r, θ, z)dz = r +2 − +1 +12rµ +φ +ϵ +∂p1 +∂θ + O +�φ +ϵ +�2 +. +(31b) +Thus, having solved the pressure distribution due to the misalignment p1 from Eq. (28), the depth-averaged velocity +components can be calculated from Eq. (31), which can in turn be used to advect the solute concentration. +B. +Numerical methods +We perform numerical simulations of the coupled transport equations described in the previous section. Recall +that we seek solutions of the water transport and associated viscosity measurements in a misaligned parallel-plate +rheometer that is valid at small gap heights and in the limit of negligible inertia. The numerical approach for solving +these systems is as follows: +1. Solve Eq. (28) for the pressure perturbation due to misalignment subject to the boundary condition p → 0 at +r = 1. +2. Calculate the depth-averaged radial and azimuthal velocity components from Eq. (31). +3. Advance the water concentration profile in time by numerically integrating Eq. (30) for one or more timesteps. +4. Calculate the new viscosity and diffusivity fields and iterate back to Step 1. +We attempt a numerical implementation of this process using a finite-difference implementation in MATLAB. However, +several complications emerge due to the extremely large Peclet numbers in the system. In particular, the diffusion +of the water concentration field is so slow that the field effectively propagates with a very sharp front. In order to +resolve this and avoid spurious oscillations in the concentration field, we use slope-limited finite differencing based +on the minmod limiter function to switch to first-order spatial differencing at the steep gradient [36]. This avoids +the oscillations that result in a pure second-order differencing scheme and still allows nearly second-order accuracy +in space globally. The more serious difficulty that we encountered with this numerical approach is solving Eq. (28) +for the perturbation pressure field. These solutions do not behave nicely due to the sharp viscosity gradients on the +right-hand side of the equation. We were not able to resolve this issue using slope-limiters. +As an alternative approach and to illustrate the qualitative dynamics that can be expected with a misaligned upper +plate, we instead assume a constant viscosity model for the purposes of calculating the velocity profile, since this + +22 +FIG. 14. Depth-averaged water concentration profile at t∗ = 3600 s as a function of misalignment for csat = 0.5 and Ω = 0.4 +rad/s. Results correspond to φ/ϵ values of (a) 0.0 (perfectly aligned), (b) 0.1, (c) 0.2, (d) 0.5, (e) 0.75, and (f) 0.95 (plates are +nearly contacting). As can be seen, as the misalignment increases, the concentration profile becomes no longer axisymmetric, +and there is a significant increase in total water transport into the fluid layer from the outer edge due to the misalignment-driven +secondary flows. +solution is well-behaved. Note that Eq. (28) has an analytical solution when µ = 1 which is given by +p1(r, θ) = −3 +4r +� +−1 + r2� +sin θ. +(32) +We use this theoretical result at constant viscosity to calculate the depth-averaged velocities in the misaligned rheome- +ter and use these to update the concentration profile. When calculating the effective measured viscosity and torque, +we always use the viscosity distribution that corresponds to the concentration profile. This limitation to a velocity +profile based on constant viscosity is a clear limitation of our results, but nevertheless they capture qualitative features +of the experiments that the axisymmetric model could not predict, and we leave a full solution with evolving velocity +profiles based on spatial variation of viscosity to future work. +C. +Results +Here, we introduce the numerical results achieved for the misaligned rheometer system based on the methodology +described in the previous section. First, we highlight the role of the misalignment on the water concentration field +in Figure 14. +Here, the depth-averaged water concentration profiles are shown at t∗ = 3600 s as a function of +misalignment for csat = 0.5 with misalignment ranging from φ/ϵ = 0 to 0.95, where 0 is the perfectly aligned parallel- +plate case, and 1 is the limit where the plates come into contact at one edge. As can be seen, the misaligned cases all +show a non-axisymmetric concentration profile. This is due to the non-axisymmetric secondary velocity components +due to the misalignment. In particular, the radial component ¯ur transports water towards or away from the outer +edge, and is ¯ur ∼ φ +ϵ +∂p1 +∂r . With p1 ∼ sin θ, this represents a radially inward flow on one half of the rheometer and a +radially outward flow on the other half and explains why in Figure 14 the concentration profile appears to be pulled in +from the right edge and pushed towards the left edge. Furthermore, this secondary velocity component is proportional +to φ/ϵ, so doubling the degree of misalignment doubles the radial advective fluxes in both directions. +In order to quantify the effect of misalignment on the measured viscosity values, simulations were performed across +a range of φ/ϵ and csat values. The final measured viscosity values from these simulations are presented in Figure 15. + +23 +FIG. 15. Final measured viscosity values µf/µi at t∗ = 3600 s as functions of the misalignment φ/ϵ and csat at an angular +rotation speed of 0.4 rad/s. +Dashed lines correspond to the 1D pure-diffusion limit. Results asymptotically approach the +1D diffusion-dominated limit as φ/ϵ → 0. +Furthermore, results show a steep drop-off in measured viscosity values as the +misalignment increases. +FIG. 16. Comparison of the final measured viscosity values at t∗ = 3600 s for the models corresponding to each of the three +regimes: (1) the 1D axisymmetric, pure-diffusion limit (dashed lines), (2) the axisymmetric, inertial regime (dot-dashed lines), +and (3) the misaligned, inertialess, small gap limit (solid lines). Results correspond to angular rotation speeds of (a) 0.4 rad/s +and (b) 1.0 rad/s. Misaligned cases are calculated with a misalignment angle of 0.0005 rad. All cases were performed with a +rheometer of radius R = 2.5 cm. +Here, the results correspond to an angular speed of 0.4 rad/s. The dashed lines in the figure correspond to the +1D diffusion-dominated regime, and the results asymptotically approach these limits as φ/ϵ → 0. Furthermore, the +results show a steep dropoff in measured final viscosities at large misalignments, which possibly explains the sharp +decrease in measured viscosity in the experiments at small gap heights (e.g., Figure 2b) that was not captured in the +axisymmetric model (e.g., Figure 12). +Finally, a comparison between all of the three different proposed models (i.e., the 1D pure diffusion limit, the +axisymmetric inertial limit, and the misaligned inertialess small gap limit) is shown in Figure 16. Here, the dashed +lines indicate the 1D diffusion-dominated limit, dot-dashed lines correspond to the inertial, axisymmetric regime, and +the solid lines show the results for the misaligned, inertialess, small-gap limit. Results were calculated for angular +rotation speeds of (a) 0.4 rad/s and (b) 1.0 rad/s. The results show that at small gap heights, the misalignment + +24 +effects dominate the inertial effects. This becomes more clear when considering that the radial secondary velocity due +to inertial effects in an axisymmetric case is O(Re), whereas the secondary velocity components due to misalignment +are O(φ/ϵ). For fixed angular rotation speed, the secondary velocities must dominate the inertial secondary velocities +as the gap height decreases. Furthermore, the results also show that the opposite is true at large gap heights. For +a fixed misalignment angle and rotation speed, φ/ϵ decreases as the gap height increases, whereas the inertial effects +increase, such that the large gap regime is dominated by inertial effects. This cross-over explains the non-monotonic +relationship between µf and gap height reported in Figure 2b. Thus, there is a critical ϵ value at which the measured +viscosity values switch from being misalignment-dominated to inertia-dominated. Numerical simulations using the +previously described models and numerical methods can be used to estimate this transition, as shown in Figure 16. +VIII. +CONCLUSIONS +In this paper, we have considered the measurement of the viscosity of glycerol in a parallel-plate rheometer. Intu- +itively, it can be anticipated that the viscosity must decrease over time due to the hygroscopic nature of the fluid as it +absorbs water vapor from the atmosphere. Based on an initial understanding of the fluid dynamics in a parallel-plate +rheometer for a constant viscosity flow, an axisymmetric model of the flow predicts that the dynamics should be +purely limited by diffusion in the thin-gap limit and become independent of gap height. However, a sharp drop-off +in measured viscosity values was observed experimentally at small gap heights, which motivated us to reconsider the +fluid dynamics in the system and led to the hypothesis that plate misalignment could drive additional secondary flows +that might affect the transport of the water concentration throughout the system. Ultimately, theoretical models +and numerical simulations of the coupled dynamics and measured viscosity values were achieved in three different +regimes: (1) the 1D inertialess, diffusion-dominated regime, (2) the axisymmetric inertial regime, and (3) the mis- +aligned, inertialess, thin-gap regime. Results confirmed that there are two types of secondary flows that can exist +in such systems. The first of these is O(Re) and corresponds to the secondary inertially driven flows in a perfectly +aligned axisymmetric parallel-plate rheometer. The other secondary flow is O(φ/ϵ) and is driven by the O(φ/ϵ) plate +misalignment. Assuming a fixed misalignment φ, then as the gap height decreases, the O(φ/ϵ) misalignment flow will +inevitably dominate the O(Re) inertially driven flow. +Based on the results here via comparison between experiments and numerical simulations, as well as between +simulations with parallel and misaligned plates, we argue that the sharp decrease in measured viscosity is attributable +to the secondary flows induced by plate misalignment. The mechanism by which the misalignment results in a faster +decrease in viscosity seems to be that the secondary velocities pull the relatively high water concentration away from +the outer boundary, which steepens the concentration gradient at the outer boundary, resulting in an increased mass +flux of water into the glycerol, which subsequently lowers the viscosity of the glycerol. These results have relevance +not only to the measurement of the viscosity of glycerol solutions (for which care must be taken to ensure all the +possible transport mechanisms are understood), but also for our understanding of the flow in parallel-plate rheometers +more generally. We have shown that misalignment effects in particular can have a surprising critical influence over +the mass transport in such systems, especially as the gap height becomes small. Furthermore, based on these results, +it is plausible that such viscosity measurements of glycerol in a parallel-plate rheometer could potentially be used as +a technique to quantify the degree of misalignment in the rheometer, although we leave a practical investigation and +demonstration of this technique for future work. Finally, we note that additional complications can arise in such a +system, such as the potential for the emergence of instabilities due to viscosity gradients. We observed evidence for +such effects in our numerical results at high angular rotation speeds in some cases. For example, Figure 17 shows a +time series of the concentration profile for a case with Ω = 10 rad/s, csat = 0.2, and h0 = 2 mm. It is known that +viscosity stratification in shear flows can lead to instability in certain regimes [37, 45]. However, the coupled viscosity +distribution and velocity profile in the rheometer system are highly nonlinear, and the flow cannot be analyzed in +terms of simple viscosity-stratified layers of fluid. Furthermore, at the high rotation speeds and gap heights where +we observed this instability, it is likely that other assumptions in our proposed models will break down, especially +the assumed flat interface at the outer boundary and the predefined slip boundary conditions there. Thus, we leave +a detailed study of these intriguing instabilities for future work. +Declaration of interests: The authors report no conflict of interest. +Author contributions: J.T.A. developed the theory and performed the numerical simulations. A.P., A.G., and +S.S. performed the experiments. J.T.A., S.S., and H.A.S. conceived the experiment and wrote the manuscript. The + +25 +FIG. 17. Apparent instability/oscillation in the concentration profile field due to viscosity gradients from numerical results +with Ω = 10 rad/s, csat = 0.2, and h0 = 2 mm. +authors would like to acknowledge Ian Jacobi for helpful discussions regarding the experimental setups. +[1] E. Andablo-Reyes, R. Hidalgo-´Alvarez, and J. de Vicente. A method for the estimation of the film thickness and plate tilt +angle in thin film misaligned plate–plate rheometry. J. Non-Newton. Fluid Mechanics, 165(19-20):1419–1421, 2010. +[2] E. Andablo-Reyes, J. de Vicente, and R. Hidalgo-Alvarez. 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Fluid Mech., 27(2):337–352, 1967. + diff --git a/c9E_T4oBgHgl3EQf0hwq/content/tmp_files/load_file.txt b/c9E_T4oBgHgl3EQf0hwq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df5baba46423facd3c7616b0865e6dd3544a26ed --- /dev/null +++ b/c9E_T4oBgHgl3EQf0hwq/content/tmp_files/load_file.txt @@ -0,0 +1,1213 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf,len=1212 +page_content='Viscosity measurements of glycerol in a parallel-plate rheometer exposed to atmosphere Jesse T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Ault∗ Center for Fluid Mechanics, Brown University, Providence, Rhode Island 02912, USA Sangwoo Shin Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA Allan Garcia, Antonio Perazzo, and Howard A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Stone Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Glycerol is a hygroscopic fluid that spontaneously absorbs water vapor from the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For applications involving glycerol, care must be taken to avoid exposure to humidity, since its viscosity decreases quickly as water is absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We report experimental measurements of the viscosity of glycerol in a parallel-plate rheometer where the outer interface is exposed to atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The mea- surements decrease with time as water is absorbed from the atmosphere and transported throughout the glycerol via diffusion and advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Measured viscosities drop faster at higher relative humidi- ties, confirming the role of hygroscopicity on the transient viscosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The rate of viscosity decrease shows a non-monotonic relationship with the rheometer gap height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This behavior is explained by considering the transition from diffusion-dominated transport in the narrow gap regime to the large gap regime where transport is dominated by inertia-driven secondary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Numerical simulations of the water absorption and transport confirm this non-monotonic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The experimental viscosity measurements show unexpectedly fast decreases at very small gap heights, violating the parallel-plate, axisymmetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We propose that this drop-off may be due to misalignment in the rheometer that becomes non-negligible for small gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Theoretical considerations show that sec- ondary flows in a misaligned rheometer dominate the typical secondary inertial flows in parallel-plate rheometers at small gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, simulations in a misaligned parallel-plate system demonstrate the same sharp drop-off in viscosity measurements at small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This modeling can be used to estimate the gap height where misalignment effects dominate the transient glycerol viscosity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' INTRODUCTION A rotational rheometer is a device in which one component rotates relative to another in order to induce a shear on the fluid placed in between the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Through a characterization of the torques and forces that result as a function of rotation rate, rheological properties of the fluids can be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the last several decades, rheometry has emerged as an essential tool for studying the fluid dynamics of complex fluids for measuring properties ranging from the dynamic viscosity of Newtonian fluids to the viscoelastic responses of non-Newtonian fluids [11, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Owing to their precision and versatility, rheometers have been utilized to investigate the rheology and mechanics of a wide range of viscous and viscoelastic fluids such as polymer melts, gels, suspensions, cells, bacterial biofilms, lipid vesicle solutions, food products, cosmetics, pharmaceuticals, and many others [9, 12, 17, 20, 29, 39, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Rheometry has also been used for evaluating drag reduction, since these techniques can also be used to characterize slip lengths such as those generated by nanostructured surfaces [5, 7, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Among various configurations, parallel-plate rheometry is a system that is commonly used for highly viscous and viscoelastic fluids due to the ability to carefully control the gap height and the spatially uniform confinement in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' From a fluid dynamics perspective, a parallel-plate rheometer generates a shear-driven fluid velocity that is primarily in the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the limit of slow rotation speed or narrow gap heights, the velocity profile in such a system is simply (ur, uθ, uz) = (0, Ωrz/h0, 0), where Ω is the angular velocity of the upper plate, h0 is the gap height between the plates, and (r, θ, z) are the cylindrical coordinates with the origin located at the center of the bottom plate (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', Bird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [4], Middleman [32] and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In addition to the primary azimuthal flow, secondary recirculating flows exist due to inertial effects for any finite angular rotation speed [19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The first experimental evidence of these secondary flows was achieved by Garner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' ∗ jesse ault@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='08329v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='flu-dyn] 19 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We measure the viscosity of glycerol in a parallel-plate rheometer exposed to atmosphere at the outer fluid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Due to the strong hygroscopic nature of glycerol, the water vapor present in the atmosphere is absorbed by the glycerol at the outer boundary of the rheometer at a mass vapor flux of jw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The absorption of water by glycerol leads to a change in the fluid properties over time including a reduction in the fluid viscosity, which leads to a transient reduction in the effective viscosity measured by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [21] in a study of the rheological properties of a hydrocarbon-type micellar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The secondary flow profile in a parallel-plate rheometer was found by Savins and Metzner more than half a century ago and has been well-described by various authors (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', Denn [15], Savins and Metzner [38]), where the radial velocity (except near the turning regions) is given as ur(r, z) = −ρΩ2h2 0r 12µ � 4 5 � z h0 � − 9 5 � z h0 �2 + � z h0 �4� , (1) where ρ is the fluid density and µ0 is the fluid viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Notably, the radial flow becomes increasingly important as the gap height or rotation speed increase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', ur/uθ ∼ Ωh2 0/ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The secondary fluid dynamics can play a surprisingly significant role in a rheometer both by altering the torque measurements and by triggering instabilities or driving fluid mixing, especially in the case of non-homogeneous fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For example, Jacobi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [26] studied the effect of radial flow on the viscosity measurements in parallel-plate and cone-and-plate settings when the target fluid is stratified with an immiscible fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In this case, the authors found that the radial flow can distort the fluid interface, leading to drastically different torque measurements and even fluid dewetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Another situation in which the secondary recirculation plays a key role is in the case of an initially homogeneous fluid that becomes non-homogeneous during the measurement procedure due to mass transfer that occurs at the exposed outer edge, such as by solute sorption or solvent evaporation/condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This is particularly important for parallel-plate rheometry since inhomogeneities alter the stress profiles, such that small changes in the fluid composition at the outer edge can have significant impacts on the torque measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This secondary flow is expected to play a significant role in the viscosity measurements of glycerol over time, since any water absorbed at the outer edge of the rheometer can be transported radially inwards by the secondary flow, redistributing the relatively lower viscosity glycerol (where the water fraction is higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Since the torque in a parallel-plate rheometer is primarily generated near the outer edge of the system, this redistribution of the lower-viscosity fluid must have direct consequences on the measured torque value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Previously, we reported that the strong hygroscopic nature of glycerol can cause unreliable viscosity measurements in a cone-and-plate rheometer due to the continuous vapor absorption from the outer edge [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Motivated by this observation, here we present a systematic study on the transient measurement of the viscosity of glycerol using a parallel-plate rheometer (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As the secondary recirculating flow effectively disperses the absorbed water throughout the glycerol layer, we find that the rate of decrease of the measured viscosity is a complex function of the rheometer gap height, angular velocity, and the relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' While the viscosity generally decreases over time as water is absorbed, we find that the rate of decrease is a non-monotonic function of the gap height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Using the theoretical solutions for the flow profile in a parallel-plate rheometer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (1)) along with numerical simulations, we show that this non-monotonic behavior is consistent with existing theory provided the gap height is not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the limit of very small gap heights, we find that the behavior of the transient viscosity measurements is inconsistent with the existing theory for a parallel-plate rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We hypothesize that misalignment effects in the rheometer result in non-negligible secondary flows at small gap heights that are responsible for this discrepancy in the measured viscosity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' By developing new theoretical solutions and computational simulations for the misaligned parallel- plate geometry, we show that this hypothesis is consistent with the measured viscosity data and that the misaligned rheometer model can predict the viscosity measurements across the full range of gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' EXPERIMENTAL VISCOSITY MEASUREMENTS Here, we consider the transient viscosity measurements of glycerol in a parallel-plate rheometer where the outer fluid interface is exposed to the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Due to the hygroscopic nature of glycerol, it will absorb water vapor from the atmosphere at the outer boundary as shown in Figure 1, which will subsequently lead to a local reduction in the viscosity of the fluid and a net reduction of the torque measured by the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the measured viscosity of glycerol in a parallel-plate rheometer is expected to decrease with time when exposed to atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Intuitively, the rate of decrease should depend on the water concentration/flux experienced at the outer boundary, which is influenced by the relative humidity in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The rate of decrease should also depend on the secondary recirculating flows in the rheometer, since these redistribute the relatively less viscous fluid where the water concentration is higher, thereby altering the stress distribution and total torque experienced by the upper plate of the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Experimental methods Glycerol was purchased from Sigma-Aldrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Viscosity measurements were performed using a stress-controlled rheometer (Physica MCR 301, Anton Paar) with a parallel-plate configuration (plate diameter = 50 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The rheometer was placed inside an acrylic chamber in which the relative humidity (RH) was controlled using multiple vapor sources and a stream of dehumidified air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' RH was constantly monitored using a digital hygrometer (VWR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Experiments were performed at 23◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Experimental results The experimental results for the transient viscosity measurements are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' First, the transient viscosity measurements of glycerol in a relative humidity environment of 53% are presented in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The experiments were performed with a rheometer with plate radius R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 cm at an angular velocity of Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results are normalized by the initial viscosity µi to account for any small amount of water absorption by the glycerol while setting up the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, the measured viscosities decrease at varying rates depending on the gap thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These trends are all monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, the rate of viscosity decrease is seen to be a non-monotonic function of the gap thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' That is, decreasing the gap height from h0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 mm to h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 mm results in the viscosity dropping at a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, subsequently decreasing the gap height results in the viscosity dropping at a faster and faster rate, until the viscosity drops sharply for the smallest gap height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, experimental results for the final measured viscosity µf/µi after t = 60 minutes are shown in Figure 2b as a function of gap height h0 for two different relative humidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, with RH = 72% the viscosities are seen to drop faster than with RH = 45% due to the increased mass flux of water vapor to the glycerol from the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In addition, the final measured viscosity values also show the same non-monotonic behavior as in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' That is, decreasing h0 first leads to an increase in µf/µi, followed by a sharp decrease for h0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Next, the transient decrease in measured viscosity values are shown in Figure 2c as functions of relative humidity at a rotation speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s and a gap height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here the results show a monotonic relationship in which increasing the relative humidity leads to a faster decrease in viscosity measurements due to the increased flux of water into the glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, the transient measured viscosities are shown in Figure 2d for varying rotation speeds at a gap height of 1 mm and a relative humidity of 54%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here again a non-monotonic relationship with Ω is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' At first, increasing the rotation speed from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s leads to a slower decrease in viscosity, while subsequently increasing the rotation speed to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s leads to a much faster decrease in the measured viscosity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The key observations from the experimental measurements are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Increased relative humidity leads to faster viscosity decrease for all measured cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The rate of viscosity decrease varies non-monotonically with the gap height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' That is to say, for different gap heights, the measured viscosity decreases at a different rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' There is a sharp decrease in the measured final viscosity for very small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Observation 1 above is a natural and intuitive result, since the hygroscopic nature of glycerol leads it to absorb more water from the atmosphere at higher humidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Since the viscosity of a glycerol–water mixture varies monotonically with the water mass fraction, the rate of viscosity decrease can be expected to vary monotonically with the relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, an intuitive explanation for observations 2 and 3 is not immediately obvious without a more careful consideration of the fluid dynamics and the coupled transport of dissolved water in the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Experimental viscosity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (a) Transient decrease in the measured viscosity as a function of time, normalized by the initial viscosity µi with Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s and R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Experiments were performed at a relative humidity of RH = 53%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (b) Measured final viscosities µf of glycerol at t = 3600 s normalized by µi with Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s and R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Experiments were performed at relative humidities of RH = 45% and RH = 72%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (c) Normalized transient viscosity measurements over time for varying relative humidities at a gap height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='1 mm and a rotation rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (d) Normalized transient viscosity measurements over time for varying rotation rates at a gap height of 1 mm and a relative humidity of 54%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' following sections, we will seek a physical explanation for these two experimentally observed behaviors using theory and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' FLUID DYNAMICS OF A GLYCEROL–WATER MIXTURE Before proceeding to analyze the specific fluid dynamics and transient viscosity evolution in a parallel-plate rheome- ter, in this section we first introduce all of the relevant governing physics that applies to such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This includes the governing Navier-Stokes equations for variable viscosity fluids, the advection-diffusion equation for the absorbed water concentration with variable diffusivity, the empirical relationships for the coefficients of viscosity and diffusivity of glycerol-water mixtures, as well as the saturation concentration of absorbed water in glycerol as a function of the relative humidity that will be used in the modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Navier-Stokes equations for a variable viscosity fluid In a glycerol-water mixture, the viscosity is strongly dependent on the local mass fraction of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In systems with a homogeneous concentration of absorbed water, the constant viscosity form of the Navier-Stokes equations is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, when gradients in water concentration exist in the system, due to circumstances such as mixing streams or the absorption of water at a gas-liquid interface, the viscosity of the fluid must be treated as a function of time and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note that for glycerol-water mixtures, the density is a weak function of the water concentration, ranging from approximately 1260 kg/m3 for pure glycerol down to 1000 kg/m3 for pure water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, we neglect this variation and use the density of pure glycerol, assuming the water mass fraction does not get too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding continuity equation for an incompressible flow is simply ∇∗ · u∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In such a system, extra stresses arise in the fluid that are related to the gradients of viscosity, and the appropriate form of the Navier-Stokes equations (for a Newtonian constitutive equation) is: ∇∗ · u∗ = 0 and ρDu∗ Dt∗ = −∇∗p∗ + µ∗∇∗2u∗ + ∇∗µ∗ · ∇∗u∗ + ∇∗µ∗ · (∇∗u∗)T , (2) where ρ is the fluid density, µ is the fluid viscosity, and T denotes the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, ∗’s denote dimensional variable quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For the case of the flow in a rheometer system, we will represent the flow using cylindrical coordinates, and we nondimensionalize the governing equations with r = r∗ R , z = z∗ h0 , ur = u∗ r ΩR, uθ = u∗ θ ΩR, uz = u∗ z Ωh0 , p = p∗ µ0ΩR2/h2 0 , µ = µ∗ µ0 , and t = Ωt∗, (3) where µ0 is a reference viscosity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With these nondimensionalizations, the component form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (2) in cylindrical coordinates is given by Re �∂ur ∂t + ur ∂ur ∂r + uθ r ∂ur ∂θ − uθ2 r + uz ∂ur ∂z � = −∂p ∂r + µ �ϵ2 r ∂ ∂r � r∂ur ∂r � + ϵ2 r2 ∂2ur ∂θ2 +∂2ur ∂z2 − ϵ2 ur r2 − 2ϵ2 r2 ∂uθ ∂θ � + 2ϵ2 ∂µ ∂r ∂ur ∂r + ϵ2 r ∂µ ∂θ �1 r ∂ur ∂θ + ∂uθ ∂r − uθ r � + ∂µ ∂z �∂ur ∂z + ϵ2 ∂uz ∂r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (4a) Re �∂uθ ∂t + ur ∂uθ ∂r + uθ r ∂uθ ∂θ + uθur r + uz ∂uθ ∂z � = −1 r ∂p ∂θ + µ �ϵ2 r ∂ ∂r � r∂uθ ∂r � + ϵ2 r2 ∂2uθ ∂θ2 +∂2uθ ∂z2 − ϵ2 uθ r2 + 2ϵ2 r2 ∂ur ∂θ � + ϵ2 ∂µ ∂r �∂uθ ∂r + 1 r ∂ur ∂θ − uθ r � + 2ϵ2 r ∂µ ∂θ �1 r ∂uθ ∂θ + ur r � + ∂µ ∂z �∂uθ ∂z + ϵ2 r ∂uz ∂θ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (4b) Re ϵ2 �∂uz ∂t + ur ∂uz ∂r + uθ r ∂uz ∂θ + uz ∂uz ∂z � = −∂p ∂z + µ �ϵ4 r ∂ ∂r � r∂uz ∂r � + ϵ4 r2 ∂2uz ∂θ2 + ϵ2 ∂2uz ∂z2 � + ∂µ ∂r � ϵ4 ∂uz ∂r + ϵ2 ∂ur ∂z � + 1 r ∂µ ∂θ �ϵ4 r ∂uz ∂θ + ϵ2 ∂uθ ∂z � + 2ϵ2 ∂µ ∂z ∂uz ∂z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (4c) where the Reynolds number is defined as Re = ρΩh2 0/µ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' and the gap aspect ratio is ϵ = h0/R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' which is typically small in a parallel-plate rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding continuity equation is given by 1 r ∂ ∂r (rur) + 1 r ∂uθ ∂θ + ∂uz ∂z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (5) Along with boundary conditions, these equations govern the motion of variable viscosity fluids in a parallel-plate rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The typical boundary conditions for such a system include no-slip conditions at the lower (stationary) and upper (rotating) plates, as well as a stress-free condition at r = 1 at the gas-liquid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Water absorption and transport Along with the equations governing the fluid dynamics in the previous section, the system further requires a transport equation to model the absorption and transport of water in the glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, this transport can be modeled with an advection-diffusion equation that is given by ∂c ∂t∗ = ∇∗ · (D∗∇∗c) − u∗ · ∇∗c, (6) where c is the mass fraction of water in the glycerol, D∗ is the diffusivity of water in glycerol, and u∗ is the dimensional fluid velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, D∗ is a spatially/temporally varying function of the local water concentration c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Using the same nondimensionalizations as above, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (6) in cylindrical coordinates becomes Pe ϵ2 ∂c ∂t = ϵ2 r ∂ ∂r � rD ∂c ∂r � + ϵ2 r2 ∂ ∂θ � D ∂c ∂θ � + ∂ ∂z � D ∂c ∂z � − Pe ϵ2 � ur ∂c ∂r + uθ r ∂c ∂θ + uz ∂c ∂z � , (7) where D = D∗/D0 is the nondimensional diffusivity with reference value D0 and Pe = ΩR2/D0 is the Peclet number representing the ratio of the timescale for diffusion of water in the radial direction to the convective timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Examining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (7), the diffusion in the z-direction is O(ϵ−2) larger than the radial diffusion due to the separation in length scales: in the low-inertia, thin-gap limit, the water concentration will be approximately uniform in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Along with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (7), the absorbed water concentration must satisfy certain boundary conditions in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, the concentration satisfies a no-flux condition at both the upper and lower plates of the rheometer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' ∂c ∂z = 0 in the parallel-plate case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In addition, a boundary condition for the water concentration is needed at the outer glycerol/air interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In general, this condition could be represented by a water flux condition such as by −D ∂c ∂r �� r=1 = jw, where the flux of water jw could be a function of the local water concentration in the glycerol at the interface as well as the water vapor concentration and distribution in the air near the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The solution of such a flux will typically require also solving the water vapor transport problem in the surrounding environment, since these transport processes are coupled at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For example, the transport of water vapor in the surrounding atmosphere may be affected by the rotation speed of the rheometer, which can drive flow in the surrounding air that may alter the vapor transport at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, we neglect these effects and assume that the water vapor transport in the atmosphere is fast relative to the water concentration transport in the glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This assumption is valid due to the substantially higher diffusivity of water vapor in air than absorbed water in glycerol, provided the recirculation in the rheometer is not significantly fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, we assume that the water mass fraction at r = 1 instantaneously reaches its saturation value based on the local relative humidity in the surrounding air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Considering both Equations (4) and (7), we see how the dynamics of the problem are fully coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the fluid problem, the Navier-Stokes equations are coupled to the water transport problem via the dependence of viscosity on water mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The water transport problem is further coupled to the Navier-Stokes equations through the dependence on the flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, the transport is further complicated by the dependence of diffusivity on water mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Due to this two-way coupling and the empirical nature of both the µ(c) and D(c) relationships, it is difficult to seek theoretical solutions to the coupled dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, here we primarily rely on numerical simulations to solve the coupled transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Physical properties of glycerol–water mixtures Detailed empirical formulas for the viscosity of a glycerol–water mixture have been proposed by Cheng [6] which are valid for water mass concentrations in the range of 0–100% and for temperatures ranging from 0 to 100◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The viscosity of a glycerol–water mixture at 22◦C varies from around µg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='1 Pa·s for pure glycerol down to µw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='96 mPa·s for pure water, spanning a range of approximately three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, we choose a reference viscosity value corresponding to that of pure glycerol µ0 = µg, such that the nondimensional viscosity varies from an initial condition of 1 down to as low as ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='73 × 10−4 if the water mass fraction were to approach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The empirical relationship determined between nondimensional viscosity and water concentration used here is shown in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The diffusivity of water in glycerol is also a function of the water mass fraction, and so D∗ is expected to evolve as a function of both position and time as more water is absorbed and transported throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' An empirical relationship for the diffusivity of water in glycerol has also been developed for mixtures at 25 ◦C by D’Errico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (a) Empirical nondimensional viscosity and diffusivity relationships used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (b) Saturation concentration of water in glycerol and corresponding specific gravity as functions of relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [16], and is given by D∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='024 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='91xg 1 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5xg 10−9 m2 s , (8) where xg is the mole fraction of glycerol that is related to c via xg = Mw(1 − c) Mw(1 − c) + Mgc, (9) where Mw is the molar mass of water and Mg is the molar mass of glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For reference, the diffusivity of water in pure glycerol (c = 0) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='341 × 10−11 m2/s, which increases up to approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='024 × 10−9 m2/s as the water mass fraction approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here we again choose a reference value equal to the diffusivity in pure glycerol D0 = Dg, such that D varies from 1 at c = 0 up to approximately 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This empirical relationship from D’Errico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [16] is also shown in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, the saturation concentration of water in glycerol as a function of relative humidity is needed for the absorption boundary condition at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These values were measured by the Glycerine Producers’ Association and are given in Table 15 of Glycerine Producers’ Association and others [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For reference, these values are plotted in Figure 3b along with the specific gravity as functions of relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With the full governing equations and empirical relationships for the physical parameters described above, we can now move on to consider the coupled fluid dynamics and water concentration transport in a rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the following section we first review the classical result for the axisymmetric parallel-plate rheometer with constant viscosity before moving on to cases with variable viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note on the assumption of constant density Before moving on, we briefly comment on the assumption of constant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As water is absorbed into the glycerol, the resulting density gradients introduce the possibility for buoyancy-driven flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' An estimate for the magnitude of such effects is given by considering that a vertical change in density ∆ρ implies a radial pressure gradient on the order of ∆ρgh0/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We can balance this radial pressure gradient with a radial viscous stress gradient µub/h2 0, where ub is the characteristic magnitude of the buoyancy-driven flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, we have ub ∼ ∆ρgh3 0 µR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For buoyancy effects to be negligible, we need ub to be small relative to the magnitude of the inertial secondary-velocity components which are O � ρΩ2h2 0R µ � , as shown in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, we need ∆ρgh3 0 µR ≪ ρΩ2h2 0R µ −→ ∆ρ ρ ≪ Ω2R2 gh0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (10) 8 The left-hand side of this inequality has a maximum value of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2 when the saturation water concentration approaches 100%, and so it will typically take a smaller value depending on the relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The right-hand side of the inequality depends on the system parameters, but a typical value with Ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 cm, and h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 mm is approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the buoyancy-driven flow can be expected to be less than 10% of the inertial secondary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' At larger gap heights and smaller rotation speeds, this inequality suggests that buoyancy effects become relatively more important, and could even dominate the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, in the very slow rotation case, the dynamics are nearly 1D, such that the density variation is almost entirely in the radial direction and the negligible vertical density gradient should not affect the radial pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' So, Equation (10) should be considered only as an estimate of the importance of buoyancy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' CLASSICAL RESULT: AXISYMMETRIC FLOW WITH CONSTANT VISCOSITY Before moving on to consider the full coupled dynamics of glycerol absorbing water in a parallel-plate rheometer, we first review the classical Newtonian, constant viscosity flow solution in a parallel-plate rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Understanding this flow is important because it illustrates the types of secondary flows we should expect in the rheometer, and also provides some initial insights into the non-monotonic relationship between measured viscosities and rotation speed and gap height in the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The axisymmetric parallel-plate rheometer has a well-known solution that has been previously described by multiple authors (see, for example Bird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [4], Middleman [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, we briefly reproduce this calculation in our notation for consistency with later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We consider a parallel-plate rheometer with radius R and gap thickness h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The lower plate is stationary and the upper plate rotates at a rotation speed of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We model the system using cylindrical coordinates with the origin located at the center of the bottom plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the governing equations are the axisymmetric and constant viscosity forms of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding boundary conditions are no-slip at both the upper and lower plates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (ur,axi, uθ,axi, uz,axi) = (0, 0, 0) at z = 0 and (ur,axi, uθ,axi, uz,axi) = (0, r, 0) at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For small gap heights ϵ ≪ 1 a solution for the velocity and pressure can be sought in the form of an expansion in powers of ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Up to O(ϵ4), this axisymmetric solution is given by ur,axi(r, z) = − 1 12rRe z(z − 1) � −4 5 + z + z2 � + O(ϵ4), (11a) uθ,axi(r, z) = rz − rRe2z 6300 (8 + z3(35 − 63z + 20z3)) + O(ϵ4), (11b) uz,axi(z) = 1 30Re z2(z − 1)2(2 + z) + O(ϵ4), (11c) paxi(r) = 3r2 20 Re + 1 30Re ϵ2z(4 − 9z + 5z3) + O(ϵ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (11d) Here, the subscript ‘axi’ denotes the axisymmetric case, and the expression for ur,axi is equivalent to the result first presented by Savins and Metzner [38], which was given above in dimensional form as equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The primary flow is the uθ,axi = rz component with an O(Re2) correction, while the leading-order secondary flows in the r- and z-directions are both O(Re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Since the flow of interest is axisymmetric, the primary velocity components of interest for redistributing absorbed species at the outer edge of the rheometer are the secondary velocity components, especially the radial component, since this will transport absorbed water from the outer edge inwards through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A visualization of the secondary velocity components is given in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the uz,axi component shows that there is an upward drift that is independent of r throughout the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The radial component shows that in the upper-half of the gap the flow is directed radially outwards, and in the lower-half of the gap the flow is directed radially inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Keep in mind that the theoretical solution presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (11) must break down near the outer edge of the rheometer where r → 1, since the lubrication approximation fails in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the true system, near the outer edge the outward-traveling flow in the upper-half of the rheometer must turn downwards for continuity and turn around to then travel radially inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The width of this turning region should be O(ϵ), and thus is progressively more confined at the outer edge as the gap height decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This effect is not captured by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (11), although it may have an important role in the transport of absorbed water in the glycerol/water system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With this picture of the secondary flows, we can hypothesize an explanation for the non-monotonic behavior of the measured viscosity with gap height and rotation speed in the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' First, in the slow rotation speed or small gap-height limit, absorbed water can only transport radially inwards via diffusion which is quite slow because the radial fluid velocity is O(Re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In reality, the diffusive transport will proceed faster than expected based on the diffusivity of water in pure glycerol, because the absorbed water increases the diffusion coefficient as it is absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, even using the diffusivity value that results for c → 1, the radial diffusion remains a slow and inefficient process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, in the low-Re limit, the absorbed water remains highly confined near the outer edge of the rheometer, 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Secondary velocity components ur,axi and uz,axi in a parallel-plate rheometer in the small-gap limit (ϵ ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The radial velocity proceeds outward along the upper-half of the gap, reverses at the outer edge, and proceeds radially inward along the lower-half of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The z-component shows an upward drift along the middle of the gap that is independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Both secondary velocities are O(Re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results correspond to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' except over impractically long experimental timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' On the one hand, this confines the relatively low viscosity fluid at the outer edge of the rheometer, where the torque is primarily generated, but it limits the amount of water that is absorbed since the concentration gradient is relatively diffuse at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As Re increases, the secondary flow begins to pull some of the absorbed water radially inward, steepening the gradient at r = 1 and increasing the total flux of water into the system, while still leaving the absorbed water relatively confined at the outer edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This results in a greater amount of absorbed water near the outer edge of the rheometer and a faster decrease in the measured viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As Re continues to increase, the secondary flows continue to more strongly pull the absorbed water away from the outer edge, further increasing the amount of absorbed water via the steeper gradient at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, it is plausible that above a certain Re the secondary flows become sufficient enough to pull the absorbed water radially inwards a distance that is sufficiently far from the outer edge such that the net effect on the measured torque begins to lessen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' That is, despite the fact that more water is absorbed, this low viscosity fluid is redistributed towards the inner part of the rheometer where the effect on the torque and measured viscosity is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the constant viscosity, axisymmetric results can provide one possible explanation for the non-monotonic behavior seen between the measured viscosity and the gap height and rotation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, in the high-Re limit, the absorbed water concentration can be expected to be well-mixed, promoting a rapid flux of water into the glycerol by maintaining a strong gradient at r = 1 and rapidly redistributing the low viscosity fluid throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In such a regime, more care should be taken with determining the flux boundary condition, since the rate of water transport in the glycerol may approach the rate of water vapor transport in the outer flow problem where depletion of water vapor near the interface may limit the available flux into the glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As a quick point of reference, with the definition of Reynolds number given by Re = ρgΩh2 0/µg, with the characteristic density and viscosity based on values for pure glycerol, the experimental results presented above in Figure 2 have Reynolds numbers ranging from ∼ 1×10−6 up to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' While these values seem small, recall that the dimensional viscosity can vary by over three orders-of-magnitude, such that a locally defined Reynolds number could be significantly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, we note that the constant viscosity parallel-plate model is apparently inconsistent with the experimental observation of rapidly decreasing viscosity measurements at very small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the limit of Re ≪ 1, the secondary flows in a parallel-plate rheometer are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In this case, the transport equation simplifies to a purely 1D radial diffusion problem, and the evolution of water concentration becomes independent of both the gap height and rotation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, the viscosity distribution likewise is independent of h0 and Ω, which is inconsistent with the sharp decrease in µf/µi seen in Figure 2b at very small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This will motivate us later in the paper to consider the potential role of misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' First, we examine in more detail the one-dimensional diffusive limit with variable viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' ONE-DIMENSIONAL DIFFUSIVE LIMIT WITH VARIABLE VISCOSITY First, we consider the evolution of the viscosity distribution and measured effective viscosity of glycerol absorbing water in the inertialess, one-dimensional diffusive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For small Reynolds numbers and gap heights, the axisymmetric form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (7) becomes Pe∂c ∂t = 1 r ∂ ∂r � rD ∂c ∂r � + 1 ϵ2 ∂ ∂z � D ∂c ∂z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (12) Considering that the water concentration boundary condition at r = 1 is independent of z, along with the no-flux conditions at the upper and lower plates, when ϵ ≪ 1 it must be the case that c is approximately independent of z, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (12) further simplifies to Pe∂c ∂t = 1 r ∂ ∂r � rD ∂c ∂r � , (13) which is simply a 1D radial diffusion equation with variable diffusivity D(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In this regime, a better choice for the characteristic time scale would be the characteristic radial diffusion time R2/D0, the use of which would yield the same equation without the Pe factor on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For consistency, we continue to use the convective 1/Ω timescale as the characteristic timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the only boundary conditions that are needed are symmetry at r = 0 and the saturation water mass fraction at r = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', c(r = 1) = csat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As the water concentration evolves, the anticipated viscosity measurement from the rheometer can be predicted through the use of the viscosity distribution as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A parallel-plate rheometer cannot measure the viscosity distribution throughout the fluid layer, but rather simply infers an effective viscosity µ∗ eff by measuring the total torque exerted on the upper plate as it spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In dimensional form, the azimuthal velocity at small gap heights is u∗ θ = Ωr∗z∗/h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This velocity profile is valid regardless of the viscosity distribution since c is a function of r and t only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The total torque experienced by the upper plate is then given by T = 2π � 0 R � 0 µ∗Ω h r∗3 dr∗ dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (14) If the viscosity is constant and uniform, the total torque on the upper plate is then T = πµ∗ΩR4 2h −→ µ∗ = 2hT πΩR4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (15) The rheometer assumes a constant viscosity fluid and reports the “effective” viscosity of the fluid that is calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (15) based on the measured torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the experimental system, the initial condition is assumed to be pure glycerol, such that Tinit = πµgΩR4/(2h), and we have µ∗ eff µg = T(t) Tinit = 2h πµgΩR4 2π � 0 R � 0 µ∗Ω h r∗3 dr∗ dθ −→ µeff = 2 π 2π � 0 1 � 0 µr3 dr dθ, (16) which simplifies to µeff = 4 1 � 0 µr3 dr (17) for axisymmetric flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, we perform 1D transient simulations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (12) using the finite-difference method with second-order accuracy in space and first-order accuracy in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Convergence studies were performed in space and time to verify the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Using these simulations, we compute the effective nondimensional viscosity over time as water is absorbed at the outer edge and diffuses radially inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We perform these simulations over a range of csat values which reproduces the effect of varying relative humidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' First, for comparison with experiments, the results are simulated for one hour to determine the degree of viscosity decrease that can be achieved via pure diffusion over the experimental timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These results are shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, diffusion alone is sufficient to generate a significant decrease in 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Transient nondimensional viscosity measurements in the inertialess, 1D, axisymmetric regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (a) Results simulated over 3600 seconds for comparison with experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results show substantial decreases in measured viscosities at large csat values, but not as significant as those seen in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Figure inset shows the final nondimensional viscosity µf versus csat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (b) Results extended to much longer times to show the final saturation of the glycerol, which corresponds to the curves leveling off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Clearly, higher values of csat reach saturation more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' measured viscosities over this timescale, although not to the degree seen in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For example, consider the experimental results in Figure 2c, which were performed at a Reynolds number of Re = ρΩh2 0/µ0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='6×10−6 and aspect ratio of ϵ = h0/R = 4 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Clearly, in such a regime the inertialess, 1D model would be expected to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, the experimental results show a much larger decrease in viscosity over this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The RH = 72% results (corresponding to approximately csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='386) drop to around µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='38, and the RH = 45% results (corresponding to csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='185) drop to around µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, in the 1D limit, the corresponding numerical predictions for csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 and csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2 only decrease to around µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='78 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='86, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the experiments show a much larger decrease in viscosity over this timescale than the 1D model with pure diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, the results shown in Figure 5a are clearly still evolving over this timescale, whereas in the long-time limit we expect all of the glycerol to homogenize at the saturation concentration based on the relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Therefore, we extend these results to much longer times in Figure 5b, which shows the measured effective viscosities level off as the water concentration saturates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here we see the influence of the variable diffusivity on the timescale for the diffusive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With the characteristic diffusivity D0, the timescale for the process would be expected to be t∗ = O(R2/D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, as can be seen, most of the cases have fully saturated well before this timescale, especially at larger csat values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This is due to the enhanced diffusion at larger water concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In fact, a much better prediction for the timescale of this 1D diffusive process is to use the diffusivity based on csat, which we call Dsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The rescaled results are shown in Figure 6, which shows that for each case the water concentration in the glycerol has fully saturated over the timescale t∗ = O(R2/Dsat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For each case, two regimes can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the early times, the water concentration in the glycerol is non-uniform, and so the diffusive transport in the domain proceeds with a spatially varying diffusivity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' At late times, the water concentration throughout the system has nearly equilibrated at around the saturation concentration, such that the diffusion coefficient is nearly uniform and the results all decay exponentially with the same rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This constant can be simply calculated by considering a 1D radial diffusion problem with constant diffusivity (since this is nearly the case at long times), where the transport in dimensional form is governed by ∂c ∂t∗ = Dsat 1 r∗ ∂ ∂r∗ � r∗ ∂c ∂r∗ � with ∂c ∂r∗ ���� r∗=0 = 0 and c(r∗ = R) = csat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (18) The solution to this is given by c(r∗, t∗) = csat + ∞ � n=1 ane−Dsatt∗λ2 nJ0(λnr), (19) 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Rescaled effective viscosities in the 1D, inertialess, axisymmetric limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For each case the water concentration fully saturates approximately over the timescale t∗ = O(R2/Dsat), which is consistent with diffusion primarily occurring at the saturation concentration diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' At late times, the rescaled viscosities all approach the saturation values exponentially with a rate constant of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='78, consistent with the 1D theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' where the an are coefficients that depend on the initial condition, and J0 is the zeroth-order Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The λn eigenvalues here are the roots of J0 divided by the radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, at late times we see that c − csat ∼ exp � −χ2 1t∗Dsat/R2� , where χ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='40483 is the first root of the J0 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus we see the χ2 1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='7832 exponential decay seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The previous results are strictly valid in the inertialess (Re ≪ 1), small gap (ϵ ≪ 1), and axisymmetric limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These calculations are significantly simplified compared to the solution for the inertial regime since (1) the water concentration profile can no longer be assumed to be independent of z due to the secondary velocity components and (2) the fluid velocity profiles must be recalculated continuously as the concentration profile evolves while taking into account the spatial variations in viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Nevertheless, it is clear that we must extend our results to the inertial regime, since the 1D diffusion-dominated results cannot reproduce the same degree of viscosity decrease over the timescale of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These simulations are pursued in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' INERTIAL REGIME WITH VARIABLE VISCOSITY Having explored the purely diffusion-dominated 1D axisymmetric regime in the previous section, we now extend our results to the inertial, axisymmetric regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Recall that in the inertial regime, the coupled dynamics are governed by four dimensionless parameters, which are: Pe = ΩR2 D0 , Re = ρΩh2 0 µ0 , ϵ = h0 R , and csat, (20) whereas in the diffusion-dominated case the dynamics are governed only by csat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the system is governed by a relatively large parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, note that the ratio µ0/(ρD0) = µg/(ρDg) is fixed for glycerol, and the Peclet number can be written as Pe = ΩR2 D0 = �ρΩh2 0 µ0 � �R2 h2 0 � � µ0 ρD0 � = Re ϵ2 � µ0 ρD0 � , (21) so that the Peclet number is uniquely determined by the choice of Re and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Sample computational mesh design for the inertial, axisymmetric simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The grid has been coarsened by a factor of 3 in the r- and z-directions for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Local mesh refinement is used near r = 1 to resolve the water concentration boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (a) Top-down view of the axisymmetric wedge mesh geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (b) Side view of the wedge mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (c) Zoom in of the local refinement near r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Several extra layers of very thin cells exist on the right-hand side which are difficult to see in order to resolve sharp concentration gradients that can occur at the boundary when inertial effects come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Numerical methods Numerical simulations were performed using OpenFOAM [43] with an axisymmetric wedge-shaped mesh geometry with a wedge angle of 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Local mesh refinement was used near r = 1 to resolve the water concentration boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A sample mesh design is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Simulations were performed using a custom in-house solver that iteratively updates the water concentration profile for 100 timesteps using a timestep of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='01 seconds using second-order backward time-stepping and then recalculates the new steady-state velocity/pressure profiles using the SIMPLE algorithm [3, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, the solver assumes that the fluid velocity does not change much during one timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Convergence tests were performed to confirm that recalculating the velocity every 100 timesteps had a negligible impact on the calculated results compared to re-solving every timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The solver assumes that the velocity/pressure profiles are quasi-steady and only evolve when the water-concentration profile changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The SIMPLE algorithm was used with relative pressure and velocity tolerances of 1 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Convergence tests also confirmed the results were insensitive to these tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, grid resolution convergence tests were performed, and a final base grid of 375 × 30 cells in the r × z directions was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The cells within the region from r = 1 − 2ϵ to 1 were all further refined by one level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, the final layer of cells at r = 1 was further refined by halving three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Using this grid, convergence tests indicate that the errors due to spatial discretization should be less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Torque measurements were calculated by integrating the wall shear stress over the upper plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results Using the numerical methods described in the previous section, simulations were performed across a range of gap heights, Re, and relative humidities (through their proxy csat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Before introducing the final measured viscosity values for comparison with the experiments, we first present results illustrating the evolution and dynamics of the water concentration field in the glycerol over a range of gap heights and rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A comparison of the evolving water concentration profiles at various rotation speeds is shown in Figure 8, and we give the results in dimensional form to make more clear the relationship of the changes to the experimental results presented earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These simulations were performed at csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2, and ϵ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 × 10−3 m)/(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 × 10−2 m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='04 with rotation speeds of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, (c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, and (d) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding nondimensional parameters for these cases are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the Reynolds number ranges from Re = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='58 × 10−4 up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='15 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This seems counter-intuitive, since even the smallest rotation speed case shows some deviation from a purely 1D, diffusive transport, as can be seen by the concentration variation in the z-direction, whereas the relatively small Reynolds numbers suggest inertial effects should be small for all of these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, consider that u∗ r,axi ∼ ΩRRe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Then the characteristic time for convection in the radial direction is τconv,rad = (Ω Re)−1, while the characteristic time for diffusion in the radial direction is τdiff,rad = R2/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, an appropriate radial Peclet number is Perad = τdiff,rad τconv,rad = ΩR2Re D0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These values are also tabulated in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, even for the smallest angular velocity case with Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s, the radial Peclet number is still greater than O(103), increasing up to O(106) at 10 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, even at relatively small Reynolds numbers, the radial transport will be dominated by convection due to the relatively low diffusivity coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For reference, Table I also tabulates the nondimensional parameters based on the viscosity and diffusivity values associated with the saturation concentration csat rather than reference values based on pure glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the Reynolds numbers are increased while both the convective and diffusive timescales are decreased due to the reduced viscosity and increased diffusivity at increased water concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Numerical results for the evolving water concentration profile c over time for different rotation speeds Ω at csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2, and ϵ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 × 10−3 m)/(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 × 10−2 m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the angular speeds are (a) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s, (b) Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, (c) Ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, and (d) Ω = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding nondimensional parameters are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the Reynolds number (based on the saturation viscosity rather than µg) ranges from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='71×10−3 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='243 as the role of secondary (inertial) flows clearly grows with Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Summary of the simulation parameters used in Figures 8 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The timescale values τconv and τdiff have units of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the parameters with ‘sat’ subscripts are calculated based on the fluid properties at the appropriate saturation mass fraction of water, and parameters without this subscript are calculated based on the fluid properties of pure glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Examining the transport dynamics in Figure 8, we see that the transport of water concentration is consistent with the axisymmetric, constant viscosity flow picture described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, in the constant viscosity case, flow proceeds radially outward along the upper plate, turns downward, and then flows radially inward along the lower plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This is shown in more detail in Figure 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the arrows are color-coded and scaled by the magnitude of the secondary velocity components |u∗ sec|, and the background is color-coded by the water concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, u∗ sec is the velocity field on the slice in the r- and z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, the water begins to diffuse inwards from the outer edge, but then the secondary velocity pulls the absorbed water down along the outer edge and then 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Detailed look at the glycerol/water dynamics for the case corresponding to Figure 8d taken at t = 1 × 104 (see Table I for all relevant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (a) Secondary velocity vectors colored and scaled by the magnitude of the secondary velocity superimposed on a colormap of the water concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, the water begins to diffuse inward from the outer boundary, where the secondary flow pulls the absorbed water downward and then radially inward along the bottom plate, leading to a steep concentration gradient at the outer edge as the rotation speed is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (b) Full water concentration profile over a full axisymmetric cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (c-f) Nondimensional diffusivity coefficient, viscosity, radial velocity component, and z velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' radially inwards along the bottom plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This creates a very thin boundary layer region near the outer edge of the rheometer, which gets thinner as Perad increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The full solute concentration profile corresponding to Figure 9a is shown in 9b, and the corresponding dimensionless diffusivity, viscosity, radial velocity, and z velocity are shown in 9c-f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This case corresponds to the parameters previously shown in Figure 8d with the parameters shown in Table I, and all results are at the nondimensional time t = Ωt∗ = (10 rad/s)(1000 s) = 1 × 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen in Figure 9, the regions of high water concentration correspond to the regions of increased diffusivity and decreased viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The radial velocity component resembles the flow for the axisymmetric constant viscosity case with outward radial flow along the upper half of the domain and inward radial flow along the lower half, except that the magnitude of both is increased throughout the extent of the low viscosity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, at the front of the propagating front of water concentration, there is a steep gradient in viscosity that corresponds to an upward secondary velocity due to the viscosity gradients as seen in Figure 9d,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, a last illustration of the water concentration dynamics at higher csat is presented in Figure 10 as a function of gap height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These results show the evolution of the water concentration profile over time at csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 and Ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s for gap heights ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='05 mm to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Again, all of the relevant nondimensional parameters are given in Table I based on both the pure glycerol reference values and the saturation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the Reynolds 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Numerical results for the evolving water concentration profile c over time for different gap heights h0 at csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5, and Ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the gap heights are (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='05 mm, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='1 mm, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2 mm, (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 mm, (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='75 mm, (f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 mm, (g) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='25 mm, (h) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 mm, and (i) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding nondimensional numbers are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Over this parameter range, the Reynolds numbers (based on the saturation viscosity) range from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='30 × 10−3 up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='68 and the gap aspect ratio ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='002 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, these cases capture the full transition from the 1D, diffusive limit, up to the inertial regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' number based on the saturation parameters ranges from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='30 × 10−3 up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='68, representing a transition from the inertialess regime into the moderate inertial regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The radial Peclet numbers based on the saturation properties remain relatively high, increasing from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='46 × 104 up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='34 × 107 as the gap height increases, suggesting that the radial transport of water is dominated by convection in these regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Nevertheless, the transport at the smallest gap heights is approximately 1D, suggesting that the Perad threshold for this transition is at a relatively large magnitude in this particular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note that the qualitative picture of the water concentration evolution is different in Figures 8 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, Figure 10f corresponds to the same gap height and rotation speed as Figure 8c, except with an increased csat value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' While this seems like a relatively minor change, the corresponding µsat value is an order of magnitude smaller at csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5, which leads to an order of magnitude stronger secondary flows in the region of locally low viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This generates an enhanced mixing that leads to a more homogeneously propagating front of water concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This can be visualized in Figure 11 for the case corresponding to Figure 10f at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, the local secondary recirculation in the low-viscosity region completely dominates the expected global secondary recirculation for the axisymmetric, constant viscosity case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In fact, that global velocity field (the axisymmetric constant viscosity solution) is negligible on the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This enhanced local recirculation at larger csat values explains why the water propagates as a more uniform front in that regime, as opposed to being pulled down and inward along the lower plate as was illustrated in Figure 9a for a smaller csat value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, having characterized and visualized the coupled transport dynamics in the parallel-plate, axisymmetric, inertial regime, we now calculate the measured effective viscosities in these simulations to see if the proposed model can fully capture the trends seen in the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The final measured dimensionless viscosities µf at t∗ = 3600 s are presented in Figure 12 as functions of gap aspect ratio ϵ, saturation concentration csat, and angular rotation speed Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The corresponding angular rotation speeds in the figure are (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, and (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the figure, the dashed lines correspond to the predictions of the 1D, axisymmetric, diffusion-dominated results previously described in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note that in the experiments, the reported viscosities were nondimensionalized by µ∗ i , the initial measured viscosity at t = 0, and in the simulations the reported viscosities have been nondimensionalized by µg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, several key relationships and trends emerge from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' First, we see clearly that in every case, the results approach the 1D diffusion-dominated limit as ϵ → 0 for constant Ω, and they appear to also approach this limit as Ω → 0 for constant ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For a given Ω, deviations from this limit increase as ϵ increases, due to enhanced inertial effects, as well as for increased csat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This latter trend is also due to 17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Enhanced secondary recirculation in the low-viscosity region corresponding to larger csat values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These results illustrate the enhanced mixing effect that is seen at early times with csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 for the parameters shown in Figure 10f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' At large values of csat, the local viscosity drops in regions of large c to such a degree that the local recirculation dominates the expected secondary motions for constant viscosity, axisymmetric flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' an increase in inertial effects, although indirectly through a decreasing in the local viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, increasing Ω clearly leads to more significant deviations from the 1D limit due to increasing secondary inertial flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Comparing these results with the experimental results, the axisymmetric inertial simulations do seem to capture many features of the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, we generally see decreased µf values at larger gap heights and larger csat values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', RH values), which are consistent with Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Specifically, the numerical results in Figure 12a correspond to the same rotation speed (Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s) as Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Similar trends are seen (except at small gap heights, which will be discussed below), with slightly less significant decreases in viscosity in the simulations compared to the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Increasing the rotation speed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s in the simulations shows more significant viscosity decreases than the experimental results at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' So the experimental results at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s agree well quantitatively with numerical predictions slighty above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' One trend seen in the experiments that we do not see in the simulations is the non-monotonic relationship between viscosity decrease and angular rotation rate seen in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, we do see evidence of a non-monotonic relationship with increasing inertial effects in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, at Ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s with csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='1 (orange curve in Figure 12d), the final measured viscosity first decreases and then increases with increasing gap height, demonstrating that the axisymmetric case can demonstrate such trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The most significant experimental result that these simulations cannot explain is the large decrease in measured viscosity values at small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In fact, one of the most consistent results of the axisymmetric, inertial simulations is the approach to the 1D, diffusion-dominated regime at small gap heights for any angular rotation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, these axisymmetric simulations apparently fail to account for some effect that becomes dominant at small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We hypothesize that this is due to misalignment effects that only become significant at very small gap heights in practical parallel-plate rheometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the next section, we perform additional simulations based on a misaligned geometry in an attempt to validate this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' ROLE OF MISALIGNMENT In the previous section, we clearly saw that the axisymmetric, inertial, variable viscosity model fails to account for the sharp decrease in measured viscosity at small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, we must consider what possible sources of error could account for these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A variety of experimental challenges exist for performing accurate measurements with a rheometer, such as underfilling of the parallel-plate gap, instrument inertia, and surface tension effects [19, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In addition to these, there are practical sources of error associated with the mechanical uncertainties in the rheometer itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A key source of these errors comes from deviations in the geometry of the gap containing the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These errors in the gap geometry could arise from non-parallelism, non-concentricity, non-flatness of the plates, non-zero 18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Compilation of all measured final dimensionless viscosities µf at t∗ = 3600 s from the axisymmetric, inertial, variable viscosity simulations for comparison with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Dashed lines indicate the 1D, inertialess, diffusion-dominated results described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results are plotted separately by rotation speed with values of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s, and (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Clearly, deviations from the diffusion-dominated limit increase with gap height and angular rotation speed due to the increase of inertial secondary flows, as well as with increasing csat due to local reductions in viscosity (and consequent increases in inertial effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' slip lengths at the upper or lower plates, edge effects at the outer edge of the rheometer, or errors in the gap- zeroing procedure [1, 2, 10, 13, 25, 28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' One reason the discussion of these sources of error arose was due to the experimental observation that as gaps decreased below several hundred microns, measured viscosities began to have systematic errors, typically decreasing with the gap height as also shown in Figure 2b [33, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Based on these observations, a variety of studies suggest that a key factor in this discrepancy in our measurements and simulations could be the misalignment of the rotating plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Although it is commonly assumed that the plates are perfectly aligned, a number of reports indicate that a small but finite misalignment is prevalent in parallel-plate and cone-and-plate rheometers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In fact, the gap height can vary over 50 µm across a few centimeters in a parallel- plate rheometer due to the non-parallelism in the gap, causing a significant error in the viscosity measurements in narrow-gap, high-shear-rate experiments [14, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This is due to the fact that the misalignment introduces additional lubrication forces in the fluid layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A variety of semi-empirical techniques have been developed to account for these systematic errors at small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For example, a simple linear approximation has been proposed in which a simple gap error is defined to correct the measured values [10, 14, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Another technique involves using ultrasound time-of-flight measurements to detect the varying thickness of the fluid layer in the case of misalignment, which can 19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Misaligned parallel-plate rheometer geometry and coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The upper plate is misaligned by a small deflection angle φ and rotates at angular speed Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With z nondimensionalized by h0 and r nondimensionalized by R, the z-coordinate defining the upper plate is h(r, θ, φ) = 1 + φϵ−1r cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note that for small angles, the angle φ can range from 0 to a maximum of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' be used to calculate the degree of misalignment [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A numerical solution of the flow in a misaligned parallel-plate rheometer was also presented by Andablo-Reyes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Also, Clasen [8] introduced a system that can self correct non-parallelism to a degree using hydrodynamic lubrication forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, a theoretical description of the velocity and stress profiles in a slightly misaligned cone-and-plate rheometer was achieved by Dudgeon and Wedgewood [18] using a domain perturbation study in the limit of zero Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In this section, we consider the role of misalignment on the transport of absorbed water throughout the glycerol, and the effects of this misalignment on the measured viscosity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In general, such an analysis would need fully three-dimensional simulations in a misaligned rheometer geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We considered performing such simulations, but found them to be intractable due to the extremely high computational cost of performing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, they require 2-3 orders of magnitude more grid cells than the axisymmetric simulations in order to resolve the water concentration boundary layer at the outer edge of the rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, all of the meshing techniques we tried that would maintain this resolution at the outer edge ultimately resulted in very high aspect ratio cells at some point in the domain that affect resolution and greatly increase the number of iterations needed to solve the velocity/pressure profile with the SIMPLE algorithm, which must be repeated continuously as the water concentration field evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For these reasons, we consider a simplified, depth-averaged case that is valid in the limit of small gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This model and the corresponding simulations and results will be described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Theory Here, we consider a misaligned parallel-plate rheometer with radius R and gap thickness h(r, θ, φ), where the upper plate is slightly tilted by the small angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Once again, the lower plate is stationary, and the upper plate rotates at a rotation speed of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The coordinate system and problem setup are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The boundary conditions for the system are u = 0 at z = 0 (on the lower plate), and ur = r cos θ sin θ sin φ tan φ, (22a) uθ = r � cos2 θ sec φ + cos φ sin2 θ � , (22b) uz = −rϵ−1 sin θ sin φ, (22c) at z = h(r, θ, φ) = 1 + φϵ−1r cos θ (the upper plate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note that these simplify to ur = uz = 0 and uθ = r at z = 1 as in the axisymmetric case when φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For small angles, the angle φ can range from 0 to a maximum of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, φ/ϵ 20 ranges from 0 to 1, and small values of φ/ϵ correspond to small plate deflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, φ/ϵ = 0 corresponds to the case of no misalignment, and φ/ϵ = 1 corresponds to the case where the plates come in contact at one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Further, recall that as before we generally also need to apply boundary conditions at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In a practical experiment, this boundary condition represents a fluid-air interface that is typically not flat and experiences surface tension effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, in the small gap limit, we lose the ability to impose such a boundary condition, and we note that this contributes to the error in velocity/pressure profiles in the O(ϵ) region near r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The governing equations are again the Navier-Stokes equations with variable viscosity and the continuity equation, which are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (4) and (5), respectively, as well as the water concentration advection–diffusion equation given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As mentioned above, the numerical simulation of the full system of coupled equations in a well-resolved 3D geometry is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In the limit of narrow gap heights ϵ ≪ 1 and negligible inertia Re ≪ 1, the Navier-Stokes equations simplify to 0 = −∂p ∂r + µ∂2ur ∂z2 + ∂µ ∂z ∂ur ∂z , (23a) 0 = −1 r ∂p ∂θ + µ∂2uθ ∂z2 + ∂µ ∂z ∂uθ ∂z , (23b) 0 = ∂p ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (23c) Furthermore, in the thin gap limit, the water concentration can be assumed to be approximately uniform in the depth direction, which gives ∂c ∂z ≈ 0 and ∂µ ∂z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This gives 0 = −∂p ∂r + µ∂2ur ∂z2 , 0 = −1 r ∂p ∂θ + µ∂2uθ ∂z2 , 0 = ∂p ∂z , and 1 r ∂ ∂r (rur) + 1 r ∂uθ ∂θ + ∂uz ∂z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (24) Note here that c(r, θ, t), µ(r, θ, t) and p(r, θ, t) in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With the fact that ∂p/∂z = 0 in this limit, the next leading-order form of the z-component of the Navier-Stokes equations becomes 0 = µ∂2uz ∂z2 + ∂µ ∂r ∂ur ∂z + 1 r ∂µ ∂θ ∂uθ ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (25) By examining the form of the gap height distribution h(r, θ, φ) = 1 + φϵ−1r cos θ we see that the magnitude of the perturbation is O(φ/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This suggests the use of a solution given by ur(r, θ, z, t) = �φ ϵ � ur,1(r, θ, z, t) + �φ ϵ �2 ur,2(r, θ, z, t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' , (26a) uθ(r, θ, z, t) = rz + �φ ϵ � uθ,1(r, θ, z, t) + �φ ϵ �2 uθ,2(r, θ, z, t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' , (26b) uz(r, θ, z, t) = �φ ϵ � uz,1(r, θ, z, t) + �φ ϵ �2 uz,2(r, θ, z, t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' , (26c) p(r, θ, t) = �φ ϵ � p1(r, θ, t) + �φ ϵ �2 p2(r, θ, t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' , (26d) which is valid in the limit φ/ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Substituting this expansion into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (24) and (25) and applying the boundary conditions gives ur,1(r, θ, z, t) =z(z − 1) 2µ ∂p1 ∂r , (27a) uθ,1(r, θ, z, t) = − r2z cos θ + z(z − 1) 2rµ ∂p1 ∂θ , (27b) uz,1(r, θ, z, t) = − z2 12r2µ � 6r3µ2 sin θ − (2z − 3) �∂µ ∂θ ∂p1 ∂θ + r2 ∂µ ∂r ∂p1 ∂r � +(2z − 3)µ �∂2p1 ∂θ2 + r �∂p1 ∂r + r∂2p1 ∂r2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (27c) 21 This procedure also yields a PDE governing the pressure distribution that is given by µ �∂2p1 ∂θ2 + r �∂p1 ∂r + r � 6rµ sin θ + ∂2p1 ∂r2 ��� = ∂µ ∂θ ∂p1 ∂θ + r2 ∂µ ∂r ∂p1 ∂r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (28) With some known distribution of viscosity in the system, a numerical solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (28) yields the pressure distribution in the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This in turn can be used to calculate the velocity profiles from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (26) and (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The velocity profiles can then be used to update the water concentration distribution via the advection–diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With the assumption that c is independent of z (valid in the small gap limit), the solute transport equation becomes Pe∂c ∂t = 1 r ∂ ∂r � rD ∂c ∂r � + 1 r2 ∂ ∂θ � D ∂c ∂θ � − Pe � ur ∂c ∂r + uθ r ∂c ∂θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (29) Since c is independent of z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' we consider solving the depth-averaged version of this equation instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' which is simply Pe∂c ∂t = 1 r ∂ ∂r � rD ∂c ∂r � + 1 r2 ∂ ∂θ � D ∂c ∂θ � − Pe � ¯ur ∂c ∂r + ¯uθ r ∂c ∂θ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (30) where bars denote depth-averaged quantities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' ¯c = c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' ¯D = D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' and ¯ur = 1 1 + φ ϵ r cos θ 1+ φ ϵ r cos θ � z=0 ur(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' z)dz = − 1 12µ φ ϵ ∂p1 ∂r + O �φ ϵ �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (31a) ¯uθ = 1 1 + φ ϵ r cos θ 1+ φ ϵ r cos θ � z=0 uθ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' z)dz = r 2 − 1 12rµ φ ϵ ∂p1 ∂θ + O �φ ϵ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (31b) Thus, having solved the pressure distribution due to the misalignment p1 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (28), the depth-averaged velocity components can be calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (31), which can in turn be used to advect the solute concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Numerical methods We perform numerical simulations of the coupled transport equations described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Recall that we seek solutions of the water transport and associated viscosity measurements in a misaligned parallel-plate rheometer that is valid at small gap heights and in the limit of negligible inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The numerical approach for solving these systems is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (28) for the pressure perturbation due to misalignment subject to the boundary condition p → 0 at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Calculate the depth-averaged radial and azimuthal velocity components from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Advance the water concentration profile in time by numerically integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (30) for one or more timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Calculate the new viscosity and diffusivity fields and iterate back to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We attempt a numerical implementation of this process using a finite-difference implementation in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, several complications emerge due to the extremely large Peclet numbers in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, the diffusion of the water concentration field is so slow that the field effectively propagates with a very sharp front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In order to resolve this and avoid spurious oscillations in the concentration field, we use slope-limited finite differencing based on the minmod limiter function to switch to first-order spatial differencing at the steep gradient [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This avoids the oscillations that result in a pure second-order differencing scheme and still allows nearly second-order accuracy in space globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The more serious difficulty that we encountered with this numerical approach is solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (28) for the perturbation pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These solutions do not behave nicely due to the sharp viscosity gradients on the right-hand side of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We were not able to resolve this issue using slope-limiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As an alternative approach and to illustrate the qualitative dynamics that can be expected with a misaligned upper plate, we instead assume a constant viscosity model for the purposes of calculating the velocity profile, since this 22 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Depth-averaged water concentration profile at t∗ = 3600 s as a function of misalignment for csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 and Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results correspond to φ/ϵ values of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 (perfectly aligned), (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='1, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2, (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5, (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='75, and (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='95 (plates are nearly contacting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, as the misalignment increases, the concentration profile becomes no longer axisymmetric, and there is a significant increase in total water transport into the fluid layer from the outer edge due to the misalignment-driven secondary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' solution is well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (28) has an analytical solution when µ = 1 which is given by p1(r, θ) = −3 4r � −1 + r2� sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' (32) We use this theoretical result at constant viscosity to calculate the depth-averaged velocities in the misaligned rheome- ter and use these to update the concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' When calculating the effective measured viscosity and torque, we always use the viscosity distribution that corresponds to the concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This limitation to a velocity profile based on constant viscosity is a clear limitation of our results, but nevertheless they capture qualitative features of the experiments that the axisymmetric model could not predict, and we leave a full solution with evolving velocity profiles based on spatial variation of viscosity to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results Here, we introduce the numerical results achieved for the misaligned rheometer system based on the methodology described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' First, we highlight the role of the misalignment on the water concentration field in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the depth-averaged water concentration profiles are shown at t∗ = 3600 s as a function of misalignment for csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 with misalignment ranging from φ/ϵ = 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='95, where 0 is the perfectly aligned parallel- plate case, and 1 is the limit where the plates come into contact at one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' As can be seen, the misaligned cases all show a non-axisymmetric concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This is due to the non-axisymmetric secondary velocity components due to the misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In particular, the radial component ¯ur transports water towards or away from the outer edge, and is ¯ur ∼ φ ϵ ∂p1 ∂r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' With p1 ∼ sin θ, this represents a radially inward flow on one half of the rheometer and a radially outward flow on the other half and explains why in Figure 14 the concentration profile appears to be pulled in from the right edge and pushed towards the left edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, this secondary velocity component is proportional to φ/ϵ, so doubling the degree of misalignment doubles the radial advective fluxes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' In order to quantify the effect of misalignment on the measured viscosity values, simulations were performed across a range of φ/ϵ and csat values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The final measured viscosity values from these simulations are presented in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 23 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Final measured viscosity values µf/µi at t∗ = 3600 s as functions of the misalignment φ/ϵ and csat at an angular rotation speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Dashed lines correspond to the 1D pure-diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results asymptotically approach the 1D diffusion-dominated limit as φ/ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, results show a steep drop-off in measured viscosity values as the misalignment increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Comparison of the final measured viscosity values at t∗ = 3600 s for the models corresponding to each of the three regimes: (1) the 1D axisymmetric, pure-diffusion limit (dashed lines), (2) the axisymmetric, inertial regime (dot-dashed lines), and (3) the misaligned, inertialess, small gap limit (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results correspond to angular rotation speeds of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s and (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Misaligned cases are calculated with a misalignment angle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0005 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' All cases were performed with a rheometer of radius R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the results correspond to an angular speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The dashed lines in the figure correspond to the 1D diffusion-dominated regime, and the results asymptotically approach these limits as φ/ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, the results show a steep dropoff in measured final viscosities at large misalignments, which possibly explains the sharp decrease in measured viscosity in the experiments at small gap heights (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', Figure 2b) that was not captured in the axisymmetric model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, a comparison between all of the three different proposed models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', the 1D pure diffusion limit, the axisymmetric inertial limit, and the misaligned inertialess small gap limit) is shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Here, the dashed lines indicate the 1D diffusion-dominated limit, dot-dashed lines correspond to the inertial, axisymmetric regime, and the solid lines show the results for the misaligned, inertialess, small-gap limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results were calculated for angular rotation speeds of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='4 rad/s and (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='0 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The results show that at small gap heights, the misalignment 24 effects dominate the inertial effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This becomes more clear when considering that the radial secondary velocity due to inertial effects in an axisymmetric case is O(Re), whereas the secondary velocity components due to misalignment are O(φ/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For fixed angular rotation speed, the secondary velocities must dominate the inertial secondary velocities as the gap height decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, the results also show that the opposite is true at large gap heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For a fixed misalignment angle and rotation speed, φ/ϵ decreases as the gap height increases, whereas the inertial effects increase, such that the large gap regime is dominated by inertial effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' This cross-over explains the non-monotonic relationship between µf and gap height reported in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, there is a critical ϵ value at which the measured viscosity values switch from being misalignment-dominated to inertia-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Numerical simulations using the previously described models and numerical methods can be used to estimate this transition, as shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have considered the measurement of the viscosity of glycerol in a parallel-plate rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Intu- itively, it can be anticipated that the viscosity must decrease over time due to the hygroscopic nature of the fluid as it absorbs water vapor from the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Based on an initial understanding of the fluid dynamics in a parallel-plate rheometer for a constant viscosity flow, an axisymmetric model of the flow predicts that the dynamics should be purely limited by diffusion in the thin-gap limit and become independent of gap height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, a sharp drop-off in measured viscosity values was observed experimentally at small gap heights, which motivated us to reconsider the fluid dynamics in the system and led to the hypothesis that plate misalignment could drive additional secondary flows that might affect the transport of the water concentration throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Ultimately, theoretical models and numerical simulations of the coupled dynamics and measured viscosity values were achieved in three different regimes: (1) the 1D inertialess, diffusion-dominated regime, (2) the axisymmetric inertial regime, and (3) the mis- aligned, inertialess, thin-gap regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Results confirmed that there are two types of secondary flows that can exist in such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The first of these is O(Re) and corresponds to the secondary inertially driven flows in a perfectly aligned axisymmetric parallel-plate rheometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The other secondary flow is O(φ/ϵ) and is driven by the O(φ/ϵ) plate misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Assuming a fixed misalignment φ, then as the gap height decreases, the O(φ/ϵ) misalignment flow will inevitably dominate the O(Re) inertially driven flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Based on the results here via comparison between experiments and numerical simulations, as well as between simulations with parallel and misaligned plates, we argue that the sharp decrease in measured viscosity is attributable to the secondary flows induced by plate misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The mechanism by which the misalignment results in a faster decrease in viscosity seems to be that the secondary velocities pull the relatively high water concentration away from the outer boundary, which steepens the concentration gradient at the outer boundary, resulting in an increased mass flux of water into the glycerol, which subsequently lowers the viscosity of the glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' These results have relevance not only to the measurement of the viscosity of glycerol solutions (for which care must be taken to ensure all the possible transport mechanisms are understood), but also for our understanding of the flow in parallel-plate rheometers more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We have shown that misalignment effects in particular can have a surprising critical influence over the mass transport in such systems, especially as the gap height becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, based on these results, it is plausible that such viscosity measurements of glycerol in a parallel-plate rheometer could potentially be used as a technique to quantify the degree of misalignment in the rheometer, although we leave a practical investigation and demonstration of this technique for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Finally, we note that additional complications can arise in such a system, such as the potential for the emergence of instabilities due to viscosity gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' We observed evidence for such effects in our numerical results at high angular rotation speeds in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' For example, Figure 17 shows a time series of the concentration profile for a case with Ω = 10 rad/s, csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2, and h0 = 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' It is known that viscosity stratification in shear flows can lead to instability in certain regimes [37, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' However, the coupled viscosity distribution and velocity profile in the rheometer system are highly nonlinear, and the flow cannot be analyzed in terms of simple viscosity-stratified layers of fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Furthermore, at the high rotation speeds and gap heights where we observed this instability, it is likely that other assumptions in our proposed models will break down, especially the assumed flat interface at the outer boundary and the predefined slip boundary conditions there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Thus, we leave a detailed study of these intriguing instabilities for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Declaration of interests: The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Author contributions: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' developed the theory and performed the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' performed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' conceived the experiment and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' The 25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Apparent instability/oscillation in the concentration profile field due to viscosity gradients from numerical results with Ω = 10 rad/s, csat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content='2, and h0 = 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' authors would like to acknowledge Ian Jacobi for helpful discussions regarding the experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Andablo-Reyes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Hidalgo-´Alvarez, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' de Vicente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' A method for the estimation of the film thickness and plate tilt angle in thin film misaligned plate–plate rheometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Non-Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Fluid Mechanics, 165(19-20):1419–1421, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Andablo-Reyes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' de Vicente, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Hidalgo-Alvarez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' On the nonparallelism effect in thin film plate–plate rheometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Rheol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=', 55(5):981–986, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' [3] I.' metadata={'source': 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Bocquet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Tabeling, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Manneville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Comment on “Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E_T4oBgHgl3EQf0hwq/content/2301.08329v1.pdf'} +page_content=' Lett.' metadata={'source': 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--git a/cNFPT4oBgHgl3EQfBjTq/content/tmp_files/2301.12986v1.pdf.txt b/cNFPT4oBgHgl3EQfBjTq/content/tmp_files/2301.12986v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6660168a0e9b2caab80f5978d5a54fbe42806e3e --- /dev/null +++ b/cNFPT4oBgHgl3EQfBjTq/content/tmp_files/2301.12986v1.pdf.txt @@ -0,0 +1,770 @@ +MOSAIC, a comparison framework for machine learning models +Matt´eo Papin +matpapin0@gmail.com +´Ecole 42, 75017 Paris, France +Yann Beaujeault-Taudi`ere +yann.beaujeault-taudiere@ijclab.in2p3.fr +Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France +Laboratoire Leprince-Ringuet (LLR), ´Ecole polytechnique, CNRS/IN2P3, 91120 Palaiseau, France +Fr´ed´eric Magniette +frederic.magniette@llr.in2p3.fr +Laboratoire Leprince-Ringuet (LLR), ´Ecole polytechnique, CNRS/IN2P3, 91120 Palaiseau, France +Abstract +We introduce MOSAIC, a Python program for machine learning models. Our framework +is developed with in mind accelerating machine learning studies through making imple- +menting and testing arbitrary network architectures and data sets simpler, faster and less +error-prone. MOSAIC features a full execution pipeline, from declaring the models, data +and related hyperparameters within a simple configuration file, to the generation of ready- +to-interpret figures and performance metrics. It also includes an advanced run manage- +ment, stores the results within a database, and incorporates several run monitoring options. +Through all these functionalities, the framework should provide a useful tool for researchers, +engineers, and general practitioners of machine learning. +MOSAIC is available from the dedicated PyPI repository. +Keywords: Open-source software, network optimization, classification, regression, Python, +neural networks +1. Introduction +Conceiving optimal machine learning models, especially with artificial neural networks, is +known to be a complex art. Although substantial work has been performed, not only on +the mathematical understanding of neural networks’ (NNs) approximation power (Hornik +et al., 1989; Cybenko, 1989; K˚urkov´a, 1991, 1992; Rolnick and Tegmark, 2018), but also +on the heuristic (Lin et al., 2017; Ojha et al., 2017) and numerical (Sagun et al., 2018; Li +et al., 2018) sides, no practically useful bounds, nor any useful recipe, is available in order +to a priori construct NNs that achieve a pre-determined degree of performance for a given +class of problems. There is no straightforward method to choose the hyperparameters of a +neural network to optimize the number of used parameters. In practice, given a machine +learning task, only prior experience give clues on which models should be used, and how +the set of hyperparameters must be tuned to reach good (let alone optimal) performances. +For a simple model like a multi-layer perceptron (MLP) (Minsky and Papert, 1969), the +shape of the network, the number of hidden layers and the number of neurons inside each +of them has a great influence on the result. +For more complicated techniques, such as convolutional neural networks (Zhang et al., +1988; LeCun et al., 1989, 1998), the number of hyperparameters increases as new quantities +come to play, such kernel size, padding, stride, pooling function, etc. +Very often, the +strategy consists in choosing enormous networks, typically using millions of parameters, to +©2023 Matt´eo Papin, Yann Beaujeault-Taudi`ere and Fr´ed´eric Magniette. +arXiv:2301.12986v1 [cs.LG] 30 Jan 2023 + +Papin, Beaujeault-Taudi`ere and Magniette +solve the problem. Although such large number of parameters is probably too much in +most situations, this gives an insurance of bringing enough expressive power to tackle the +complexity of the targeted problematic. +The problem of this “brute-force” approach is that it requires a lot of labeled input +data in order to truly constrain the parameters of the model. Furthermore, this introduces +the need for a lot of computing resources along the training process and the use of the +model. In addition, the difficulty of optimizing a neural network increases dramatically +with its size, further augmenting the computational resources in order to attain satisfactory +performances. As shown on the left part of Figure 1, when a statistical model is trained +on data, two kinds of error occur. These are driven by the respective complexity of the +data and the model. If the model is simpler than the data, i.e., the number of parameter is +smaller than what is required in order to satisfactorily learn the data, a bias error is pre- +eminent, due to the model underfitting the data by learning only its roughest features. On +the opposite, in the case of too complex models, a new type of error appears, the so-called +variance error. It corresponds to the parameters being too numerous with respect to the +data size, so that they end up learning irrelevant features of the training data set, such +as noise or unnecessarily moments of the data distribution, which in turn deteriorates the +generalization power of the network, causing the test losses to rise. An optimal model is in +the middle, where the two kinds of errors balance and result in a minimal total error. This +is called the bias-variance tradeoff (Belkin et al., 2019). +Thus, the problem of using models that are too large with respect to the data is the +induction of large variance of the validation loss. The effect of this overfitting can be seen +in the middle panel of Figure 1. +Most of the modern implementations fight against overfitting by using a tremendous +quantity of data to lower the variance. +As seen at the right of Figure 1, in that case, +the variance becomes negligible and, provided the model is sufficiently expressive, it can +effectively handle the complexity of the problem. +Model complexity +0 +2 +4 +6 +8 +10 +Error +Total +Bias +Variance +Optimum +Epochs +0 +2 +4 +6 +8 +10 +Error +Train error +Test error +Model complexity +0 +2 +4 +6 +8 +10 +Error +Total +Bias +Variance +Figure 1: Schematic illustration of the performance of NNs’ losses according to the relative +complexities of the model and data. Left: bias-variance tradeoff, typical of a +situation where data complexity ≫ model complexity. Middle: overfitting (model +complexity ≫ data complexity). Right: well-balanced network and data (model +complexity ∼ data complexity). +2 + +MOSAIC, a comparison framework for machine learning models +In a context where the resources are limited, like when porting a network on reconfig- +urable electronics (FPGA) or on huge data sets that requires long treatment times, it can +be interesting to tune carefully the network in order to reduce the number of parameters +without reducing the global performance. Furthermore, in the case of more sophisticated +techniques, like graph convolutional networks (GCN) (Kipf and Welling, 2016), another key +problem is present. Namely, that there are plenty of different algorithms available (Wu +et al., 2019), all of them having properties that can hardly be predicted accurately. In that +case, it is interesting to test extensively the different algorithms to choose the best one for +a dedicated problem. +In both cases, it is highly interesting to test a large collection of models on a data set and +to compare their performance. This is a subject that is well-covered by the machine learning +community. Plenty of documentation explains how different classes of models compare and +their specificities (). But the problem of most of the approaches consists in their “one-shot” +aspect. Indeed, a set of concurrent models are tested and compared at the same time, and +when a new model or data set is added to the collection, all the corresponding tests have +to be run again. +In this work, we try to ease to the maximum the solution of such problems, with in view +a long-term approach through the development of useful software. We have developed a +framework, called MOSAIC, allowing the sequential test of various architectures on multiple +data sets. It provides a database system to store the results and is well-adapted to test +hypotheses all along a research project. All the models are evaluated with different sets +of hyperparameters, and the run results are uploaded on-the-fly to a database for future +comparison. Mainly based on PyTorch (Paszke et al., 2019), it is very generic and can be +adapted almost effortlessly to any class of model and data. Our framework also provides +several tools to compare effectively the models. +2. The MOSAIC framework +MOSAIC is an open-source Python framework which provides all the facilities to test exten- +sively machine learning models on arbitrary data. It is based on the concept of pipelines, +which fully represent a treatment chain including data, data formatting, and model training, +all of which possibly including different parts. All the hyperparameters are also included in +the pipeline. They are generated by a subpart of the system, via interpreting a configuration +file. Once generated, they will be executed, possibly in a parallel way. A standard training +procedure is applied using an Adam optimizer (Kingma and Ba, 2017), and at every step, a +training and testing error is computed. At the end of the training, the performance and dif- +ferent indicators are stored in the database. The model parameters and the learning curves +are stored in output files. During all the execution process, an advanced error monitoring +is performed. Each pipeline can easily be executed multiple times to fill the database with +different models and different hyperparameters values. At any time, the analysis module +can be used to extract the desired information from the database and produce comparison +plots. Figure 2 shows the entire workflow of the framework. +3 + +Papin, Beaujeault-Taudi`ere and Magniette +Figure 2: Global scheme of the MOSAIC framework. Pipelines are generated from a configu- +ration file. These pipelines are executed in parallel, and the results of the training +are stored in files and in a database. In a second step, an analysis module queries +the database to produce the comparison plots. +3. Pipelines generation +The pipelines are generated from a configuration file (see A.1). We use the ini format, +where parameters are written in text within a section indicated by a word between square +brackets. Two mandatory sections describe the different parameters of the system itself. +The first one, named MONITOR, includes all the parameters for the run manager, includ- +ing the requirement for GPUs, the level of parallelism, the cache path and size and the +multiplicity. This last parameter allows executing multiple instances of the same pipeline +in order to collect statistics. +The second one, named PROCESS, gives the hyperparameters of the training: initial +learning rate, loss function, and so on. It also contains the paths to the classes implement- +ing custom metrics, such as user-defined loss functions. Two crucial elements of this section +are the data and model schemes describing the pipeline. They are a sequence of actions +that respectively refer to the data set building pipeline and the model. For example, we +could have a sequence ‘‘Data loading | enrichment | normalization’’ for the data +set. Additionally, one may combine simple models into more complex ones. For example +‘‘Convolution1 | pooling | Convolution2 | polling | readout | mlp’’. The dif- +ferent parts of the pipeline scheme must be implemented as classes and have a dedicated +section. +As an illustration, a section for a funnel-shaped MLP can be represented by +[mlp_funnel] +type = mlp +class = mlp_funnel +path_to_class = ./mosaic/share/mlp.py +length = 4 +width = {2-4},8 +The type must be the name present in the scheme. The class and path indicate what class +to import. The other parameters are specific to the class. In the simplest configurations, +the values are a single number or string. It can also be, as is here the case for the “width” +4 + +Pipeline +execution +Result +process +files +Configuration +Pipeline +Run +file +generator +manager +Plot +module +Pipeline +Service +Database +files +moduleMOSAIC, a comparison framework for machine learning models +parameter, a range of values. +In that case, multiples pipelines will be generated, that +correspond to all the different values allowed by the range (here, 2, 3, 4, and 8). +One +must be cautious when giving ranges to multiple parameters, because each cross-possibility +among the different class-specific parameters will generate a pipeline, and their total number +can grow quite large if several wide ranges are used simultaneously. Thus, for large-scale +exploratory studies, it is desirable that the users already have some insight on the relevant +ranges of each parameter, or coarse-grain the range of explored values, possibly iteratively +refining them after assessing the performances with the help of the plotting tools provided +by MOSAIC (see Sec. 5). All the generated pipelines are stored in JSON files to be executed +directly or later. +A convenient corollary of structuring the runs into pipelines is that it results in minimal +length, non-ambiguous characterization of a run. +This is particularly useful for gener- +ating the plot labels. For instance, the four funnel-shaped MLPs defined in the example +[mlp section] can be represented as mlp funnel(4,2), mlp funnel(4,3), mlp funnel(4,4) +and mlp funnel(4,8). +4. Run and performance indicators +When the pipelines are generated, they can be run by the core of the framework, the run +manager. Every element of the pipeline must be implemented by a Python class providing +the required methods. The class API is very similar to the Torch.nn.module class. A +forward method provides the evaluation of the data by the model, and a backward method +(which can be implicit) is used for computing the partial derivatives. +Another method +should provide the parameter values of the model. Two methods implement the saving and +loading of the model in a file (it can be any format, not only a PyTorch tensor). A last +method provides information to the other elements of the pipeline. +To execute a pipeline, every element is called sequentially, and a communication between +them is organized following two modalities. First, the result of every element is sent directly +to the second. There is no constraint on the format of this return value, except for the last +element of the data set model, which must provide a training and testing data loaders (in +PyTorch format). +Another way of communication between the elements is based on the info method. +Any class can, via this method, output a dictionary of valued parameters which is made +available to every subsequent classes, especially for initialization. For example, the data +set class should provide the shape of the inputs for the initialization of the neural network +input layer. The names of the class objects must be standardized across the classes entering +a pipeline. +During its execution, the model defined by a given pipeline is trained following a tradi- +tional training loop, led by an Adam optimizer. Every pipeline is an independent process, +communicating with a service module to access to the different services (cache, database, +etc.). The run manager follows the progression of the processes and launches new ones as +soon as computing resources are available. +At the end of the training loop, various performance indicators, which we detail in +Subsection 4.1, are calculated and stored in the database for further analysis. +5 + +Papin, Beaujeault-Taudi`ere and Magniette +4.1 Performance indicators +We have implemented six performance indicators in the framework. Each of these provides a +valuable, global performance metric, and their collection can be used to efficiently compare +the models. In the following, the total number of epochs will be denoted n, a given epoch +will be indexed by the integer e = 1, . . . , n, and the loss functions at the epoch e will be +written Ltrain/test(e). +The first two metrics are the final test and train losses. +These are the traditional, +and usually considered most important, indicators to compare the performance of different +models. +Due to the random nature of the parameters’ initialization and the stochastic +nature of the Adam optimizer, it is often interesting to try multiple times the same pipeline. +This allows users to acquire statistics over the (data, model) couple, as allowed by the +multiplicity parameter described earlier. When the multiplicity of a given pipeline is not +too large, it is often sufficient to compare only the train and test losses across its different +executions in order to identify well-performing runs. +The third indicator we have implemented measures the overfitting of the model. We +have chosen the difference of the last train and test error, normalized by the global range +of these errors: +overfitting = +| Ltest(n) − Ltrain(n) | +max +� +Ltrain/test(n) +� +− min +� +Ltrain/test(n) +�. +(1) +The smaller the indicator is, the lower the overfitting. +Fourth, the framework also implements a convergence rate indicator, which gives clues +on whether the convergence is reached after the training. For this, we consider the last ten +percents of the training loss curve, and carry out two operations. First, the gradients over +these points is calculated using a first-order finite difference, and the average gradient is +computed as a difference of averages: +slope(e) = Ltrain(e) − Ltrain(e − 1), +(2) +slope = +1 +⌈n/20⌉ +� +� +� +n +� +e=n−⌈n/20⌉ +Ltrain(e) − +n−⌈n/20⌉ +� +e=n−⌈n/10⌉ +Ltrain(e) +� +� +� . +(3) +The slopes thus obtained give a rough idea about the convergence. Averaging over the +two halves of the last ten percents smoothens sudden jumps of the loss that might occur +from one epoch to the following one, which ultimately increases the quality of this metric. In +addition, it guarantees that the slope is asymptotically vanishing, which reflects that beyond +a certain number of epochs, it is reasonable to expect that the networks’ performances will +on average stay constant up to negligible variations. Second, since the gradients altogether +incorporate information about the possible fluctuations of the loss during the last epochs, +we split them into two parts, according to whether they are larger or smaller than the +average. The first (resp. second) ones correspond to relative increases (resp. decreases) of +the loss, and are thus counted as statistical biases of the final slope towards larger (resp. +smaller) values. For each subset, the standard deviation is calculated: +6 + +MOSAIC, a comparison framework for machine learning models +σ+ = σ({slope(k) such that slope(k) ≥ slope and k ≥ n − ⌈n/10⌉}), +(4) +σ− = σ({slope(k) such that slope(k) ≤ slope and k ≥ n − ⌈n/10⌉}). +(5) +This indicator is finally presented as slope +σ+ +σ−. A negative value indicates that the loss +function is likely to decrease if a few epochs are added, whereas a positive or very small +slope hints that the model is unlikely to improve and might have reached a local minimum. +In the former case, a re-run mechanism allows continuing the training by restarting the +corresponding pipeline from the last obtained results. +Fifth, we have implemented a trainability indicator. The idea is to measure how fast a +given instance of the model has been trained on the data set. For this purpose, we consider +the integral of the train loss curve with respect to the x-axis: +trainability = +n +� +e=1 +Ltrain(e). +(6) +If the integral is small, it means that the model has learned fast. Conversely, a high +trainability hints that the model is learning the data slowly. This criterion can be used to +choose between two models (or classes thereof) otherwise presenting the same performances. +If the trainability is lower for one of them, it has been easier to train. +Following the same idea of selecting equally performing models, a last indicator is simply +the training time. This indicator is often of lesser importance since it does not reflect the +bare performances of the models; moreover, it depends on the machine running the training. +If the test is performed on a single machine or in a homogeneous environment, then this +value bears more relevance, and can be used in contexts where the full execution of the +pipeline is critical, e.g., when the quantity of data to treat becomes a limiting factor. +5. Result analysis +All the results of the runs are stored in a database. +We have chosen the SQLite3 file +format, in order to benefit from the power of the SQL language and the flexibility of the file +storage. SQL allows us to be very accurate on the choice of the models that participate in +our studies, up to the point one can easily and quickly query batches of data or cherry-pick +specific runs. +The analysis modules provide two kind of plots. The first one is a collection of train and +test losses plots along the epochs, as represented on Figure 3. Each plot also includes the +performance metrics introduced in 4.1, except for the runtime. The plots can be selected by +a SQL request. By default, all the plots are grouped in a single PDF file, but the number +of pages can be very important if the study includes numerous models. +In particular, +overfitting and convergence indicators can be used to select problematic trainings. +The other kind of plot produced by MOSAIC is called meta-plots. They are composed of +different curves representing any indicator in ordinate and any parameter in abscissa. Figure +4 shows an example of such format, in the form of violin plots. In such plots, the discrete re- +sults condition a Gaussian kernel density distribution (Hastie and Tibshirani, 2017), which +allows estimating a continuous probability density function from a finite number of runs. +7 + +Papin, Beaujeault-Taudi`ere and Magniette +0 +10 +20 +30 +40 +50 +0.20 +0.25 +0.30 +0.35 +overfit 0.16 +trainability 0.21 +slope -1.374E-072.701E +08 +3.638E +08 +1:dataset_OR(1,20,0.8) +mlp_brick(100,2):603 +0 +10 +20 +30 +40 +50 +0.05 +0.10 +0.15 +0.20 +overfit 0.97 +trainability 0.22 +slope -1.249E-032.673E +05 +1.787E +05 +2:dataset_OR(1,20,0.05) +mlp_brick(2,2):15 +0 +2 +4 +6 +8 +0.200 +0.225 +0.250 +0.275 +overfit 0.19 +trainability 0.22 +slope -2.236E-030.000E + 00 +0.000E + 00 +3:dataset_OR(1,20,0.8) +mlp_brick(2,2):15 +0 +10 +20 +30 +40 +50 +0.00 +0.05 +0.10 +0.15 +overfit 0.00 +trainability 0.00 +slope -2.803E-083.624E +09 +3.636E +09 +4:dataset_OR(1,500,0.8) +mlp_brick(2,2):15 +Figure 3: To check the convergence of the models, a PDF file can be generated containing all +the configuration curves of the different selected models. The different indicators +are in the legend (over-fitting, trainability and slope). The train and test loss +curves are represented by the solid blue line and dashed red line, respectively. +With this format and specific examples, one quickly identifies that run 1 converges +to too high values, run 2 overfits, run 3 is likely not converged yet, and run 4 +shows excellent convergence. +This presentation allows displaying all the available information of a pipeline’s results, mak- +ing it extremely convenient to visually and finely compare different hyper-parametrizations. +However, probability distribution are poorly suited for comparison of a large amount of +models at the same time. +When one wants to compare several kinds of models, line plots are particularly useful, +as each curve condenses the most information into single data points. Figure 5 depicts +two examples of such meta-plots, comparing here two otherwise identical MLPs that differ +only by their shape (funnel versus brick). Here, each point is an average over all strictly +identical pipelines, whereas the error bars represent one standard deviation. As illustrated +on this figure, it is often interesting to choose the number of parameters as abscissa, in +order to compare the quality of the models depending on their inherent complexity. Indeed, +the number of parameters is a good indication of the needed time for training and of the +needed resources to implement the network on a resource-constrained environment (FPGA +or neuromorphic chips, for instance). As an illustration, one quickly reads from Fig. 5 that +for this data set, brick-shaped MLPs clearly outperform the funnel-shaped ones. +8 + +MOSAIC, a comparison framework for machine learning models +21 +37 +56 +nb_params +0.002 +0.003 +0.004 +0.005 +0.006 +0.007 +0.008 +test_loss +dataset_OR(1,30,0.8) | mlp_funnel(3,3) +dataset_OR(1,30,0.8) | mlp_funnel(3,2) +dataset_OR(1,30,0.8) | mlp_funnel(3,4) +21 +37 +57 +nb_params +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +0.0035 +0.0040 +0.0045 +test_loss +dataset_OR(1,30,0.8) | mlp_brick(3,2) +dataset_OR(1,30,0.8) | mlp_brick(3,4) +dataset_OR(1,30,0.8) | mlp_brick(3,3) +Figure 4: Example of comparison plots (“meta-plot”). The abscissa is the number of pa- +rameters of the model, and the ordinate is the error made on the test data set. +Such plots allow quickly comparing in detail how the performances of a given +class of models, here funnel-shaped (left) and brick-shaped (right) MLPs, evolve +with the number of parameters. +20 +25 +30 +35 +40 +45 +50 +55 +nb_params +0.002 +0.004 +0.006 +0.008 +test_loss +dataset_OR(1,0.8) | mlp_brick() +dataset_OR(1,0.8) | mlp_funnel() +20 +25 +30 +35 +40 +45 +50 +55 +nb_params +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +overfit +dataset_OR(1,0.8) | mlp_brick() +dataset_OR(1,0.8) | mlp_funnel() +Figure 5: Example of comparison plots. The abscissa is the number of parameters of the +model, and the ordinate is the error made on the test data set (left) and the +overfit (right). The style of the error (left: bars, right: filled) is controlled by a +parameter in the .ini configuration file. Note that in the captions, the section +of the pipeline describing the model’s length and width is automatically adjusted +to the plotted data. +To get meaningful results from these plots, it is often useful to group different models, +or different values of hyperparameters, when studying their performance. This allows a +finer selection of the hyperparameters and their values. To do this, a special parameter, +named key, can be added in the meta-plots configuration file (see A.2). The keys are regular +9 + +Papin, Beaujeault-Taudi`ere and Magniette +expressions that are stored in the database. This structure eases the grouping by simplifying +the SQL request. In Figure 5, for example, the two plots are obtained by displaying on a +single figure pipelines corresponding to different depth and shapes. +The implemented indicators can be inoperant in very specific conditions. In order to +allow any specific case, it is possible to choose any field from the database that has been +filled by any class during the call of the info method. This way, any specific analysis can +be performed without restriction of the predefined indicators. +6. Installation and use of the MOSAIC program +MOSAIC is freely available as a Python package from the PyPI package manager (pyp), +which eases its installation. The PyPI page of the project is https://pypi.org/project/ml- +mosaic/. The framework provides a complete text interface for each functionality, includ- +ing pipelines generation, run start and monitoring, re-running of incomplete trainings and +graphics generation (basic and advanced). A full documentation of the interface and its +multiple parameters, and a tutorial explaining step by step how to set up a comparison +experiment, are available on the project’s web page, https://llrogcid.in2p3.fr/the-mosaic- +framework/. Both are also present into the downloaded package, as doc.md and tutorial.md. +7. Conclusion and perspectives +MOSAIC is an open-source framework designed to carry out compared performance studies +on machine learning models. It provides all the necessary tools to integrate the testing +logic in the long term of a feasibility study. It also includes advanced tools to compare +the models and select the optimal ones. Such a key-in-hand framework should be useful to +a wide range of users, ranging from novices to experimented practitioners, by drastically +reducing the amount of time spent in developing and benchmarking their machine learning +programs. +In future releases, an important improvement will be to support the use of clusters +through calls to batch systems. We are presently working on the integration of the condor +(Thain et al., 2005) system to distribute the process over a GPU-equiped cluster. 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Example of configuration files +The following two sections give exhaustive examples of configuration files. +A.1 Pipeline configuration file +[PROCESS] +lr = 1e-2 +epochs = 200 +loss_function = MSELoss +data_scheme = dataset_generator, data_augmentation, data_transformation +pipeline_scheme = convolution, readout, mlp +run_files_path = .runs +[MONITOR] +need_gpu = True +gpu_available = cuda:1, cuda:2 +nb_processus = 8 +multiplicity = 4 +cache_database_path = ./cache.db +cache_size = 1G +[dataset_gen] +type = dataset_generator +class = dataset_generator +path_to_class = ./path/to/dataset/dataset_generator.py +batch_size = 64 +data_size = 10000 +train_prop = 0.9 +key = data_{class} +[data_augmentation] +type = data_augmentation +class = data_augmentation +path_to_class = ./path/to/data_augmentation/data_augmentation.py +data_size = 1000 +key = data_{class} +[data_transformation] +type = data_transformation +class = transform +path_to_class = ./path/to/data_transformation/data_transformation.py +style = normalisation, standardisation +key = data_{class} +[convolution] +13 + +Papin, Beaujeault-Taudi`ere and Magniette +type = convolution +class = convolution +path_to_class = ./path/to/convolution/convolution.py +pooling = True, False +nb_convolution = {3, 5}, 8 +key = conv_{class} +[readout_1] +type = readout +class = simple_readout +path_to_class = ./path/to/readout/readout.py +key = readout_{class} +[readout_2] +type = readout +class = complex_readout +path_to_class = ./path/to/readout/readout.py +key = readout_{class} +[mlp_funnel] +type = mlp +class = mlp_funnel +path_to_class = ./path/to/mlp/mlp_funnel.py +length = 5, {6-8} +width = {2-4} +key = mlp_{class} +[mlp_brick] +type = mlp +class = mlp_brick +path_to_class = ./path/to/mlp/mlp_brick.py +length = 5, 6, 7, 8 +width = {2-4} +key = mlp_{class} +A.2 Meta-plots configuration file +[global] +file_title = output.plot +[plot_1] +abscissae = nb_params +ordinates = test_loss +include_keys = class, data_size, length +14 + +MOSAIC, a comparison framework for machine learning models +include_values = [dataset_OR], [30, 60], 3 +excludes = width, length, data_size +plot_type = line +errorbars_style = bars +15 + diff --git a/cNFPT4oBgHgl3EQfBjTq/content/tmp_files/load_file.txt b/cNFPT4oBgHgl3EQfBjTq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8bc00ccc44867928b95a721c24e201f139fdf2b7 --- /dev/null +++ b/cNFPT4oBgHgl3EQfBjTq/content/tmp_files/load_file.txt @@ -0,0 +1,616 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf,len=615 +page_content='MOSAIC, a comparison framework for machine learning models Matt´eo Papin matpapin0@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='com ´Ecole 42, 75017 Paris, France Yann Beaujeault-Taudi`ere yann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='beaujeault-taudiere@ijclab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='fr Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Laboratoire Leprince-Ringuet (LLR), ´Ecole polytechnique, CNRS/IN2P3, 91120 Palaiseau, France Fr´ed´eric Magniette frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='magniette@llr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='fr Laboratoire Leprince-Ringuet (LLR), ´Ecole polytechnique, CNRS/IN2P3, 91120 Palaiseau, France Abstract We introduce MOSAIC, a Python program for machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Our framework is developed with in mind accelerating machine learning studies through making imple- menting and testing arbitrary network architectures and data sets simpler, faster and less error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' MOSAIC features a full execution pipeline, from declaring the models, data and related hyperparameters within a simple configuration file, to the generation of ready- to-interpret figures and performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It also includes an advanced run manage- ment, stores the results within a database, and incorporates several run monitoring options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Through all these functionalities, the framework should provide a useful tool for researchers, engineers, and general practitioners of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' MOSAIC is available from the dedicated PyPI repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Keywords: Open-source software, network optimization, classification, regression, Python, neural networks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Introduction Conceiving optimal machine learning models, especially with artificial neural networks, is known to be a complex art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Although substantial work has been performed, not only on the mathematical understanding of neural networks’ (NNs) approximation power (Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Cybenko, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' K˚urkov´a, 1991, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Rolnick and Tegmark, 2018), but also on the heuristic (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Ojha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2017) and numerical (Sagun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2018) sides, no practically useful bounds, nor any useful recipe, is available in order to a priori construct NNs that achieve a pre-determined degree of performance for a given class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' There is no straightforward method to choose the hyperparameters of a neural network to optimize the number of used parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In practice, given a machine learning task, only prior experience give clues on which models should be used, and how the set of hyperparameters must be tuned to reach good (let alone optimal) performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For a simple model like a multi-layer perceptron (MLP) (Minsky and Papert, 1969), the shape of the network, the number of hidden layers and the number of neurons inside each of them has a great influence on the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For more complicated techniques, such as convolutional neural networks (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 1989, 1998), the number of hyperparameters increases as new quantities come to play, such kernel size, padding, stride, pooling function, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Very often, the strategy consists in choosing enormous networks, typically using millions of parameters, to ©2023 Matt´eo Papin, Yann Beaujeault-Taudi`ere and Fr´ed´eric Magniette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='12986v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='LG] 30 Jan 2023 Papin, Beaujeault-Taudi`ere and Magniette solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Although such large number of parameters is probably too much in most situations, this gives an insurance of bringing enough expressive power to tackle the complexity of the targeted problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The problem of this “brute-force” approach is that it requires a lot of labeled input data in order to truly constrain the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Furthermore, this introduces the need for a lot of computing resources along the training process and the use of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In addition, the difficulty of optimizing a neural network increases dramatically with its size, further augmenting the computational resources in order to attain satisfactory performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' As shown on the left part of Figure 1, when a statistical model is trained on data, two kinds of error occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' These are driven by the respective complexity of the data and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' If the model is simpler than the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', the number of parameter is smaller than what is required in order to satisfactorily learn the data, a bias error is pre- eminent, due to the model underfitting the data by learning only its roughest features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' On the opposite, in the case of too complex models, a new type of error appears, the so-called variance error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It corresponds to the parameters being too numerous with respect to the data size, so that they end up learning irrelevant features of the training data set, such as noise or unnecessarily moments of the data distribution, which in turn deteriorates the generalization power of the network, causing the test losses to rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' An optimal model is in the middle, where the two kinds of errors balance and result in a minimal total error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This is called the bias-variance tradeoff (Belkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Thus, the problem of using models that are too large with respect to the data is the induction of large variance of the validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The effect of this overfitting can be seen in the middle panel of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Most of the modern implementations fight against overfitting by using a tremendous quantity of data to lower the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' As seen at the right of Figure 1, in that case, the variance becomes negligible and, provided the model is sufficiently expressive, it can effectively handle the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Model complexity 0 2 4 6 8 10 Error Total Bias Variance Optimum Epochs 0 2 4 6 8 10 Error Train error Test error Model complexity 0 2 4 6 8 10 Error Total Bias Variance Figure 1: Schematic illustration of the performance of NNs’ losses according to the relative complexities of the model and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Left: bias-variance tradeoff, typical of a situation where data complexity ≫ model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Middle: overfitting (model complexity ≫ data complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Right: well-balanced network and data (model complexity ∼ data complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 2 MOSAIC, a comparison framework for machine learning models In a context where the resources are limited, like when porting a network on reconfig- urable electronics (FPGA) or on huge data sets that requires long treatment times, it can be interesting to tune carefully the network in order to reduce the number of parameters without reducing the global performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Furthermore, in the case of more sophisticated techniques, like graph convolutional networks (GCN) (Kipf and Welling, 2016), another key problem is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Namely, that there are plenty of different algorithms available (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2019), all of them having properties that can hardly be predicted accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In that case, it is interesting to test extensively the different algorithms to choose the best one for a dedicated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In both cases, it is highly interesting to test a large collection of models on a data set and to compare their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This is a subject that is well-covered by the machine learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Plenty of documentation explains how different classes of models compare and their specificities ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' But the problem of most of the approaches consists in their “one-shot” aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Indeed, a set of concurrent models are tested and compared at the same time, and when a new model or data set is added to the collection, all the corresponding tests have to be run again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In this work, we try to ease to the maximum the solution of such problems, with in view a long-term approach through the development of useful software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' We have developed a framework, called MOSAIC, allowing the sequential test of various architectures on multiple data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It provides a database system to store the results and is well-adapted to test hypotheses all along a research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' All the models are evaluated with different sets of hyperparameters, and the run results are uploaded on-the-fly to a database for future comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Mainly based on PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2019), it is very generic and can be adapted almost effortlessly to any class of model and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Our framework also provides several tools to compare effectively the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The MOSAIC framework MOSAIC is an open-source Python framework which provides all the facilities to test exten- sively machine learning models on arbitrary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It is based on the concept of pipelines, which fully represent a treatment chain including data, data formatting, and model training, all of which possibly including different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' All the hyperparameters are also included in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' They are generated by a subpart of the system, via interpreting a configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Once generated, they will be executed, possibly in a parallel way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A standard training procedure is applied using an Adam optimizer (Kingma and Ba, 2017), and at every step, a training and testing error is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' At the end of the training, the performance and dif- ferent indicators are stored in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The model parameters and the learning curves are stored in output files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' During all the execution process, an advanced error monitoring is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Each pipeline can easily be executed multiple times to fill the database with different models and different hyperparameters values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' At any time, the analysis module can be used to extract the desired information from the database and produce comparison plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Figure 2 shows the entire workflow of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 3 Papin, Beaujeault-Taudi`ere and Magniette Figure 2: Global scheme of the MOSAIC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Pipelines are generated from a configu- ration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' These pipelines are executed in parallel, and the results of the training are stored in files and in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In a second step, an analysis module queries the database to produce the comparison plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Pipelines generation The pipelines are generated from a configuration file (see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' We use the ini format, where parameters are written in text within a section indicated by a word between square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Two mandatory sections describe the different parameters of the system itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The first one, named MONITOR, includes all the parameters for the run manager, includ- ing the requirement for GPUs, the level of parallelism, the cache path and size and the multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This last parameter allows executing multiple instances of the same pipeline in order to collect statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The second one, named PROCESS, gives the hyperparameters of the training: initial learning rate, loss function, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It also contains the paths to the classes implement- ing custom metrics, such as user-defined loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Two crucial elements of this section are the data and model schemes describing the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' They are a sequence of actions that respectively refer to the data set building pipeline and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For example, we could have a sequence ‘‘Data loading | enrichment | normalization’’ for the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Additionally, one may combine simple models into more complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For example ‘‘Convolution1 | pooling | Convolution2 | polling | readout | mlp’’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The dif- ferent parts of the pipeline scheme must be implemented as classes and have a dedicated section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' As an illustration, a section for a funnel-shaped MLP can be represented by [mlp_funnel] type = mlp class = mlp_funnel path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/mosaic/share/mlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py length = 4 width = {2-4},8 The type must be the name present in the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The class and path indicate what class to import.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The other parameters are specific to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In the simplest configurations, the values are a single number or string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It can also be, as is here the case for the “width” 4 Pipeline execution Result process files Configuration Pipeline Run file generator manager Plot module Pipeline Service Database files moduleMOSAIC, a comparison framework for machine learning models parameter, a range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In that case, multiples pipelines will be generated, that correspond to all the different values allowed by the range (here, 2, 3, 4, and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' One must be cautious when giving ranges to multiple parameters, because each cross-possibility among the different class-specific parameters will generate a pipeline, and their total number can grow quite large if several wide ranges are used simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Thus, for large-scale exploratory studies, it is desirable that the users already have some insight on the relevant ranges of each parameter, or coarse-grain the range of explored values, possibly iteratively refining them after assessing the performances with the help of the plotting tools provided by MOSAIC (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' All the generated pipelines are stored in JSON files to be executed directly or later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A convenient corollary of structuring the runs into pipelines is that it results in minimal length, non-ambiguous characterization of a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This is particularly useful for gener- ating the plot labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For instance, the four funnel-shaped MLPs defined in the example [mlp section] can be represented as mlp funnel(4,2), mlp funnel(4,3), mlp funnel(4,4) and mlp funnel(4,8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Run and performance indicators When the pipelines are generated, they can be run by the core of the framework, the run manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Every element of the pipeline must be implemented by a Python class providing the required methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The class API is very similar to the Torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='module class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A forward method provides the evaluation of the data by the model, and a backward method (which can be implicit) is used for computing the partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Another method should provide the parameter values of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Two methods implement the saving and loading of the model in a file (it can be any format, not only a PyTorch tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A last method provides information to the other elements of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' To execute a pipeline, every element is called sequentially, and a communication between them is organized following two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' First, the result of every element is sent directly to the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' There is no constraint on the format of this return value, except for the last element of the data set model, which must provide a training and testing data loaders (in PyTorch format).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Another way of communication between the elements is based on the info method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Any class can, via this method, output a dictionary of valued parameters which is made available to every subsequent classes, especially for initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For example, the data set class should provide the shape of the inputs for the initialization of the neural network input layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The names of the class objects must be standardized across the classes entering a pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' During its execution, the model defined by a given pipeline is trained following a tradi- tional training loop, led by an Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Every pipeline is an independent process, communicating with a service module to access to the different services (cache, database, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The run manager follows the progression of the processes and launches new ones as soon as computing resources are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' At the end of the training loop, various performance indicators, which we detail in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='1, are calculated and stored in the database for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 5 Papin, Beaujeault-Taudi`ere and Magniette 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='1 Performance indicators We have implemented six performance indicators in the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Each of these provides a valuable, global performance metric, and their collection can be used to efficiently compare the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In the following, the total number of epochs will be denoted n, a given epoch will be indexed by the integer e = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' , n, and the loss functions at the epoch e will be written Ltrain/test(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The first two metrics are the final test and train losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' These are the traditional, and usually considered most important, indicators to compare the performance of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Due to the random nature of the parameters’ initialization and the stochastic nature of the Adam optimizer, it is often interesting to try multiple times the same pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This allows users to acquire statistics over the (data, model) couple, as allowed by the multiplicity parameter described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' When the multiplicity of a given pipeline is not too large, it is often sufficient to compare only the train and test losses across its different executions in order to identify well-performing runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The third indicator we have implemented measures the overfitting of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' We have chosen the difference of the last train and test error, normalized by the global range of these errors: overfitting = | Ltest(n) − Ltrain(n) | max � Ltrain/test(n) � − min � Ltrain/test(n) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' (1) The smaller the indicator is, the lower the overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Fourth, the framework also implements a convergence rate indicator, which gives clues on whether the convergence is reached after the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For this, we consider the last ten percents of the training loss curve, and carry out two operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' First, the gradients over these points is calculated using a first-order finite difference, and the average gradient is computed as a difference of averages: slope(e) = Ltrain(e) − Ltrain(e − 1), (2) slope = 1 ⌈n/20⌉ � � � n � e=n−⌈n/20⌉ Ltrain(e) − n−⌈n/20⌉ � e=n−⌈n/10⌉ Ltrain(e) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' (3) The slopes thus obtained give a rough idea about the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Averaging over the two halves of the last ten percents smoothens sudden jumps of the loss that might occur from one epoch to the following one, which ultimately increases the quality of this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In addition, it guarantees that the slope is asymptotically vanishing, which reflects that beyond a certain number of epochs, it is reasonable to expect that the networks’ performances will on average stay constant up to negligible variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Second, since the gradients altogether incorporate information about the possible fluctuations of the loss during the last epochs, we split them into two parts, according to whether they are larger or smaller than the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' second) ones correspond to relative increases (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' decreases) of the loss, and are thus counted as statistical biases of the final slope towards larger (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' smaller) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For each subset, the standard deviation is calculated: 6 MOSAIC, a comparison framework for machine learning models σ+ = σ({slope(k) such that slope(k) ≥ slope and k ≥ n − ⌈n/10⌉}), (4) σ− = σ({slope(k) such that slope(k) ≤ slope and k ≥ n − ⌈n/10⌉}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' (5) This indicator is finally presented as slope σ+ σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A negative value indicates that the loss function is likely to decrease if a few epochs are added, whereas a positive or very small slope hints that the model is unlikely to improve and might have reached a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In the former case, a re-run mechanism allows continuing the training by restarting the corresponding pipeline from the last obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Fifth, we have implemented a trainability indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The idea is to measure how fast a given instance of the model has been trained on the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' For this purpose, we consider the integral of the train loss curve with respect to the x-axis: trainability = n � e=1 Ltrain(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' (6) If the integral is small, it means that the model has learned fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Conversely, a high trainability hints that the model is learning the data slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This criterion can be used to choose between two models (or classes thereof) otherwise presenting the same performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' If the trainability is lower for one of them, it has been easier to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Following the same idea of selecting equally performing models, a last indicator is simply the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This indicator is often of lesser importance since it does not reflect the bare performances of the models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' moreover, it depends on the machine running the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' If the test is performed on a single machine or in a homogeneous environment, then this value bears more relevance, and can be used in contexts where the full execution of the pipeline is critical, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', when the quantity of data to treat becomes a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Result analysis All the results of the runs are stored in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' We have chosen the SQLite3 file format, in order to benefit from the power of the SQL language and the flexibility of the file storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' SQL allows us to be very accurate on the choice of the models that participate in our studies, up to the point one can easily and quickly query batches of data or cherry-pick specific runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The analysis modules provide two kind of plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The first one is a collection of train and test losses plots along the epochs, as represented on Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Each plot also includes the performance metrics introduced in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='1, except for the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The plots can be selected by a SQL request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' By default, all the plots are grouped in a single PDF file, but the number of pages can be very important if the study includes numerous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In particular, overfitting and convergence indicators can be used to select problematic trainings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The other kind of plot produced by MOSAIC is called meta-plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' They are composed of different curves representing any indicator in ordinate and any parameter in abscissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Figure 4 shows an example of such format, in the form of violin plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In such plots, the discrete re- sults condition a Gaussian kernel density distribution (Hastie and Tibshirani, 2017), which allows estimating a continuous probability density function from a finite number of runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 7 Papin, Beaujeault-Taudi`ere and Magniette 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='35 overfit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='16 trainability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='21 slope -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='374E-072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='701E 08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='638E 08 1:dataset_OR(1,20,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) mlp_brick(100,2):603 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='20 overfit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='97 trainability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='22 slope -1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) mlp_brick(2,2):15 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='15 overfit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='00 trainability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='00 slope -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='803E-083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='624E 09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='636E 09 4:dataset_OR(1,500,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) mlp_brick(2,2):15 Figure 3: To check the convergence of the models, a PDF file can be generated containing all the configuration curves of the different selected models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The different indicators are in the legend (over-fitting, trainability and slope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The train and test loss curves are represented by the solid blue line and dashed red line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' With this format and specific examples, one quickly identifies that run 1 converges to too high values, run 2 overfits, run 3 is likely not converged yet, and run 4 shows excellent convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This presentation allows displaying all the available information of a pipeline’s results, mak- ing it extremely convenient to visually and finely compare different hyper-parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' However, probability distribution are poorly suited for comparison of a large amount of models at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' When one wants to compare several kinds of models, line plots are particularly useful, as each curve condenses the most information into single data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Figure 5 depicts two examples of such meta-plots, comparing here two otherwise identical MLPs that differ only by their shape (funnel versus brick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Here, each point is an average over all strictly identical pipelines, whereas the error bars represent one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' As illustrated on this figure, it is often interesting to choose the number of parameters as abscissa, in order to compare the quality of the models depending on their inherent complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Indeed, the number of parameters is a good indication of the needed time for training and of the needed resources to implement the network on a resource-constrained environment (FPGA or neuromorphic chips, for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' As an illustration, one quickly reads from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 5 that for this data set, brick-shaped MLPs clearly outperform the funnel-shaped ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 8 MOSAIC, a comparison framework for machine learning models 21 37 56 nb_params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='008 test_loss dataset_OR(1,30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_funnel(3,3) dataset_OR(1,30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_funnel(3,2) dataset_OR(1,30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_funnel(3,4) 21 37 57 nb_params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='0045 test_loss dataset_OR(1,30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_brick(3,2) dataset_OR(1,30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_brick(3,4) dataset_OR(1,30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_brick(3,3) Figure 4: Example of comparison plots (“meta-plot”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The abscissa is the number of pa- rameters of the model, and the ordinate is the error made on the test data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Such plots allow quickly comparing in detail how the performances of a given class of models, here funnel-shaped (left) and brick-shaped (right) MLPs, evolve with the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 20 25 30 35 40 45 50 55 nb_params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='008 test_loss dataset_OR(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_brick() dataset_OR(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_funnel() 20 25 30 35 40 45 50 55 nb_params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='012 overfit dataset_OR(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_brick() dataset_OR(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='8) | mlp_funnel() Figure 5: Example of comparison plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The abscissa is the number of parameters of the model, and the ordinate is the error made on the test data set (left) and the overfit (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The style of the error (left: bars, right: filled) is controlled by a parameter in the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='ini configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Note that in the captions, the section of the pipeline describing the model’s length and width is automatically adjusted to the plotted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' To get meaningful results from these plots, it is often useful to group different models, or different values of hyperparameters, when studying their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This allows a finer selection of the hyperparameters and their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' To do this, a special parameter, named key, can be added in the meta-plots configuration file (see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The keys are regular 9 Papin, Beaujeault-Taudi`ere and Magniette expressions that are stored in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This structure eases the grouping by simplifying the SQL request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In Figure 5, for example, the two plots are obtained by displaying on a single figure pipelines corresponding to different depth and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The implemented indicators can be inoperant in very specific conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In order to allow any specific case, it is possible to choose any field from the database that has been filled by any class during the call of the info method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' This way, any specific analysis can be performed without restriction of the predefined indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Installation and use of the MOSAIC program MOSAIC is freely available as a Python package from the PyPI package manager (pyp), which eases its installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The PyPI page of the project is https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='org/project/ml- mosaic/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The framework provides a complete text interface for each functionality, includ- ing pipelines generation, run start and monitoring, re-running of incomplete trainings and graphics generation (basic and advanced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A full documentation of the interface and its multiple parameters, and a tutorial explaining step by step how to set up a comparison experiment, are available on the project’s web page, https://llrogcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='fr/the-mosaic- framework/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Both are also present into the downloaded package, as doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='md and tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Conclusion and perspectives MOSAIC is an open-source framework designed to carry out compared performance studies on machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It provides all the necessary tools to integrate the testing logic in the long term of a feasibility study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It also includes advanced tools to compare the models and select the optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Such a key-in-hand framework should be useful to a wide range of users, ranging from novices to experimented practitioners, by drastically reducing the amount of time spent in developing and benchmarking their machine learning programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In future releases, an important improvement will be to support the use of clusters through calls to batch systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' We are presently working on the integration of the condor (Thain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=', 2005) system to distribute the process over a GPU-equiped cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' It should speed up considerably the global process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Acknowledgments The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-21-CE31-0030, project OGCID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' The authors acknowledge finan- cial support from the P2IO LabEx (ANR-10-LABX-0038).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 10 MOSAIC, a comparison framework for machine learning models References URL https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Belkin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='00596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Tanida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Itoh, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Ichioka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Shift-invariant pattern recognition neural network and its optical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' In Proceedings of annual conference of the Japan Society of Applied Physics, pages 2147–2151, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' 12 MOSAIC, a comparison framework for machine learning models Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' Example of configuration files The following two sections give exhaustive examples of configuration files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='1 Pipeline configuration file [PROCESS] lr = 1e-2 epochs = 200 loss_function = MSELoss data_scheme = dataset_generator, data_augmentation, data_transformation pipeline_scheme = convolution, readout, mlp run_files_path = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='runs [MONITOR] need_gpu = True gpu_available = cuda:1, cuda:2 nb_processus = 8 multiplicity = 4 cache_database_path = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='db cache_size = 1G [dataset_gen] type = dataset_generator class = dataset_generator path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/dataset/dataset_generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py batch_size = 64 data_size = 10000 train_prop = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='9 key = data_{class} [data_augmentation] type = data_augmentation class = data_augmentation path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/data_augmentation/data_augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py data_size = 1000 key = data_{class} [data_transformation] type = data_transformation class = transform path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/data_transformation/data_transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py style = normalisation, standardisation key = data_{class} [convolution] 13 Papin, Beaujeault-Taudi`ere and Magniette type = convolution class = convolution path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/convolution/convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py pooling = True, False nb_convolution = {3, 5}, 8 key = conv_{class} [readout_1] type = readout class = simple_readout path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/readout/readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py key = readout_{class} [readout_2] type = readout class = complex_readout path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/readout/readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py key = readout_{class} [mlp_funnel] type = mlp class = mlp_funnel path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/mlp/mlp_funnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py length = 5, {6-8} width = {2-4} key = mlp_{class} [mlp_brick] type = mlp class = mlp_brick path_to_class = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='/path/to/mlp/mlp_brick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='py length = 5, 6, 7, 8 width = {2-4} key = mlp_{class} A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='2 Meta-plots configuration file [global] file_title = output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} +page_content='plot [plot_1] abscissae = nb_params ordinates = test_loss include_keys = class, data_size, length 14 MOSAIC, a comparison framework for machine learning models include_values = [dataset_OR], [30, 60], 3 excludes = width, length, data_size plot_type = line errorbars_style = bars 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfBjTq/content/2301.12986v1.pdf'} diff --git a/ddE2T4oBgHgl3EQfbAcC/content/tmp_files/2301.03879v1.pdf.txt b/ddE2T4oBgHgl3EQfbAcC/content/tmp_files/2301.03879v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b86621af1f2c08bb2090306693d3f77bb511ebc --- /dev/null +++ b/ddE2T4oBgHgl3EQfbAcC/content/tmp_files/2301.03879v1.pdf.txt @@ -0,0 +1,598 @@ +arXiv:2301.03879v1 [hep-lat] 10 Jan 2023 +KEK-TH-2488, RIKEN-iTHEMS-Report-23 +Numerical studies on the finite-temperature CP +restoration in 4D SU(N) gauge theory at 휽 = 흅 +Akira Matsumoto,푎,∗ Kohta Hatakeyama,푏 Mitsuaki Hirasawa,푐 Masazumi Honda,푎,푑 +Jun Nishimura푏,푒 and Atis Yosprakob 푓 +푎RIKEN iTHEMS, +2-1 Hirosawa, Wako, Saitama 351-0198, Japan +푏KEK Theory Center, High Energy Accelerator Research Organization, +1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan +푐Sezione di Milano Bicocca, Istituto Nazionale di Fisica Nucleare, +Piazza della Scienza, 3, I-20126 Milano, Italy +푑Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, +Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan +푒The Graduate University for Advanced Studies, SOKENDAI, +1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan +푓 Department of Physics, Niigata University, Niigata 950-2181, Japan. +E-mail: akira.matsumoto@riken.jp, khat@post.kek.jp, +mitsuaki.hirasawa@mib.infn.it, masazumi318@gmail.com, jnishi@post.kek.jp, +ayosp@phys.sc.niigata-u.ac.jp +Recent studies on the ’t Hooft anomaly matching condition have suggested a nontrivial phase +structure in 4D SU(푁) gauge theory at 휃 = 휋. In the large-푁 limit, it has been found that CP +symmetry at 휃 = 휋 is broken in the confined phase, while it restores in the deconfined phase, which +is indeed one of the possible scenarios. However, at small 푁, one may find other situations that are +consistent with the consequence of the anomaly matching condition. Here we investigate this issue +for 푁 = 2 by direct lattice calculations. The crucial point to note is that the CP restoration can be +probed by the sudden change of the tail of the topological charge distribution at 휃 = 0, which can +be seen by simulating the theory at imaginary 휃 without the sign problem. Our results suggest that +the CP restoration at 휃 = 휋 occurs at temperature higher than the deconfining temperature unlike +the situation in the large-푁 limit. +The 39th International Symposium on Lattice Field Theory (Lattice2022), +8-13 August, 2022 +Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto +1. +Introduction +The non-perturbative effect of the topological theta term in quantum field theories has been +studied as a long-standing problem. Recently, the phase structure of 4D pure Yang-Mills (YM) +theory with a theta term has attracted a lot of attention. There was a novel progress on application +of ’t Hooft anomaly matching [1, 2], which suggests that the 4D SU(푁) pure YM theory cannot +have a unique trivial vacuum at 휃 = 휋. Indeed, this statement is consistent with the known phase +diagram for large 푁, where the CP symmetry at 휃 = 휋 is spontaneously broken in the confined +phase. On the other hand, the phase diagram for small 푁, in particular 푁 = 2, is not determined +yet. It is possible that the SU(2) YM theory has a qualitatively different phase structure. +Thus, it is an interesting challenge to investigate the phase structure by a first-principle method. +Since the effect of the theta term is genuinely non-perturbative, the SU(푁) YM theory with a theta +term should be analyzed by non-perturbative methods. However, usual Monte Carlo simulations of +the lattice gauge theory including the theta term is difficult due to the sign problem. +We propose a new method to probe the critical behavior at 휃 = 휋 based on the topological +charge distribution 휌(푞) at 휃 = 0. The crucial point of the method is that the expectation value ⟨푄⟩휃 +of topological charge for any 휃 is completely determined by the distribution 휌(푞). Thanks to this +property, we can investigate the behavior of ⟨푄⟩휃 indirectly by the information at 휃 = 0. In order +to see the temperature dependence of the distribution 휌(푞) clearly, we introduce the imaginary 휃 +parameter, which can enhance the tail structure of 휌(푞). +2. +Identifying the CP restoration +The application of ’t Hooft anomaly matching condition [1, 2] to the 4D pure YM theory +suggests that the phase structure at 휃 = 휋 should be nontrivial. There are a lot of possible phase +structures which agree with this condition. Here we consider two kinds of phase transition. One +is the deconfinement transition at 푇 = 푇dec(휃 = 휋), which corresponds to breaking of 푍푁 center +symmetry. Note that the deconfining temperature 푇dec(휃) depends on 휃 in general. The other +transition is the restoration of CP symmetry at 푇 = 푇CP, which is broken at low temperature. +For large 푁, these two transitions occur at the same temperature 푇CP = 푇dec(휋). Namely, the CP +symmetry is recovered simultaneously with the deconfinement transition. In this case, either the +푍푁 center symmetry or the CP symmetry is broken at any temperature. Thus, it is consistent with +the anomaly matching condition. +It is interesting to explore whether the theory with 푁 = 2 has a similar phase diagram. In fact, +the numerical study of 4D SU(2) YM theory by the subvolume method [3, 4] shows an indication +of the CP broken phase at low temperature. It is also confirmed that the instanton gas phase, which +is CP symmetric, appears at high temperature. Thus, the restoration of CP symmetry is expected +to occur also for 푁 = 2. However, the relation between the two critical temperatures 푇CP and +푇dec(휋) can be different. The anomaly matching condition for these two temperatures requires that +푇CP ≥ 푇dec(휋). The reason is that the CP symmetry can be broken not only in the confined phase +but also in the deconfined phase. The overlap of the CP broken phase and the 푍2 broken phase is +allowed. There is a related study of these two critical temperatures by using the super YM theory +[5]. Interestingly, the result shows 푇CP > 푇dec(휋) only for 푁 = 2, but 푇CP = 푇dec(휋) for 푁 ≥ 3. +2 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto +It is worth trying to numerically investigate the CP restoration temperature 푇CP for the 4D SU(2) +YM theory and compare it with the deconfining temperature 푇dec(휋). However, the usual Monte +Carlo simulation at 휃 = 휋 suffers from the sign problem since the theta term is purely imaginary. +In this section we introduce a new method to determine 푇CP without direct simulation at 휃 = 휋. +First, we explain the property of topological charge 푄. Since the topological charge is a CP +odd operator, its expectation value ⟨푄⟩휃 can be an order parameter of CP symmetry. +⟨푄⟩휃 = −푖 휕 +휕휃 log 푍휃 +(1) +If CP symmetry at 휃 = 휋 is spontaneously broken, ⟨푄⟩휃 should be discontinuous there: +Δ푄 = +���⟨푄⟩휃=휋−휖 − ⟨푄⟩휃=휋+휖 +��� +� +> 0 +: CP broken, += 0 +: CP restored. +(2) +Thus, the CP restoration temperature 푇CP can be regarded as a temperature at which Δ푄 vanishes. +To determine 푇CP, we need to investigate the temperature dependence of Δ푄. However, it is difficult +to directly evaluate Δ푄 due to the sign problem. But we can also study it from another direction. +Let us note that the partition function 푍휃 and the topological charge distribution 휌(푞) at 휃 = 0 are +related via +휌(푞) = 1 +푍0 +∫ +푑퐴 훿(푞 − 푄)푒−푆푔 = 1 +푍0 +∫ +푑휃 +2휋 푒−푖휃푞 푍휃. +(3) +Since the distribution 휌(푞) is a Fourier transform of the partition function 푍휃, +푍휃 = +∫ +푑퐴 푒−푆푔+푖휃푄 = 푍0 +∫ +푑푞 푒푖휃푞 휌(푞), +(4) +we find that the expectation value ⟨푄⟩휃 at any 휃 is completely determined by 휌(푞) as +⟨푄⟩휃 = −푖 휕 +휕휃 log +∫ +푑푞 푒푖휃푞 휌(푞) = +∫ +푑푞 푞푒푖휃푞 휌(푞) +∫ +푑푞 푒푖휃푞 휌(푞) +. +(5) +This is nothing but the reweighting formula by using the information at 휃 = 0. Thus, to calculate +⟨푄⟩휃 around 휃 ∼ 휋, we need exponentially large amount of statistics. However, our goal is not to +determine the complete 휃 dependence of ⟨푄⟩휃 but to probe the critical temperature 푇CP. In fact, we +do not need the complete information of 휌(푞) in that case. It is enough to determine whether Δ푄 +is zero or not from 휌(푞). +In this study, we propose to use the expectation value ⟨푄⟩ ˜휃 of the topological charge at an +imaginary theta 휃 = 푖 ˜휃 ( ˜휃 ∈ R) to probe the critical behavior; +⟨푄⟩ ˜휃 = 1 +푍 ˜휃 +∫ +푑퐴 푄푒−푆푔− ˜휃푄 = +∫ +푑푞 푞푒− ˜휃푞휌(푞) +∫ +푑푞 푒− ˜휃푞휌(푞) +. +(6) +In practice, we normalize it by the topological susceptibility +휒0 = 1 +푉 ⟨푄2⟩휃=0 +(7) +3 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto +at 휃 = 0 and the volume 푉, so that we just measure the ratio of two independent observables. +⟨푄⟩ ˜휃 +휒0푉 = ⟨푄⟩ ˜휃 +⟨푄2⟩0 +(8) +Let us discuss the behavior of this observable in some well known models. The first example is the +instanton gas model, for which the free energy is obtained as +퐹휃 = − log 푍휃 = 휒0푉(1 − cos 휃). +(9) +The 휃-dependence of ⟨푄⟩ 휃 /휒0푉 for real 휃 is given by the sine function +⟨푄⟩ 휃 +휒0푉 = 푖 sin 휃, +(10) +which indicates that CP symmetry at 휃 = 휋 is not broken. +Correspondingly, the imaginary-휃 +dependence turns out to be the hyperbolic sine function. +⟨푄⟩ ˜휃 +휒0푉 = − sinh ˜휃 +(11) +The second example is the Gaussian model +퐹휃 = 1 +2 휒0푉 min +푛 (휃 − 2휋푛)2, +(12) +which is known to be realized for large 푁 at low temperature. The real-휃 dependence of ⟨푄⟩ 휃 /휒0푉 +is given by +⟨푄⟩ 휃 +휒0푉 = 푖(휃 +mod 2휋) +(13) +for 푉 ≫ 1, which indicates that the CP is broken. For imaginary 휃, we find the linear behavior. +⟨푄⟩ ˜휃 +휒0푉 = − ˜휃 +(14) +We can see the clear difference between the behaviors of ⟨푄⟩ ˜휃 /휒0푉 for these two models. It behaves +as − sinh ˜휃 for the instanton gas model (CP restored), while it behaves as − ˜휃 for the Gaussian model +(CP broken). +Although the 4D SU(2) YM theory will not be as simple as these models, this +observable is still useful to investigate the CP restoration. In fact, the expectation value ⟨푄⟩ ˜휃 for +imaginary 휃 is sensitive to the tail of the distribution of 휌(푞). The imaginary theta term enhances +the contribution of large-푞 sectors because of the factor 푒− ˜휃푞 in the integrand of (6). Note that, for +these two examples, the tail of 휌(푞) behaves for 푞 ≫ 1 as follows: +휌(푞) ∼ + + +exp +� +−푞 log +2푞 +휒0푉 +� +: instanton gas, +exp +� +− +푞2 +2휒0푉 +� +: Gaussian. +(15) +4 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto +3. +4D SU(2) gauge theory with a theta term +In this study, we focus on the SU(2) pure Yang-Mills theory on the 4D Euclidean space. The +action for the gauge field 퐴푎 +휇 (푎 = 1, 2, 3) (휇 = 1, . . . , 4) is defined by +푆푔 = +1 +4푔2 +∫ +푑4푥 퐹푎 +휇휈퐹푎 +휇휈, +(16) +where 푔 is the coupling constant and 퐹푎 +휇휈 is the field strength. +퐹푎 +휇휈 = 휕휇퐴푎 +휈 − 휕휈퐴푎 +휇 − 휖 푎푏푐 퐴푏 +휇퐴푐 +휈 +(17) +The topological charge is given by +푄 = +1 +64휋2 +∫ +푑4푥 휖휇휈휌휎퐹푎 +휇휈퐹푎 +휌휎, +(18) +which takes an integer value on the compact space. +We introduce the topological theta term +푆휃 = −푖휃푄 with a parameter 휃 ∈ R, so that the total action is 푆 = 푆푔 + 푆휃. Since the partition +function +푍 = +∫ +D퐴 푒−푆푔+푖휃푄 +(19) +is invariant under the shift 휃 → 휃 + 2휋, the theory has 2휋 periodicity with respect to 휃. Since the +parameter 휃 flips its sign by the CP transformation 휃 → −휃, the theta term explicitly breaks the CP +symmetry for 휃 ≠ 0. However, thanks to the 2휋 periodicity, the CP symmetry exists also at 휃 = 휋. +Next, we define the lattice action for the numerical study. The gauge field is represented by the +link variable 푈푛,휇 ∈ SU(2). The index 푛 labels the lattice sites. The plaquette is given by +푃휇휈 +푛 += 푈푛,휇푈푛+ ˆ휇,휈푈† +푛+ ˆ휈,휇푈† +푛,휈, +(20) +where ˆ휇 represents the unit vector along the 휇-th direction. Then we define the plaquette action +with the lattice coupling constant 훽. +푆푔 = − 훽 +4 +� +푛 +� +휇≠휈 +Tr(푃휇휈 +푛 ) +(21) +Similarly, we can define the topological charge on the lattice by the so-called "clover leaf" formula, +[6] +푄cl = − +1 +32휋2 +� +푛 +1 +24 +±4 +� +휇,휈,휌,휎=±1 +˜휖휇휈휌휎Tr(푃휇휈 +푛 푃휌휎 +푛 ). +(22) +Here the orientationof the plaquette is extended tothe negative directions as well. Thecorresponding +anti-symmetric tensor ˜휖휇휈휌휎 also has negative indices, so that +1 = ˜휖1234 = − ˜휖2134 = − ˜휖−1234 = · · · . +(23) +It is known that the naively defined topological charge 푄cl does not take an integer value on the +lattice due to the discretization effect. In order to recover the topological property of the gauge +field, we need to eliminate short-range fluctuations. In fact, there are some smoothing techniques, +such as the gradient flow, stout smearing and so on. By using such a technique, we can define the +smeared topological charge so that it becomes close to an integer. In this study, we introduce the +stout smearing to the hybrid Monte Carlo simulation, which is discussed in section 4. +5 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto +4. +Stout smearing for the HMC +Since the CP symmetry at 휃 = 휋 is related to the 2휋 periodicity of 휃, the topological property +of the theory is essential in the study of the phase structure. Thus, we use the stout smearing [7] +to define the topological charge. In the hybrid Monte Carlo simulation, the drift force is used to +update the configuration. If the action has the theta term with the smeared topological charge, it +also contributes to the drift force. We can explicitly calculate the drift force from the smeared +topological charge by using stout smearing. In this section, we briefly review the stout smearing in +the hybrid Monte Carlo simulation. +Stout smearing is an iterative procedure to obtain the smeared link ˜푈푛,휇 starting from the +original link 푈푛,휇. We call the number of iterations 푁휌. +푈푛,휇 = 푈 (0) +푛,휇 → 푈 (1) +푛,휇 → · · · → 푈 (푁휌) +푛,휇 += ˜푈푛,휇. +(24) +In one (isotropic) smearing step from 푘 to 푘 + 1, the link variable 푈 (푘) +푛,휇 ∈ SU(2) is mapped to +푈 (푘+1) +푛,휇 +∈ SU(2) defined by following formulae: +푈 (푘+1) +푛,휇 += 푒푖푌푛,휇푈 (푘) +푛,휇, +(25) +푖푌푛,휇 = − 휌 +2 Tr(퐽푛,휇휏푎)휏푎, +(26) +퐽푛,휇 = 푈푛,휇Ω푛,휇 − Ω† +푛,휇푈† +푛,휇, +(27) +Ω푛,휇 = +� +휎(≠휇) +� +푈푛+ ˆ휇,휎푈† +푛+ ˆ휎,휇푈† +푛,휎 + 푈† +푛+ ˆ휇− ˆ휎,휎푈† +푛− ˆ휎,휇푈푛− ˆ휎,휎 +� +. +(28) +Here 휏푎 are the SU(2) generators in fundamental representation. The smearing step parameter +휌 > 0 should be chosen appropriately depending on the system. +In the hybrid Monte Carlo simulation, we obtain the smeared link ˜푈푛,휇 by this procedure, and +then we use ˜푈푛,휇 to calculate the topological charge (22) instead of the original link 푈푛,휇. The +topological charge given by the stout smearing +푄 := 푄cl( ˜푈) +(29) +is used in the theta term 푆휃 = −푖휃푄 as well as in measuring the observable. In the step of molecular +dynamics, we need to calculate the drift force +퐹푛,휇 = 푖휏푎퐷푎 +푛,휇푆휃 +(30) +from the theta term. Although 푆휃 is a complicated function of the original link variable 푈푛,휇, it is +possible to calculate the drift force by reversing the smearing steps (24). +5. +Result of the HMC +In this section, we show the result of the hybrid Monte Carlo simulation with the imaginary theta +term. For the stout smearing, we set 푁휌 = 40 and 휌 = 0.09 so that the topological charge is close +to an integer. In Fig. 1, we plot −⟨푄⟩ ˜휃/휒0푉 against ˜휃/휋 = 휃/푖휋 for various values of temperature +6 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +−〈Q〉 / χ0V +θ~ / π +T = 0.92Tc +T = 0.96Tc +T = 1.00Tc +T = 1.04Tc +T = 1.08Tc +T = 1.12Tc +T = 1.16Tc +T = 1.20Tc +θ~ +sinh θ~ +Nρ=40, ρ=0.09, V=203x5 +Figure 1: +The imaginary 휃 dependence of − ⟨푄⟩ ˜휃 /휒0푉 for various values of temperature in 0.92 ≤ +푇/푇dec(0) ≤ 1.20 with increments of 0.04. The green solid curve is obtained by the instanton gas approxi- +mation, which is valid at high temperature. The purple solid line is obtained by the Gaussian model. +in the range 0.9 ≤ 푇/푇dec(휃 = 0) ≤ 1.2. We found that, at high temperature, the data points are +consistent with the instanton gas approximation. On the other hand, the data points approach the +behavior of the Gaussian model at low temperature. It is convincing that the SU(2) YM theory +behaves as the instanton gas model at high temperature. However, it does not necessarily coincides +with the Gaussian model at low temperature since the situation of 푁 = 2 can be different from that +of large 푁. Nevertheless, this observable is suitable for probing the phase structure. Indeed, we can +see that the behaviors of ⟨푄⟩ ˜휃/휒0푉 change drastically slightly above the deconfining temperature +푇dec(0) at 휃 = 0. +In order to see the temperature dependence of −⟨푄⟩ ˜휃/휒0푉, we plot it against temperature at +fixed 휃/휋 = 0.75푖 in Fig. 2. The left figure is the result for 푉 = 163 × 5, and the right figure +is the result for 푉 = 203 × 5. The yellow curve shows the result of fitting by a cubic function +푓 (푥) = 푎푥3 + 푏푥2 + 푐푥 + 푑 where 푎, 푏, 푐 and 푑 are fitting parameters. The orange curve is the +derivative of 푓 (푥). We find that the derivative is the largest at around 푇peak ∼ 1.06푇dec(0). We also +find that the height of the peak grows as the spatial volume 푉s increases. +In Fig. 3(left), we plot the peak position 푇peak/푇dec(0) against 1/푉s obtained by the same +analysis for 휃/휋 = 0.6푖, 0.75푖 and 0.9푖. The significant volume dependence of 푇peak is not observed. +These results suggest that there is a phase transition around 푇/푇dec(0) ∼ 1.06. In Fig. 3(right), the +peak height of the fitting function is plotted against 푉1/3 +s +. This non-linear finite size scaling suggests +that the phase transition is of the second order or higher. +The existence of the transition indicates that the distribution 휌(푞) of the topological charge +changes drastically around푇/푇dec (0) ∼ 1.06. Assuming that CP symmetry at 휃 = 휋 is spontaneously +broken at low temperature, the drastic change of 휌(푞) should correspond to the critical behavior +of ⟨푄⟩ 휃=휋. Thus, we identify the critical temperature 푇peak as the CP restoration temperature 푇CP, +which suggests 푇CP > 푇dec(0). +7 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 + 5.5 + 0.85 0.9 0.95 + 1 + 1.05 1.1 1.15 1.2 1.25 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 +− < Q > / χ0V +f’(x) +T/Tdec(0) +f(x) +df(x)/dx +V = 163 x 5, θ / π = 0.75i + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 + 5.5 + 0.85 0.9 0.95 + 1 + 1.05 1.1 1.15 1.2 1.25 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 +− < Q > / χ0V +f’(x) +T/Tdec(0) +f(x) +df(x)/dx +V = 243 x 5, θ / π = 0.75i +Figure 2: The temperature dependence of − ⟨푄⟩ ˜휃 /휒0푉 at 휃/휋 = 0.75푖 for푉 = 163 ×5 (left) and 푉 = 203 ×5 +(right). We fit the data points by a cubic function 푓 (푥) = 푎푥3 + 푏푥2 + 푐푥 + 푑. The orange solid line shows +the result of fitting. The red solid line is derivative of 푓 (푥) with respect to 푥. + 1 + 1.02 + 1.04 + 1.06 + 1.08 + 1.1 + 1.12 + 1.14 +0.0x100 +1.0x10-4 +2.0x10-4 +3.0x10-4 +Tpeak / Tdec(0) +1 / Vs +θ / π = 0.60 i +θ / π = 0.75 i +θ / π = 0.90 i + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 0 + 4 + 8 + 12 + 16 + 20 + 24 + 28 +peak height +Vs +1/3 +θ / π = 0.60 i +θ / π = 0.75 i +θ / π = 0.90 i +0.80 Vs +1/3 +1.35 Vs +1/3 +1.74 Vs +1/3 +Figure 3: (left) The peak position 푇peak/푇dec(0) against 1/푉s for 휃/휋 = 0.6푖, 0.75푖 and 0.9푖, where 푉s is the +spacial lattice volume. (right) The peak height of 푓 ′(푥) is plotted against 푉1/3 +s +. The straight lines represent +the fit to the behavior 푎푉1/3 +s +. +6. +Summary +Recent studies on the ’t Hooft anomaly matching condition for the 4D SU(푁) gauge theory +have suggested that the phase structure at 휃 = 휋 should be nontrivial. For large 푁, it is known that +the CP symmetry at 휃 = 휋 is spontaneously broken in the confined phase, while it is restored in the +deconfined phase. However, for small 푁, a qualitatively different phase structure can be realized, +as long as the anomaly matching condition is satisfied. In this work, we investigated this issue for +푁 = 2 by hybrid Monte Carlo simulation of lattice gauge theory. We probed the restoration of the CP +symmetry by a sudden change of the topological charge distribution at 휃 = 0, which can be seen by +simulating the theory with imaginary 휃. This method is free from the sign problem. We measured +the normalized expectation value ⟨푄⟩ /휒0푉 of the topological charge as a probe of the distribution. +We found that this observable has a finite-temperature transition around 푇/푇dec(0) ∼ 1.06. +Although the deconfinement temperature 푇dec at 휃 = 휋 is not known, it is expected to be lower +than 푇dec(0). Thus, our results suggest that the CP symmetry at 휃 = 휋 is restored at the temperature +higher than the deconfinement temperature—unlike the situation at large 푁. We plan to refine this +8 + +Finite-temperature CP restoration in 4D SU(N) gauge theory +Akira Matsumoto +result by taking the continuum limit. We are also trying to extend this method to the 4D SU(3) YM +theory, in order to see a possible qualitative difference between 푁 = 2 and 푁 = 3, as suggested +from the result in super YM theory. +Acknowledgments +The computations were carried out on the PC clusters in KEK Computing Research Center and +KEK Theory Center. This work is supported by the Particle, Nuclear and Astro Physics Simulation +Program No.2021-005 (FY2021) and No.2022-004 (FY2022) of Institute of Particle and Nuclear +Studies, High Energy Accelerator Research Organization (KEK). A. M. is supported by JSPS Grant- +in-Aid for Transformative Research Areas (A) JP21H05190. M. Honda. is supported by MEXT +Q-LEAP, JST PRESTO Grant Number JPMJPR2117 and JSPS Grant-in-Aid for Transformative +Research Areas (A) JP21H05190. A. Y. is supported by a JSPS Grant-in-Aid for Transformative +Research Areas (A) JP21H05191. +References +[1] D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal, and +temperature, JHEP 05 (2017) 091 [1703.00501]. +[2] R. Kitano, T. Suyama and N. Yamada, 휃 = 휋 in 푠푢(푛)/Z_푛 gauge theories, +JHEP 09 (2017) 137 [1709.04225]. +[3] R. Kitano, N. Yamada and M. Yamazaki, Is 푁 = 2 Large?, JHEP 02 (2021) 073 +[2010.08810]. +[4] R. Kitano, R. Matsudo, N. Yamada and M. Yamazaki, Peeking into the 휃 vacuum, +Phys. Lett. B 822 (2021) 136657 [2102.08784]. +[5] S. Chen, K. Fukushima, H. Nishimura and Y. Tanizaki, Deconfinement and CP breaking at +휃 = 휋 in Yang-Mills theories and a novel phase for SU(2), Phys. Rev. D 102 (2020) 034020 +[2006.01487]. +[6] P. Di Vecchia, K. Fabricius, G. Rossi and G. Veneziano, Preliminary Evidence for U(1)-A +Breaking in QCD from Lattice Calculations, . +[7] C. Morningstar and M.J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, +Phys. Rev. D 69 (2004) 054501 [hep-lat/0311018]. +9 + diff --git a/ddE2T4oBgHgl3EQfbAcC/content/tmp_files/load_file.txt b/ddE2T4oBgHgl3EQfbAcC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a96d220631d3729b6bb07b9a132151d528cdbb65 --- /dev/null +++ b/ddE2T4oBgHgl3EQfbAcC/content/tmp_files/load_file.txt @@ -0,0 +1,358 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf,len=357 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='03879v1 [hep-lat] 10 Jan 2023 KEK-TH-2488,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' RIKEN-iTHEMS-Report-23 Numerical studies on the finite-temperature CP restoration in 4D SU(N) gauge theory at 휽 = 흅 Akira Matsumoto,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Niigata University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Niigata 950-2181,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' E-mail: akira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='matsumoto@riken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='jp, khat@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='kek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='jp, mitsuaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='hirasawa@mib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='it, masazumi318@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='com, jnishi@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='kek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='jp, ayosp@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='niigata-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='jp Recent studies on the ’t Hooft anomaly matching condition have suggested a nontrivial phase structure in 4D SU(푁) gauge theory at 휃 = 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In the large-푁 limit, it has been found that CP symmetry at 휃 = 휋 is broken in the confined phase, while it restores in the deconfined phase, which is indeed one of the possible scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, at small 푁, one may find other situations that are consistent with the consequence of the anomaly matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Here we investigate this issue for 푁 = 2 by direct lattice calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The crucial point to note is that the CP restoration can be probed by the sudden change of the tail of the topological charge distribution at 휃 = 0, which can be seen by simulating the theory at imaginary 휃 without the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Our results suggest that the CP restoration at 휃 = 휋 occurs at temperature higher than the deconfining temperature unlike the situation in the large-푁 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory (Lattice2022), 8-13 August, 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='it/ Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Introduction The non-perturbative effect of the topological theta term in quantum field theories has been studied as a long-standing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Recently, the phase structure of 4D pure Yang-Mills (YM) theory with a theta term has attracted a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' There was a novel progress on application of ’t Hooft anomaly matching [1, 2], which suggests that the 4D SU(푁) pure YM theory cannot have a unique trivial vacuum at 휃 = 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Indeed, this statement is consistent with the known phase diagram for large 푁, where the CP symmetry at 휃 = 휋 is spontaneously broken in the confined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' On the other hand, the phase diagram for small 푁, in particular 푁 = 2, is not determined yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' It is possible that the SU(2) YM theory has a qualitatively different phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, it is an interesting challenge to investigate the phase structure by a first-principle method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Since the effect of the theta term is genuinely non-perturbative, the SU(푁) YM theory with a theta term should be analyzed by non-perturbative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, usual Monte Carlo simulations of the lattice gauge theory including the theta term is difficult due to the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We propose a new method to probe the critical behavior at 휃 = 휋 based on the topological charge distribution 휌(푞) at 휃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The crucial point of the method is that the expectation value ⟨푄⟩휃 of topological charge for any 휃 is completely determined by the distribution 휌(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thanks to this property, we can investigate the behavior of ⟨푄⟩휃 indirectly by the information at 휃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In order to see the temperature dependence of the distribution 휌(푞) clearly, we introduce the imaginary 휃 parameter, which can enhance the tail structure of 휌(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Identifying the CP restoration The application of ’t Hooft anomaly matching condition [1, 2] to the 4D pure YM theory suggests that the phase structure at 휃 = 휋 should be nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' There are a lot of possible phase structures which agree with this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Here we consider two kinds of phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' One is the deconfinement transition at 푇 = 푇dec(휃 = 휋), which corresponds to breaking of 푍푁 center symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Note that the deconfining temperature 푇dec(휃) depends on 휃 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The other transition is the restoration of CP symmetry at 푇 = 푇CP, which is broken at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' For large 푁, these two transitions occur at the same temperature 푇CP = 푇dec(휋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Namely, the CP symmetry is recovered simultaneously with the deconfinement transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In this case, either the 푍푁 center symmetry or the CP symmetry is broken at any temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, it is consistent with the anomaly matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' It is interesting to explore whether the theory with 푁 = 2 has a similar phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In fact, the numerical study of 4D SU(2) YM theory by the subvolume method [3, 4] shows an indication of the CP broken phase at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' It is also confirmed that the instanton gas phase, which is CP symmetric, appears at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, the restoration of CP symmetry is expected to occur also for 푁 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, the relation between the two critical temperatures 푇CP and 푇dec(휋) can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The anomaly matching condition for these two temperatures requires that 푇CP ≥ 푇dec(휋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The reason is that the CP symmetry can be broken not only in the confined phase but also in the deconfined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The overlap of the CP broken phase and the 푍2 broken phase is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' There is a related study of these two critical temperatures by using the super YM theory [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Interestingly, the result shows 푇CP > 푇dec(휋) only for 푁 = 2, but 푇CP = 푇dec(휋) for 푁 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 2 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto It is worth trying to numerically investigate the CP restoration temperature 푇CP for the 4D SU(2) YM theory and compare it with the deconfining temperature 푇dec(휋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, the usual Monte Carlo simulation at 휃 = 휋 suffers from the sign problem since the theta term is purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In this section we introduce a new method to determine 푇CP without direct simulation at 휃 = 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' First, we explain the property of topological charge 푄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Since the topological charge is a CP odd operator, its expectation value ⟨푄⟩휃 can be an order parameter of CP symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' ⟨푄⟩휃 = −푖 휕 휕휃 log 푍휃 (1) If CP symmetry at 휃 = 휋 is spontaneously broken, ⟨푄⟩휃 should be discontinuous there: Δ푄 = ���⟨푄⟩휃=휋−휖 − ⟨푄⟩휃=휋+휖 ��� � > 0 : CP broken, = 0 : CP restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (2) Thus, the CP restoration temperature 푇CP can be regarded as a temperature at which Δ푄 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' To determine 푇CP, we need to investigate the temperature dependence of Δ푄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, it is difficult to directly evaluate Δ푄 due to the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' But we can also study it from another direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Let us note that the partition function 푍휃 and the topological charge distribution 휌(푞) at 휃 = 0 are related via 휌(푞) = 1 푍0 ∫ 푑퐴 훿(푞 − 푄)푒−푆푔 = 1 푍0 ∫ 푑휃 2휋 푒−푖휃푞 푍휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (3) Since the distribution 휌(푞) is a Fourier transform of the partition function 푍휃, 푍휃 = ∫ 푑퐴 푒−푆푔+푖휃푄 = 푍0 ∫ 푑푞 푒푖휃푞 휌(푞), (4) we find that the expectation value ⟨푄⟩휃 at any 휃 is completely determined by 휌(푞) as ⟨푄⟩휃 = −푖 휕 휕휃 log ∫ 푑푞 푒푖휃푞 휌(푞) = ∫ 푑푞 푞푒푖휃푞 휌(푞) ∫ 푑푞 푒푖휃푞 휌(푞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (5) This is nothing but the reweighting formula by using the information at 휃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, to calculate ⟨푄⟩휃 around 휃 ∼ 휋, we need exponentially large amount of statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, our goal is not to determine the complete 휃 dependence of ⟨푄⟩휃 but to probe the critical temperature 푇CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In fact, we do not need the complete information of 휌(푞) in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' It is enough to determine whether Δ푄 is zero or not from 휌(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In this study, we propose to use the expectation value ⟨푄⟩ ˜휃 of the topological charge at an imaginary theta 휃 = 푖 ˜휃 ( ˜휃 ∈ R) to probe the critical behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' ⟨푄⟩ ˜휃 = 1 푍 ˜휃 ∫ 푑퐴 푄푒−푆푔− ˜휃푄 = ∫ 푑푞 푞푒− ˜휃푞휌(푞) ∫ 푑푞 푒− ˜휃푞휌(푞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (6) In practice, we normalize it by the topological susceptibility 휒0 = 1 푉 ⟨푄2⟩휃=0 (7) 3 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto at 휃 = 0 and the volume 푉, so that we just measure the ratio of two independent observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' ⟨푄⟩ ˜휃 휒0푉 = ⟨푄⟩ ˜휃 ⟨푄2⟩0 (8) Let us discuss the behavior of this observable in some well known models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The first example is the instanton gas model, for which the free energy is obtained as 퐹휃 = − log 푍휃 = 휒0푉(1 − cos 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (9) The 휃-dependence of ⟨푄⟩ 휃 /휒0푉 for real 휃 is given by the sine function ⟨푄⟩ 휃 휒0푉 = 푖 sin 휃, (10) which indicates that CP symmetry at 휃 = 휋 is not broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Correspondingly, the imaginary-휃 dependence turns out to be the hyperbolic sine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' ⟨푄⟩ ˜휃 휒0푉 = − sinh ˜휃 (11) The second example is the Gaussian model 퐹휃 = 1 2 휒0푉 min 푛 (휃 − 2휋푛)2, (12) which is known to be realized for large 푁 at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The real-휃 dependence of ⟨푄⟩ 휃 /휒0푉 is given by ⟨푄⟩ 휃 휒0푉 = 푖(휃 mod 2휋) (13) for 푉 ≫ 1, which indicates that the CP is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' For imaginary 휃, we find the linear behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' ⟨푄⟩ ˜휃 휒0푉 = − ˜휃 (14) We can see the clear difference between the behaviors of ⟨푄⟩ ˜휃 /휒0푉 for these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' It behaves as − sinh ˜휃 for the instanton gas model (CP restored), while it behaves as − ˜휃 for the Gaussian model (CP broken).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Although the 4D SU(2) YM theory will not be as simple as these models, this observable is still useful to investigate the CP restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In fact, the expectation value ⟨푄⟩ ˜휃 for imaginary 휃 is sensitive to the tail of the distribution of 휌(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The imaginary theta term enhances the contribution of large-푞 sectors because of the factor 푒− ˜휃푞 in the integrand of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Note that, for these two examples, the tail of 휌(푞) behaves for 푞 ≫ 1 as follows: 휌(푞) ∼ \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 exp � −푞 log 2푞 휒0푉 � : instanton gas, exp � − 푞2 2휒0푉 � : Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (15) 4 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 4D SU(2) gauge theory with a theta term In this study, we focus on the SU(2) pure Yang-Mills theory on the 4D Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The action for the gauge field 퐴푎 휇 (푎 = 1, 2, 3) (휇 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' , 4) is defined by 푆푔 = 1 4푔2 ∫ 푑4푥 퐹푎 휇휈퐹푎 휇휈, (16) where 푔 is the coupling constant and 퐹푎 휇휈 is the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 퐹푎 휇휈 = 휕휇퐴푎 휈 − 휕휈퐴푎 휇 − 휖 푎푏푐 퐴푏 휇퐴푐 휈 (17) The topological charge is given by 푄 = 1 64휋2 ∫ 푑4푥 휖휇휈휌휎퐹푎 휇휈퐹푎 휌휎, (18) which takes an integer value on the compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We introduce the topological theta term 푆휃 = −푖휃푄 with a parameter 휃 ∈ R, so that the total action is 푆 = 푆푔 + 푆휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Since the partition function 푍 = ∫ D퐴 푒−푆푔+푖휃푄 (19) is invariant under the shift 휃 → 휃 + 2휋, the theory has 2휋 periodicity with respect to 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Since the parameter 휃 flips its sign by the CP transformation 휃 → −휃, the theta term explicitly breaks the CP symmetry for 휃 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, thanks to the 2휋 periodicity, the CP symmetry exists also at 휃 = 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Next, we define the lattice action for the numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The gauge field is represented by the link variable 푈푛,휇 ∈ SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The index 푛 labels the lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The plaquette is given by 푃휇휈 푛 = 푈푛,휇푈푛+ ˆ휇,휈푈† 푛+ ˆ휈,휇푈† 푛,휈, (20) where ˆ휇 represents the unit vector along the 휇-th direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Then we define the plaquette action with the lattice coupling constant 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 푆푔 = − 훽 4 � 푛 � 휇≠휈 Tr(푃휇휈 푛 ) (21) Similarly, we can define the topological charge on the lattice by the so-called "clover leaf" formula, [6] 푄cl = − 1 32휋2 � 푛 1 24 ±4 � 휇,휈,휌,휎=±1 ˜휖휇휈휌휎Tr(푃휇휈 푛 푃휌휎 푛 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (22) Here the orientationof the plaquette is extended tothe negative directions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thecorresponding anti-symmetric tensor ˜휖휇휈휌휎 also has negative indices, so that 1 = ˜휖1234 = − ˜휖2134 = − ˜휖−1234 = · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (23) It is known that the naively defined topological charge 푄cl does not take an integer value on the lattice due to the discretization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In order to recover the topological property of the gauge field, we need to eliminate short-range fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In fact, there are some smoothing techniques, such as the gradient flow, stout smearing and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' By using such a technique, we can define the smeared topological charge so that it becomes close to an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In this study, we introduce the stout smearing to the hybrid Monte Carlo simulation, which is discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 5 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Stout smearing for the HMC Since the CP symmetry at 휃 = 휋 is related to the 2휋 periodicity of 휃, the topological property of the theory is essential in the study of the phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, we use the stout smearing [7] to define the topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In the hybrid Monte Carlo simulation, the drift force is used to update the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' If the action has the theta term with the smeared topological charge, it also contributes to the drift force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We can explicitly calculate the drift force from the smeared topological charge by using stout smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In this section, we briefly review the stout smearing in the hybrid Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Stout smearing is an iterative procedure to obtain the smeared link ˜푈푛,휇 starting from the original link 푈푛,휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We call the number of iterations 푁휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 푈푛,휇 = 푈 (0) 푛,휇 → 푈 (1) 푛,휇 → · · · → 푈 (푁휌) 푛,휇 = ˜푈푛,휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (24) In one (isotropic) smearing step from 푘 to 푘 + 1, the link variable 푈 (푘) 푛,휇 ∈ SU(2) is mapped to 푈 (푘+1) 푛,휇 ∈ SU(2) defined by following formulae: 푈 (푘+1) 푛,휇 = 푒푖푌푛,휇푈 (푘) 푛,휇, (25) 푖푌푛,휇 = − 휌 2 Tr(퐽푛,휇휏푎)휏푎, (26) 퐽푛,휇 = 푈푛,휇Ω푛,휇 − Ω† 푛,휇푈† 푛,휇, (27) Ω푛,휇 = � 휎(≠휇) � 푈푛+ ˆ휇,휎푈† 푛+ ˆ휎,휇푈† 푛,휎 + 푈† 푛+ ˆ휇− ˆ휎,휎푈† 푛− ˆ휎,휇푈푛− ˆ휎,휎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (28) Here 휏푎 are the SU(2) generators in fundamental representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The smearing step parameter 휌 > 0 should be chosen appropriately depending on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In the hybrid Monte Carlo simulation, we obtain the smeared link ˜푈푛,휇 by this procedure, and then we use ˜푈푛,휇 to calculate the topological charge (22) instead of the original link 푈푛,휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The topological charge given by the stout smearing 푄 := 푄cl( ˜푈) (29) is used in the theta term 푆휃 = −푖휃푄 as well as in measuring the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In the step of molecular dynamics, we need to calculate the drift force 퐹푛,휇 = 푖휏푎퐷푎 푛,휇푆휃 (30) from the theta term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Although 푆휃 is a complicated function of the original link variable 푈푛,휇, it is possible to calculate the drift force by reversing the smearing steps (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Result of the HMC In this section, we show the result of the hybrid Monte Carlo simulation with the imaginary theta term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' For the stout smearing, we set 푁휌 = 40 and 휌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='09 so that the topological charge is close to an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 1, we plot −⟨푄⟩ ˜휃/휒0푉 against ˜휃/휋 = 휃/푖휋 for various values of temperature 6 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto 0 2 4 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='8 1 −〈Q〉 / χ0V θ~ / π T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='92Tc T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='96Tc T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='00Tc T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='04Tc T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='08Tc T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='12Tc T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='16Tc T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='20Tc θ~ sinh θ~ Nρ=40, ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='09, V=203x5 Figure 1: The imaginary 휃 dependence of − ⟨푄⟩ ˜휃 /휒0푉 for various values of temperature in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='92 ≤ 푇/푇dec(0) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='20 with increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The green solid curve is obtained by the instanton gas approxi- mation, which is valid at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The purple solid line is obtained by the Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='9 ≤ 푇/푇dec(휃 = 0) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We found that, at high temperature, the data points are consistent with the instanton gas approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' On the other hand, the data points approach the behavior of the Gaussian model at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' It is convincing that the SU(2) YM theory behaves as the instanton gas model at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, it does not necessarily coincides with the Gaussian model at low temperature since the situation of 푁 = 2 can be different from that of large 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Nevertheless, this observable is suitable for probing the phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Indeed, we can see that the behaviors of ⟨푄⟩ ˜휃/휒0푉 change drastically slightly above the deconfining temperature 푇dec(0) at 휃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In order to see the temperature dependence of −⟨푄⟩ ˜휃/휒0푉, we plot it against temperature at fixed 휃/휋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75푖 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The left figure is the result for 푉 = 163 × 5, and the right figure is the result for 푉 = 203 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The yellow curve shows the result of fitting by a cubic function 푓 (푥) = 푎푥3 + 푏푥2 + 푐푥 + 푑 where 푎, 푏, 푐 and 푑 are fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The orange curve is the derivative of 푓 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We find that the derivative is the largest at around 푇peak ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='06푇dec(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We also find that the height of the peak grows as the spatial volume 푉s increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 3(left), we plot the peak position 푇peak/푇dec(0) against 1/푉s obtained by the same analysis for 휃/휋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='6푖, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75푖 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='9푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The significant volume dependence of 푇peak is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' These results suggest that there is a phase transition around 푇/푇dec(0) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 3(right), the peak height of the fitting function is plotted against 푉1/3 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' This non-linear finite size scaling suggests that the phase transition is of the second order or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The existence of the transition indicates that the distribution 휌(푞) of the topological charge changes drastically around푇/푇dec (0) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Assuming that CP symmetry at 휃 = 휋 is spontaneously broken at low temperature, the drastic change of 휌(푞) should correspond to the critical behavior of ⟨푄⟩ 휃=휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, we identify the critical temperature 푇peak as the CP restoration temperature 푇CP, which suggests 푇CP > 푇dec(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 7 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='25 0 2 4 6 8 10 12 14 16 18 20 − < Q > / χ0V f’(x) T/Tdec(0) f(x) df(x)/dx V = 163 x 5, θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='25 0 5 10 15 20 25 30 35 − < Q > / χ0V f’(x) T/Tdec(0) f(x) df(x)/dx V = 243 x 5, θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75i Figure 2: The temperature dependence of − ⟨푄⟩ ˜휃 /휒0푉 at 휃/휋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75푖 for푉 = 163 ×5 (left) and 푉 = 203 ×5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We fit the data points by a cubic function 푓 (푥) = 푎푥3 + 푏푥2 + 푐푥 + 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The orange solid line shows the result of fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The red solid line is derivative of 푓 (푥) with respect to 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='0x100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='0x10-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='0x10-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='0x10-4 Tpeak / Tdec(0) 1 / Vs θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='60 i θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75 i θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='90 i 0 10 20 30 40 50 60 0 4 8 12 16 20 24 28 peak height Vs 1/3 θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='60 i θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75 i θ / π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='90 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='80 Vs 1/3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='35 Vs 1/3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='74 Vs 1/3 Figure 3: (left) The peak position 푇peak/푇dec(0) against 1/푉s for 휃/휋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='6푖, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='75푖 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='9푖, where 푉s is the spacial lattice volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' (right) The peak height of 푓 ′(푥) is plotted against 푉1/3 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' The straight lines represent the fit to the behavior 푎푉1/3 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Summary Recent studies on the ’t Hooft anomaly matching condition for the 4D SU(푁) gauge theory have suggested that the phase structure at 휃 = 휋 should be nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' For large 푁, it is known that the CP symmetry at 휃 = 휋 is spontaneously broken in the confined phase, while it is restored in the deconfined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' However, for small 푁, a qualitatively different phase structure can be realized, as long as the anomaly matching condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' In this work, we investigated this issue for 푁 = 2 by hybrid Monte Carlo simulation of lattice gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We probed the restoration of the CP symmetry by a sudden change of the topological charge distribution at 휃 = 0, which can be seen by simulating the theory with imaginary 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' This method is free from the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We measured the normalized expectation value ⟨푄⟩ /휒0푉 of the topological charge as a probe of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We found that this observable has a finite-temperature transition around 푇/푇dec(0) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Although the deconfinement temperature 푇dec at 휃 = 휋 is not known, it is expected to be lower than 푇dec(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Thus, our results suggest that the CP symmetry at 휃 = 휋 is restored at the temperature higher than the deconfinement temperature—unlike the situation at large 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We plan to refine this 8 Finite-temperature CP restoration in 4D SU(N) gauge theory Akira Matsumoto result by taking the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' We are also trying to extend this method to the 4D SU(3) YM theory, in order to see a possible qualitative difference between 푁 = 2 and 푁 = 3, as suggested from the result in super YM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Acknowledgments The computations were carried out on the PC clusters in KEK Computing Research Center and KEK Theory Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' This work is supported by the Particle, Nuclear and Astro Physics Simulation Program No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='2021-005 (FY2021) and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content='2022-004 (FY2022) of Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' is supported by JSPS Grant- in-Aid for Transformative Research Areas (A) JP21H05190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Honda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' is supported by MEXT Q-LEAP, JST PRESTO Grant Number JPMJPR2117 and JSPS Grant-in-Aid for Transformative Research Areas (A) JP21H05190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' is supported by a JSPS Grant-in-Aid for Transformative Research Areas (A) JP21H05191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Gaiotto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE2T4oBgHgl3EQfbAcC/content/2301.03879v1.pdf'} +page_content=' Kapustin, Z.' 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b/ddFJT4oBgHgl3EQfSCyU/content/tmp_files/2301.11498v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f20b8f02aa0b04297b82de2da1d6eeba679e024 --- /dev/null +++ b/ddFJT4oBgHgl3EQfSCyU/content/tmp_files/2301.11498v1.pdf.txt @@ -0,0 +1,2131 @@ +DRAFT VERSION JANUARY 30, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +CLASSY VI: The Density, Structure and Size of Absorption-Line Outflows in Starburst Galaxies∗ +XINFENG XU,1 TIMOTHY HECKMAN,1 ALAINA HENRY,1, 2 DANIELLE A. BERG,3 JOHN CHISHOLM,3 BETHAN L. JAMES,4 +CRYSTAL L. MARTIN,5 DANIEL P. STARK,6 MATTHEW HAYES,7 KARLA Z. ARELLANO-CÓRDOVA,3 CODY CARR,8 +MASON HUBERTY,8 MATILDE MINGOZZI,2 CLAUDIA SCARLATA,8 AND YUMA SUGAHARA9, 10 +1Center for Astrophysical Sciences, Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA +2Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +3Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA +4AURA for ESA, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +5Department of Physics, University of California, Santa Barbara, Santa Barbara, CA 93106, USA +6Steward Observatory, The University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721, USA +7Stockholm University, Department of Astronomy and Oskar Klein Centre for Cosmoparticle Physics, AlbaNova University Centre, SE-10691, Stockholm, Sweden +8Minnesota Institute for Astrophysics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA +9National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +10Waseda Research Institute for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, +Japan +Submitted to AASJournal ApJ +ABSTRACT +Galaxy formation and evolution are regulated by the feedback from galactic winds. Absorption lines provide +the most widely available probe of winds. However, since most data only provide information integrated along +the line-of-sight, they do not directly constrain the radial structure of the outflows. In this paper, we present a +method to directly measure the gas electron density in outflows (ne), which in turn yields estimates of outflow +cloud properties (e.g., density, volume filling-factor, and sizes/masses). We also estimate the distance (rn) from +the starburst at which the observed densities are found. We focus on 22 local star-forming galaxies primarily +from the COS Legacy Archive Spectroscopic SurveY (CLASSY). In half of them, we detect absorption lines +from fine structure excited transitions of Si II (i.e., Si II*). We determine ne from relative column densities of +Si II and Si II*, given Si II* originates from collisional excitation by free electrons. We find that the derived ne +correlates well with the galaxy’s star-formation rate per unit area. From photoionization models or assuming +the outflow is in pressure equilibrium with the wind fluid, we get rn ∼ 1 to 2r∗ or ∼ 5r∗, respectively, where +r∗ is the starburst radius. Based on comparisons to theoretical models of multi-phase outflows, nearly all of the +outflows have cloud sizes large enough for the clouds to survive their interaction with the hot wind fluid. Most +of these measurements are the first-ever for galactic winds detected in absorption lines and, thus, will provide +important constraints for future models of galactic winds. +Keywords: Galactic Winds (572), Galaxy evolution (1052), Galaxy kinematics and dynamics(602), Starburst +galaxies (1570), Ultraviolet astronomy (1736), Galaxy spectroscopy (2171) +1. INTRODUCTION +Corresponding author: Xinfeng Xu +xinfeng@jhu.edu +∗ Based on observations made with the NASA/ESA Hubble Space Telescope, +obtained from the Data Archive at the Space Telescope Science Institute, +which is operated by the Association of Universities for Research in As- +tronomy, Inc., under NASA contract NAS 5-26555. +Galactic winds are essential to the evolution of galaxies +and the intergalactic medium (IGM). In star-forming galax- +ies (without accreting black holes), these winds are driven +by mass, energy, and momentum supplied by star-formation, +in the form of radiation, stellar winds, and supernovae (e.g., +Veilleux et al. 2005). +The latter two result in the cre- +ation of a tenuous and energetic wind fluid that flows out +and accelerates existing gas clouds, which are observable +as warm to cold outflows (e.g., Xu et al. 2022a). +Galac- +arXiv:2301.11498v1 [astro-ph.GA] 27 Jan 2023 + +2 +XU ET AL. +tic winds and the outflows they drive are able to transport +mass/energy/momentum against the gravitational potential of +the hosts. Thus, they have been proposed to explain vari- +ous feedback effects, e.g., regulating the star formation rate +(SFR) of the host galaxy (e.g., Martin 2005; Rupke et al. +2005; Cazzoli et al. 2014; Heckman & Borthakur 2016), +chemically enriching the circum-galactic medium (CGM) +and IGM (e.g., Heckman et al. 2000; Dalcanton 2007; Martin +et al. 2012; Rubin et al. 2014; Heckman et al. 2017; Chisholm +et al. 2018), and explaining the “overcooling problem" in +cosmological simulations by reducing the baryon fractions in +galactic discs (e.g., Steidel et al. 2010; Hopkins et al. 2012). +Outflows can also clear neutral gas away from young star- +bursts and regulate the escape of Lyman continuum photons, +which is responsible for the cosmic reionization (e.g., Heck- +man et al. 2011; Chisholm et al. 2017; Hogarth et al. 2020; +Carr et al. 2021; Saldana-Lopez et al. 2022). +In the last few decades, galactic winds and outflows have +been intensely studied in the literature, especially in star- +forming and starburst galaxies, which commonly host pow- +erful outflows (see reviews in Heckman & Thompson 2017; +Rupke 2018; Veilleux et al. 2020; Nguyen et al. 2023; and +references therein). Outflows are multi-phase (e.g., Fluetsch +et al. 2021; Marasco et al. 2022), but the most abundant data +probe the warm ionized phase. This material is believed to be +accelerated by the combined momentum of a hot wind fluid +created by stellar ejecta and radiation pressure (Heckman & +Thompson 2017). The two major ways to detect the warm +ionized gas are from rest-frame UV absorption lines (e.g., +O I, Si II, Si IV, and C IV), and optical emission lines (e.g., +[O III] and Hα). Even though both emission and absorption +lines can show kinematic features that represent the outflows, +they are thought to arise from different environments (e.g., +Chisholm et al. 2016a). Emission lines are weighted towards +the denser environments (brightness scales with outflow elec- +tron density (ne) squared), while ne have been found to be +∼ 100 – 1000 cm−3 (e.g., Heckman et al. 1990; Perna et al. +2020; Marasco et al. 2022). On the contrary, absorption lines +trace lower density environments (optical depth scales with +ne). Thus, Wood et al. (2015) suggests that UV absorption +lines can trace larger-scale galactic outflows and are more re- +liable tracers of warm gas in starburst-driven outflows. +There exist well-developed ways to constrain various im- +portant outflow parameters from the absorption lines, includ- +ing outflow velocity (Vout), ionization, and column density +(NH) (e.g., Martin 2005; Rupke et al. 2005; Chisholm et al. +2015; Scarlata & Panagia 2015; Chisholm et al. 2016a; Heck- +man & Borthakur 2016; Carr et al. 2018; Xu et al. 2022a). +The strength of the absorption outflows and their potential +feedback effects can then be quantified by their mass, mo- +mentum, and energy rates, i.e., ˙Mout ∝ NHroutVout, ˙pout ∝ +NHroutV 2 +out, and ˙Eout ∝ 1 +2NHroutV 3 +out, respectively, where rout is +the assumed radius of the outflows. +The major uncertainty for these outflow rates is from rout +(e.g., Chisholm et al. 2016b). This is because these surveys +of galactic outflows in absorption lines only have integrated +spectra in a single aperture. It is not possible to measure +rout directly. Most previous studies either assume a fiducial +radius (e.g., 1 – 5 kpc in Martin 2005; Rupke et al. 2005; +Martin et al. 2012), or assume rout starts at a few times the +starburst radius (r∗, e.g., Chevalier & Clegg 1985; Heckman +et al. 2015; Chisholm et al. 2017; Carr et al. 2021; Xu et al. +2022a). Recently, Wang et al. (2020) showed that the non- +resonant emission lines are much weaker and narrower than +the corresponding absorption lines in a sample of starburst +galaxies. They suggest that observed absorbing material for +outflows could be located at radii significantly larger than r∗. +Moreover, the meaning of rout is only well-defined for the +idealized case in which the outflow is a thin bubble. In the +more general case where the outflow is continuous (i.e., rout +is a distribution), the appropriate value of rout for calculating +outflow rates will depend upon the radial variation of den- +sity and velocity in the outflow. Without knowing the radial +structure of the outflow, outflow rates are uncertain. +In addition to uncertainties in the radial structure of the +outflows probed by absorption lines, there is the long- +standing theoretical problem about the nature of outflows. +How can the absorbing material survive long enough to be +accelerated to hundreds of km/s without being shredded by +the hydro-dynamical interaction with the wind fluid (e.g., +Nguyen et al. 2023)? Recent work (e.g., Gronke & Oh 2020; +Fielding & Bryan 2022) imply that clouds exposed to an out- +flowing hot wind can either grow by accreting gas at the +cloud’s interface with the hot phase (for large clouds), or be +destroyed (for small clouds). To date, there are no good em- +pirical constraints on the cloud masses (Mcl) or radii (Rcl) in +outflows. +In this paper, we aim to shed light on a method to mea- +sure ne, the radius at which these densities apply (rn), Mcl, +and Rcl from outflow absorption lines. We focus on 22 local +star-forming galaxies selected from the COS Legacy Archive +Spectroscopy SurveY (CLASSY) atlas (Berg et al. 2022; +James et al. 2022) and Heckman et al. (2015). These galax- +ies have high signal-to-noise ratio (SNR) HST/COS spectra +which cover their rest-frame UV bands. In half of the galax- +ies, we can securely detect absorption lines from the fine +structure excited transitions of Si II, i.e., Si II*. From it, we +determine various important physical parameters of the out- +flows, including ne, rn, outflow volume filling factor, outflow +cloud sizes, and cloud masses. Since the majority of these +measurements are the first-ever for galactic winds detected +in absorption lines, we discuss their implications and what + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +3 +observational constraints they provide for future models of +galactic winds. +The structure of the paper is as follows. In Section 2, we +introduce the data and observations that are used in this pa- +per. We then describe how to measure the column density +from Si II and Si II* in Section 3 and how to derive ne from +these two quantities. In Section 4, we present results for the +ne and rn. We compare them with empirical estimates that are +commonly adopted in the literature. We also describe how to +derive several other important outflow parameters (the cloud +masses, radii, and volume filling factors). Finally, in Section +5, we discuss and compare our results with other outflow den- +sity and radius measurements in the literature. We also con- +trast our results with current outflow models in Section 5. We +conclude the paper in Section 6. +We adopt a cosmology with H0 = 69.6 km s−1 Mpc−1, Ωm = +0.286, and ΩΛ = 0.714 (Bennett et al. 2014), and we use Ned +Wright’s Javascript Cosmology Calculator website (Wright +2006). In this paper, we adopt the notation r for distances +from the starburst, and use R to represent outflow cloud radii. +2. OBSERVATIONS AND DATA REDUCTIONS +In this paper, we select galaxies from the parent sample of +the CLASSY dataset (Berg et al. 2022), which includes 45 lo- +cal star-forming galaxies (0.002 < z < 0.182). These galax- +ies are observed by the G130M+G160M+G185M/G225M +gratings on Hubble Space Telescope (HST)/Cosmic Ori- +gins Spectrograph (COS) for their rest-frame far-ultraviolet +(FUV) spectral regions. To enlarge the dynamic range of +the sample at the highest star-formation rates (SFR), we also +include five similar galaxies from the Lyman Break Analog +(LBA) sample in Heckman et al. (2015). These galaxies have +similar quality HST/COS observations as CLASSY ones. We +then apply three selection criteria: 1) the SNR per resolution +element (0.18 Å) in the continuum near 1260 Å in the rest- +frame is ≥ 5; 2) the UV half-light radius of the starburst is < +1.5′′ (so that the COS spectrum represents the majority of the +starburst); 3) an outflow has been detected (Xu et al. 2022a). +These criteria result in a final sample of 22 galaxies. +All data were reduced locally using the COS data- +reduction package CalCOS v.3.3.101, including spectra ex- +traction and wavelength calibration. We refer readers to Berg +et al. (2022) and James et al. (2022) for more details about +these data reductions and spectral coaddition procedures. We +also apply the same reductions to the LBA galaxies from +Heckman et al. (2015). Therefore, the whole sample was +reduced and processed in a self-consistent way. We have re- +sampled the spectra into bins of 0.18 Å (spectral resolution +∼ 6000 – 10000 from the blue to red end) (Xu et al. 2022a). +1 https://github.com/spacetelescope/calcos/releases +These galaxies’ redshift are derived from fitting the optical +emission lines discussed in Mingozzi et al. (2022). +3. ANALYSES +3.1. Summary of Previous Outflow Analyses +For our sample, the detailed analyses of outflow proper- +ties and their relationship to the host galaxy properties are +reported in Xu et al. (2022a). We briefly summarize the key +steps as follows. +1. Given the reduced data from CalCOS, we start with +fitting the stellar continuum of galaxies using stellar +models from Starburst99 (Leitherer et al. 1999; 2010). +We follow the methodology discussed in Chisholm +et al. (2019). We then normalize the spectra by the +best-fit stellar continuum for each galaxy. +2. For each galaxy, the final reduced HST/COS spectra +cover ∼ 1200 Å – 2000 Å in the observed frame. In +this region, various lines from galactic outflows are de- +tected as absorption troughs, from, e.g., O I λ1302, C II +λ1334, Si II multiplet (λ1190, 1193, 1260, 1304, and +1526), Si III λ1206, and Si IV λλ1393, 1402. +3. To isolate the outflowing gas component from the +static ISM, we fit a double-Gaussians model to each +absorption trough. The first Gaussian has a fixed ve- +locity center at v = 0 km s−1, which represents the static +ISM component, and the second Gaussian has a veloc- +ity center < 0 km s−1, which stands for the blueshifted +outflow component. +Since the line-spread-functions +(LSF) from HST website2 is only suitable for point +sources, we have constructed non-point source line- +spread-functions (LSF) for each galaxy and convolved +them with standard Gaussian profiles in the fitting pro- +cess. +4. To robustly measure the ionic column density (Nion) +of outflows, we apply partial coverage (PC) models +to Si II multiplet and Si IV doublet absorption troughs. +From the PC models, we have determined the optical +depths, covering fraction (CF), and Nion for Si II and +Si IV as functions of velocity. +5. We then compare the measured Nion to grids of pho- +toionization models from CLOUDY [version c17.01, +(Ferland et al. 2017)] to determine the total silicon and +hydrogen column densities, i.e., N(Si) and NH, respec- +tively. +2 https://www.stsci.edu/hst/instrumentation/cos/performance/ +spectral-resolution + +4 +XU ET AL. +6. We derive the mass/momentum/energy outflow rates +given the derived NH and Vout, while we assume rout += r∗, which we take to be the radius enclosing 90% of +the starburst FUV emission. +3.2. Measurements of Column Density from Si II* +Galactic outflows not only show absorption lines from res- +onance transitions, e.g., Si II λ1260, but also from fine struc- +ture excited transitions, e.g., Si II* λ1265 (e.g., Jaskot et al. +2019). The combination of both can be adopted to derive +the electron number density (ne) of the outflows (see Section +3.3). In this subsection, we focus on measuring N(Si II*) for +galaxies in our sample. +There are a total of six Si II* lines observable in the rest- +frame FUV. We list their important atomic information in Ta- +ble 1. We find the observed absorption lines from Si II* are +commonly weak in our galaxies. This is consistent with the +assumed low ne (∼ 10 cm−3) for typical starburst galaxies +(e.g., Xu et al. 2022a). This low ne has both pros and cons +for our analysis. On the one hand, the weaker Si II* troughs +are generally optically thin (τ ≪ 1), and we can safely mea- +sure N(Si II*) by adopting CF = 1, given the apparent optical +depth (AOD) assumption (Savage & Sembach 1991). On the +other hand, the shallow Si II* troughs are sometimes difficult +to measure, even in our high SNR HST/COS spectra. An- +other complexity is that the emission lines from Si II* (i.e., +so-called fluorescent lines) can contaminate the blue-shifted +absorption troughs of Si II*, especially when the outflow ve- +locity (Vout) is small. The steps for our fitting process of Si II* +lines and measurements of N(Si II*) are as follows: +1. We fit the fluorescent emission and fine-structure ab- +sorption lines from Si II* λ1197, 1265, 1309, and 1533 +for each galaxy simultaneously. +We exclude Si II* +λ1194 because it is commonly blended with the ab- +sorption trough from Si II λ1193. For each galaxy, we +also exclude Si II* lines that fall into a chip gap or are +contaminated by Galactic lines (e.g., Si II λ1197 can +be affected by Galactic Lyα). +2. For each Si II* absorption line, we assume it has a +Gaussian optical depth profile: +Ik(v) = e−τk(v) +τk(v) = +bk +σ +√ +2π +×exp((v−vc)2 +2σ2 +) +(1) +where k stands for the kth Si II* absorption line, Ik(v) +is the normalized intensity, τk(v) is the optical depth at +each velocity of the absorption trough, v is the velocity. +Given the AOD assumption, the optical depths of dif- +ferent Si II* absorption lines (scaled by coefficient bk) +are linked by their oscillator strength ( f) ratios (see +Table 1). The velocity center (vc) and dispersion (σ) +of the Si II* absorption lines are fixed among all Si II* +lines. These fixed values are chosen to be the same as +the median values from all Si II resonance absorption +lines (Section 3.1). This assumes that the same outflow +clouds have produced Si II and Si II* absorption lines, +which is true since both lines have close energy levels +(Section 3.3). +3. For each Si II* emission line, we model it using only +one Gaussian profile in velocity space. This is because +Si II* in our sample show weak and narrow fluorescent +emission-lines, and are inconsistent with arising from +the outflowing gas seen in absorption (i.e., broad lines). +This implies that most of the emission from the outflow +arises on scales larger than the projected COS aperture +(Wang et al. 2020). For all Si II* emission lines, we fix +vc at the systematic velocity, and σ is set to be in the +range between 0 and the median FWHM of the static +ISM component of Si II resonance lines (Section 3.1). +Their amplitudes are free parameters. +4. Then, we conduct χ2 minimization to fit all 2×N pro- +files simultaneously to the spectral regions of Si II*. +Here the 2 stands for the emission and absorption line +for each Si II*, and N equals the number of Si II* lines +that are clean and used in the fit. We adopt the fitting +routine mpfit (Markwardt 2009). +5. Finally, assuming AOD, N(Si II*) can be derived from +the best-fitted τk(v) as follows (Savage & Sembach +1991): +Nion(v) = 3.8× 1014 +fk ·λk +·τk(v) +Nion = +� +Nion(v)dv +(2) +where λk is the wavelength for the kth Si II* line +that has τk(v). Note that under the AOD assumption, +choices of different Si II* lines in Equation (2) lead to +the exact same Nion. +There are two close transitions of Si II* at ∼ 1265 Å, i.e., +Si II* λ1264.73 and λ1265.02. Both are from the same lower +energy level at 0.036 eV (= 287.24 cm−1), but have slightly +different upper energy levels due to fine structure splitting +(δE ∼ 4×10−4 eV). Since the velocity offset between these +two lines is only 69 km s−1, we can barely resolve their ab- +sorption lines in the spectra. Thus, we adopt the combined +f value in the calculations of N(Si II*) (Borguet et al. 2012). +Since Si II* λ1265.02 has ∼ 10 times smaller f value than +that of Si II* λ1264.73 (Table 1), the absorption trough is al- +ways dominated by the latter. + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +5 +Figure 1. Example of fitting to the absorption and emission lines for Si II* spectral regions for galaxy J0150+1308 (z = 0.14668). The black +and gray histograms are the data and errors, respectively. In each panel, the blueshifted outflow component for the Si II resonance line is shown +in blue (adopted from Xu et al. 2022a; see Section 3.1). The fitted absorption and emission models for Si II* are shown in green and orange, +respectively. The summed model for Si II* is shown in red. The green and orange dashed lines mark the velocity centers for the fitted absorption +and emission lines, respectively. See detailed fitting methods in Section 3.2. +Table 1. Atomic Data for the Resonance and Excited Transitions of +Si II (a) +Ions +Vac. Wave. +flk +Akl +Elow – Eup +(1) +(2) +(3) +(4) +(5) +Si II +1190.42 +2.77 × 10−1 +6.53 × 108 +0.0 - 10.41 +Si II +1193.29 +5.75 × 10−1 +2.69 × 109 +0.0 - 10.39 +Si II +1260.42 +1.22 +2.57 × 109 +0.0 - 9.84 +Si II +1304.37 +9.28 × 10−2 +3.64 × 108 +0.0 - 9.50 +Si II +1526.71 +1.33 × 10−1 +3.81 × 108 +0.0 - 8.12 +Si II* +1194.50 +7.37 × 10−1 +3.45 × 109 +0.036 - 10.41 +Si II* +1197.39 +1.50 × 10−1 +1.40 × 109 +0.036 - 10.39 +Si II* +1264.73(b) +1.09 +3.04 × 109 +0.036 - 9.84 +Si II* +1265.02(b) +1.13× 10−1 +4.73 × 108 +0.036 - 9.84 +Si II* +1309.28 +8.00 × 10−2 +6.23 × 108 +0.036 - 9.50 +Si II* +1533.45 +1.33 × 10−1 +7.52 × 108 +0.036 - 8.12 +Note. – +(a). Data are obtained from National Institute of Standards and +Technology (NIST) atomic database (Kramida et al. 2018). +(2). Vacuum wavelengths in units of Å. +(3). Oscillator strengths. +(4). Einstein A coefficients in units of s−1. +(5). Energies from lower to upper levels in units of eV. +(b). Si II* has two close transitions at ∼ 1265 Å, i.e., Si II* +λ1264.73 and λ1265.02. Both are from the same lower energy +level at 0.036 eV, but have slightly different upper energy +levels due to fine structure splitting (δE ∼ 4×10−4 eV). See +discussion in Section 3.3. +An example of the fitted Si II* spectral regions is shown +in Figure 1. The outflow components for the Si II resonance +lines are shown in blue (Section 3.1), while the fitted absorp- +tion and emission models for Si II* are shown in green and +orange, respectively. The overall model for Si II* by sum- +ming both the absorption and emission is shown in red. There +is a clear absorption trough from Si II* λ1265, and it is well- +fitted, while there is no trough seen in Si II* λ1309. This is as +expected since f1265/f1309 = 15, which leads to τ1265/τ1309 = 15 +under AOD models. Overall, we have measured N(Si II*) se- +curely in 11 out of 22 galaxies in our sample. These galaxies +that have N(Si II*) measured yield a mean N(Si II*)/N(Si II) +∼ 0.01. +In Table 3, we report the measured N(Si II) and +N(Si II*) in column 6 and 7, respectively. +3.3. Mechanisms for Generating Si II*: Collisions v.s. +Radiative Pumping +As shown in Table 1, the observed fine-structure transitions +of Si II* in FUV have lower energy levels as Elow = 0.036 eV +(= 287.24 cm−1), which is the first excited energy level of +Si II (hereafter, Si II* specifically stands for this level). Two +mechanisms can populate Si II*: 1) Collisional excitation of +the ground state of Si II by free electrons (e.g., Silva & Vie- +gas 2002; Osterbrock & Ferland 2006; Borguet et al. 2012). +In this case, a higher ne would yield a higher n(Si II*)/n(Si II) +ratio, where n(Si II*) and n(Si II) stand for the level popu- +lation of the first excited and ground state of Si II. 2) In- +direct UV pumping, i.e., the Si II ground state is excited by +absorption of a UV photon to an upper energy level, followed +by a spontaneous decay to the excited level at Si II* 287.24 +cm−1. In this case, a stronger radiation field leads to higher +n(Si II*)/n(Si II) (see, e.g., Prochaska et al. 2006). +To check if indirect UV pumping can be the dominant +mechanism, we estimate the radiation intensity G (in units of +ergs cm−2 s−1) suffered by outflows for galaxies in our sam- +ple. We first measure each galaxy’s continuum flux (λFλ) +around Si II λ1260. Then we convert it to luminosity as: +λLλ = 4πD2 +L × λFλ, where DL is the luminosity distance of +the galaxy. We conservatively assume the location of ob- + +6 +XU ET AL. +served outflowing gas is at or beyond the starburst radius +(i.e., rout > r∗), which we will show in Section 4.3 to be a +fair assumption. Finally, we derive G for each galaxy as G += λLλ/(4πr2 +out). We find the majority of galaxies in our sam- +ple have G/G0 < 103 (with two exceptions), while the mean +G/G0 is only ∼ 250. Here, G0 represents the interstellar FUV +intensity of our Milky Way (Habing 1968), which is ∼ 1.6 × +10−3 ergs cm−2 s−1. +As shown in Prochaska et al. (2006), n(Si II*)/n(Si II) +< 10−4 when G/G0 < 103. +Given our observed mean +N(Si II*)/N(Si II) = 0.013, we conclude that indirect UV +pumping commonly contribute < 10−4/0.01 = 1% of the +observed population of Si II*. +This is different from the +fine structure absorption lines detected in γ-ray bursts in +Prochaska et al. (2006), where indirect UV pumping dom- +inates because the radiation field is much stronger. +Thus, collisional excitation is the dominant mechanism for +populating n(Si II*) in our galaxies. We show the relation- +ship between level population ratio and ne in Figure 2 for +Si II. The modelled curves are calculated using the CHIANTI +database (v8.0.7, Del Zanna et al. 2015), assuming collisional +excitation under three different temperatures. The relation +is only weakly dependent on temperature. The critical den- +sity (ncr) for Si II* is defined at the position where n(Si II*) = +n(Si II). For T = 10,000 K, we get ncr ∼ 2000 cm−3. From +Figure 2, we can derive ne from the observed column den- +sity ratio of N(Si II*)/N(Si II) (e.g., Borguet et al. 2012; Xu +et al. 2019). The errors of ne are propagated from the errors +of N(Si II*)/N(Si II). +Given galaxies in our sample have N(Si II*)/N(Si II) in the +range between ∼ 0.001 and ∼ 0.1, we get ne from a few +to ∼ 100 cm−3. The derived ne for each galaxy is listed in +Table 3. For galaxies that show no absorption on N(Si II*), +we present upper limits on ne based on their upper limits of +N(Si II*) (Section 3.2). +4. RESULTS +We summarize the main notations and measured quanti- +ties in this paper at Table 2. We illustrate their details in the +following subsections. +4.1. Outflow Density Distribution and Correlations +In Table 2, we show the statistics of the derived ne for 11 +galaxies which have secure measurements of their N(Si II*). +We find outflows in the galaxies have the mean and median +value ne ∼ 23 cm−3. These values are consistent with what +has been estimated before from absorption-line data for star- +burst galaxies (e.g., ne= 19 – 34 cm−3 in Chisholm et al. +2018). In Figure 3, we show a strong positive correlation +3 In the LOS, the observed N(Si II*)/N(Si II) = n(Si II*)/n(Si II). +Figure 2. +Population ratio of Si II’s fine structure level (Elow = +287 cm−3) to the ground state (Elow = 0 cm−3) versus the electron +number density (ne) (e.g., Osterbrock & Ferland 2006). The mod- +elled curves are calculated using the CHIANTI database (v8.0.7, +Del Zanna et al. 2015), assuming collisional excitation under three +different temperatures. The green vertical line represents the me- +dian value of ne measured from galaxies in our sample. See Section +3.3 for more discussion. +Figure 3. Strong orrelations between outflow electron number den- +sity (ne) and SFR surface density. Galaxies that have ne measure- +ment (Mea.) or upper limits (UL) are shown as the red-filled or +gray-open symbols, respectively. +Kendall’s τ correlation coeffi- +cients are shown at the bottom-right corner of each panel, where +we have considered the upper limits following Akritas & Siebert +(1996). The best linear-fit to all measurements is shown as the or- +ange dashed line, and the fitted slope and intercept are shown in +the top-left corner. The blue line represents the model from Cheva- +lier & Clegg (1985) assuming the outflow gas is in pressure balance +with the wind fluid at the radius of the starburst [Equation (12) and +Section 4.3.2]. + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +7 +Table 2. Summary of the Notations and Measured Quantities +Notation +Definition +Reference +Mean +Median +STDDEV(a) +(1) +(2) +(3) +(4) +(5) +(6) +rout +Actual outflow radius distribution +Section 1 +. . . (b) +. . . +. . . +r∗ +Starburst radius of the galaxy +Section 1 +. . . +. . . +. . . +rn +Outflow radius at the derived ne from Si II* +Section 4.2 +. . . (c) +. . . +. . . +ne +Outflow electron number density +Section 4.1 +22.3 cm−3 +22.8 cm−3 +18.7 cm−3 +rphot +Outflow radius assuming photoionization +Section 4.3.1 +1.6 kpc +1.2 kpc +1.4 kpc +rram +Outflow radius assuming pressure equilibrium +Section 4.3.2 +4.2 kpc +4.1 kpc +2.0 kpc +FF +Outflow volume filling factor +Section 4.4 +0.5% +0.4% +0.4% +Rcl +Outflow cloud size +Section 4.4 +13 pc +5 pc +15 pc +Mcl +Outflow cloud mass +Section 4.4 +1.1 × 104 M⊙ +202 M⊙ +1.0 × 104 M⊙ +Note. – +(*). The first part of this table are the important notations adopted throughout the paper. The second part (beginning +with ne) shows the measured quantities (galaxies with lower and upper limits are excluded). We show the mean, +median values, and standard deviations for these quantities. +(a). Standard deviation. +(b). rout is a range or distribution, which can only be measured from spatially resolved detections of outflows. rout = rn +only when the outflow is a thin bubble (see discussions in Sections 1 and 4.3). +(c). Based on different assumptions, we can measure rn as rphot or rram specifically (illustrated in Section 4.3). +between ne with the SFR surface density. We will discuss the +implications of this below. +In all Figures in this Section, galaxies in our sample with +ne measurement or upper limits are shown as red-filled or +gray-open symbols, respectively. Kendall’s τ correlation co- +efficient (rk) and the probability of the null hypothesis (pk) +are shown at the bottom-right corner. We have taken account +of these upper limits in the Kendall τ test following Akritas +& Siebert (1996). +4.2. Interpretations of Outflow Density in Models +Since our HST/COS spectra are integrated over the whole +line-of-sight (LOS), the derived ne values for outflows also +represent mean values over the velocity profile. To better +interpret the measured ne discussed above, we consider two +common outflow models (e.g., Xu et al. 2022a) as follows. +The simplest case for outflow is an expanding thin shell +model given a mean electron number density (i.e., ne = ns) +and shell thickness (s). In this case, we have: +N(Si II*) = n(Si II*)×s +N(Si II) = n(Si II)×s +(3) +where Nion and nion represent the column and number den- +sity for a certain ion, respectively. For galactic outflows, ne +varies between ∼ 10 cm−3 and ∼ 2000 cm−3 (e.g., Chevalier +& Clegg 1985; Yoshida et al. 2019). In this range, the curve +in Figure 2 is approximately linear, so we have: +n(Si II*) +n(Si II) ≈ ne +ncr +(4) +Combining Equation (3) and (4), we get N(Si II*)/N(Si II) +≈ ne/ncr = ns/ncr. Therefore, for a thin shell outflow model, +the derived ne from N(Si II*)/N(Si II) discussed in Section 3.3 +is just the mean density in the shell. +In the second case, we consider a mass-conserving galac- +tic wind with constant velocity (e.g., Carr et al. 2021). In +this case, the outflow has a density profile n(r) = n0(r/r0)−2, +where r0 is the radius at which the outflow begins, and n0 is +the density at this radius. In this case, we have: +N(Si II) = +� ∞ +r0 +n(Si II)dr = C0 ×n0r0 +(5) +where the integration is from r0 to infinity (note n(∞) = 0) +and C0 = n(Si II)0/n0 is the conversion factor from gas num- +ber density to Si II number density at r0. C0 depends on gas +metallicity and ionization. Similarly, for Si II*, we get: +N(Si II*) = +� ∞ +r0 +n(Si II*)dr +≈ +� ∞ +r0 +n(r)2 +ncr +×C(r)dr += C0n2 +0r0 +3ncr +(6) +where in the second row we have adopted Equation (4) to +replace n(Si II*). Thus, this mass-conserving outflow model +yields N(Si II*)/N(Si II) = n0/(3ncr), i.e., the derived ne from +Section 3.3 is a third of n0. Equivalently, our derived ne cor- +responds to the gas density at rn = +√ +3r0. Hereafter, we define +rn as the radius of the outflows at which the mean ne derived +from fine-structure absorption lines above would occur. + +8 +XU ET AL. +Similarly, if we take a general form of n(r) = n0(r/r0)−γ, +we get: +N(Si II) = C0 ×n0r0 +(γ −1) +N(Si II*) = +C0n2 +0r0 +(2γ −1)ncr +N(Si II*)/N(Si II) = γ −1 +2γ −1 +n0 +ncr +(7) +Thus, in the general form, the derived ne from integrated +spectra corresponds to γ−1 +2γ−1n0. Equivalently, the derived ne is +the gas density at a radius of ( 2γ−1 +γ−1 ) +1 +γ × r0. Given the evidence +for relatively shallow radial density profiles found in outflows +(e.g., Wang et al. 2020; Burchett et al. 2021), we consider +the additional cases γ = 1.5 and 1.2, and get rn = 2.52 r0 and +5.06 r0, respectively.4 We will compare these sizes with those +estimated from our measured values of ne below. +We note that if the outflows are more complex than a gen- +eral form of n(r) = n0(r/r0)−γ, or if there is a range in density +at a given radius, the exact interpretation of our measured ne +and rout will be different and dependent on the actual form of +n(r). We do not dive in this direction, which is beyond the +scope of this paper. +4.3. Derivations of Outflow Distances from ne +For galactic outflows, their radial extent (also referred as +the “outflow distance”) can not be determined given only +LOS integrated spectra (see, e.g., Wang et al. 2020). In fact, +for a continuous outflow, there is no unique way to define +distances (e.g., one could define minimum or maximum val- +ues, or a half-mass radius, etc.). Moreover, while we have +measured a single value for ne for a continuous outflow, this +value will only apply at some specific radius in the outflow +(i.e., rn). Here we will estimate rn based on two methods as +follows. +4.3.1. Outflow Distances Assuming Photoionization +In star-forming galaxies, the ultraviolet outflow absorption +lines (e.g., from O I, Si II, Si III, Si IV) have been found to be +well-described by photoionization models instead of shock- +heating models (Chisholm et al. 2016a). In this case, we +have: +UH = +QH +4πr2 +photonHc −→ rphot = +� +QH +4πUHnHc +(8) +where UH is the ionization parameter, QH is the source emis- +sion rate of ionizing hydrogen photons, c is the speed of light, +and nH is the hydrogen number density of the outflow. On the +4 Note that these expressions diverge for γ ≤ 1, so we do not consider these +shallower profiles. +right side of the Equation (8), we show the solved formula +for outflow distance r assuming photoionization (hereafter, +rphot). +For QH, we adopt the values from spectral energy distribu- +tion (SED) fitting with UV and optical photometry described +in Berg et al. (2022). We note that the resulting QH value is +the intrinsic value, but only a portion of the ionizing photons +can reach the observed outflows due to attenuation by neutral +hydrogen and dust. Thus, we estimate the escaped ionizing +photon rate (QH,esc) as (e.g., Xu et al. 2022b): +QH,esc = QH,tot ×(1−CF)×10−0.4E(B−V)k(912) +(9) +where, for each galaxy, CF represents the covering fraction +of the static ISM component derived from the absorption line +profiles in Xu et al. (2022a), E(B−V) is the internal dust ex- +tinction (derived in Berg et al. 2022), and k(912) = 12.87 is +the extinction curve at the Lyman limit by assuming the ex- +tinction law from Reddy et al. (2016). The second term on +the right of Equation (9) represents the attenuation by neu- +tral hydrogen, where a fraction of CF around the galaxy is +covered by ISM and is generally optically thick to QH. The +third term stands for the attenuation by dust. We note that +this assumes all the extinction arises inside the starburst and +that the outflow is at least as large as the starburst. +For UH, we adopt the values determined from outflow ab- +sorption lines of Si II and Si IV as described in Xu et al. +(2022a). For nH, we approximate it as ∼ ne/1.2, which is +applicable for ionized gas, assuming ∼ 90% hydrogen and +∼ 9% helium and some metals. Overall, we can solve rphot +from Equation (8). The derived results are shown in Table +3, which are in the range of 0.2 – 5 kpc. In the left panel +of Figure 4, we compare the rphot values with r∗, which is +the commonly assumed outflow radius in the literature (e.g., +Heckman et al. 2015; Xu et al. 2022a). We see a strong cor- +relation with rphot∼ 1 to 2r∗ as we move from the smallest to +largest galaxies. +4.3.2. Outflow Distances Assuming Pressure Equilibrium +In this section, we compare our data to the simple ana- +lytic model for a starburst-driven wind by Chevalier & Clegg +(1985) (CC85). To review, CC85 model assumes that mas- +sive stars return mass and kinetic energy to the starburst +through supernova explosions and stellar winds. These ejecta +are thermalized through shocks to form a very hot region of +gas inside the starburst. This gas expands through a sonic +radius (the starburst radius r∗) and becomes a high-velocity +supersonic wind that can accelerate clouds in its path, pro- +ducing the blue-shifted absorption lines we see. This latter +gas is much denser and cooler than the wind fluid. +The CC85 model requires a density of the wind fluid at its +sonic point that depends on SFR/r2 +∗. We see this dependence +for the absorption-line gas in our data (see Figure 3), so it is + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +9 +Figure 4. Comparisons of the derived outflow distances with the starburst radius (r∗). The labels and symbols are the same as Figure 3. +Galaxies that have r as a measurement or a lower limit are shown as the red-filled or gray-open symbols, respectively. Left: Derived rphot +from photoionization outflow models (Section 4.3.1). Right: Derived rram by assuming pressure equilibrium (Section 4.3.2). We show the 1:1 +correlation as the blue lines. In both cases, the derived r correlate strongly with r∗. +worth exploring the connection between the wind fluid and +the gas we measure. We begin by comparing the pressures +implied by the densities we measure via Si II to the pressure +of the hot wind fluid predicted by CC85 at r∗. This pressure +takes two forms, the thermal pressure of the wind fluid and +its ram pressure. In convenient units, at r∗ the total (summed) +pressure is given as: +Ptot/k = 1.79×105SFR×r−2 +∗ [K cm−3] +(10) +where SFR is in units of M⊙/yr, r∗ is in units of kpc, and k is +the Boltzmann constant. If we assume that the gas we mea- +sure with Si II is in pressure equilibrium with the hot wind +fluid, we have: +Ptot = 2nekT +(11) +where ne is the density of the outflows, the factor 2 of is due +to the gas is highly ionized, and we assume T = 10,000 K, for +the gas temperature. Thus, for pressure balance with the wind +fluid, the gas densities traced by Si II at r∗ are proportional to +SFR/r2 +∗ as: +ne ≃ 9×SFR×r−2 +∗ [cm−3] +(12) +This allows us to compare the relationship predicted by the +model to the data. We show this model as the blue line in the +right panel of Figure 3. We see that the densities we measure +are about an order-of-magnitude lower than the model. This +suggests that we are measuring densities at radii significantly +larger than r∗ where pressures are lower. +In this region, the CC85 model shows that ram pressure +is dominant over thermal pressure. Direct measurements of +the radial density profiles for the optical emission-line gas +show that this material is in pressure balance with the wind +ram pressure (Lehnert & Heckman 1996), so this is plausible +for the ionized absorption-line gas as well. We can therefore +compute the location (rram) at which the observed absorption- +line gas is in pressure balance with the hot wind’s ram pres- +sure: +Pram = ˙pSFR/4πr2 +ram +(13) +where Pram is the wind ram pressure and ˙pSFR is the total +momentum flux of the wind, which equals the input momen- +tum from the starburst [reported in Xu et al. (2022a) for our +galaxies]. +Combining Equations (11) and (13), we can solve rram as: +rram = +� +˙pSFR/(8πne kT) +(14) +The derived results are shown in Table 3, and are in the +range ∼ 1 to 8 kpc. In the right panel of Figure 4, we compare +rram with r∗, which shows a strong correlation. In Figure +5, we also compare rphot (Section 4.3.1) with rram for each +object. We find a linear relation with the pressure-based sizes +being typically ∼ 5 times larger than the starburst radius. +4.3.3. Outflow Sizes: Summary +We have discussed three estimates of the radius of the +outflow at the location at which the measured density oc- +curs,based on different assumptions: +1. The first is based on an outflow with a power-law radial +density profile n(r) = n0(r/r0)−γ. It predicts our mea- +sured densities occur at the characteristic radius rn = +1.73, 2.52, and 5.06r∗ for γ = 2, 1.5, and 1.2 respec- +tively (Section 4.2). Here r∗ is the radius of the star- +burst. +2. The second assumes the gas is photoionized by a frac- +tion of the starburst ionizing flux that reaches the out- + +10 +XU ET AL. +Figure 5. +Comparisons between outflow distances (r) derived +from assuming photoionization (y-axis) and pressure equilibrium +(x-axis). The labels and symbols are the same as Figure 3. We find +the pressure-based sizes are typically ∼ 2 to 5 times larger than the +values derived from photoionization models, with the ratio decreas- +ing from the small to large cases. +flow. This estimates that the measured densities occur +at a typical distance ∼ 1 to 2r∗ (Section 4.3.1). +3. Finally, we have assumed that the gas we measure is in +pressure balance with the ram pressure of the hot wind +fluid. This estimates that the measured densities occur +at a typical distance ∼ 4 to 5r∗ (Section 4.3.2). +Given the systematic uncertainties in these estimates, we +regard this level of agreement as satisfactory. In all cases, we +are tracing the region of the outflow where densities are high +enough to measure with our technique. As explained at the +beginning of Section 4.3, the maximum extent of the outflow +could be considerably larger than rn. +4.4. Other Important Outflow Parameters +Besides the density and structure of outflows, there are var- +ious essential parameters of outflows that have rarely been +measured from observations. In this sub-section, we con- +strain these parameters for outflows in our sample. We com- +pare them with simulations of outflows and discuss the im- +plications in Section 5.3. +We start with the volume filling factor (FF) of the ob- +served outflow clouds, where we treat the absorbing material +as an ensemble of clouds (e.g., Fielding & Bryan 2022). This +yields: +FF = Ncl ×4/3πR3 +cl +AUVrn +(15) +where Ncl is the number of outflow clouds entrained in the +hot wind at the outflow distance rn, and AUV is the cross- +sectional area of the starburst UV continuum. We also have +the definition of outflow column density (NH) as: +NH = Ncl ×4/3πR3 +cl ×nH +AUV +(16) +One can estimate FF from Equations (15) and (16) as: +FF = NH +nHrn +(17) +where, in this equation, all variables on the right side can be +measured for at least part of the galaxies in our sample (see +Sections 3.1, 4.1, and 4.3). +In Xu et al. (2022a), we also derived the area covering frac- +tion of the outflow (CF) from the Si II and Si IV absorption +lines (Sections 3.1). We can rewrite CF as: +CF = βsh × Ncl ×πR2 +cl +AUV +(18) +where βsh is a coefficient between 0 and 1 to account for the +shadowing effects. This is because the projected areas by dif- +ferent outflow clouds in the LOS can overlap each other so +that their total covered area drops by the factor of βsh. This +factor depends on 1) the overall spatial distribution of out- +flow clouds; and 2) the second term of Equation (18), i.e., +the number and relative size of each cloud to AUV. In Ap- +pendix A, we show how to estimate βsh from Monte Carlo +simulations and the measured CF in Xu et al. (2022a). For +the 22 galaxies analyzed in this paper, we get βsh in the range +of ∼ 0.3 to 0.6. +For simplicity of symbols, we define CFsh = CF/βsh. From +Equations (15) and (18), we can solve the size of the outflow +clouds as: +Rcl = 3 +4 +FF +CFsh +rn +(19) +Using the above expression for FF in Equation (17), this +can be rewritten as: +Rcl = 3 +4 +NH +nHCFsh +(20) +This shows that Rcl does not depend on rn and can be com- +puted from directly measured quantities. Once we have Rcl +constrained, we can combine Equations (18) and (20) to get +Ncl as: +Ncl = CFsh × AUV +πR2 +cl += CFsh × R2 +UV +R2 +cl +(21) +where RUV is the UV size of a galaxy and we approximate +it as r∗ that we measured from the HST/COS acquisition im- +ages. Note that Ncl is also independent of rn. We find the +mean and median values of Ncl are 105.5 and 104.9, respec- +tively. + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +11 +Finally, we can estimate the average mass of the individual +outflow clouds (Mcl) by: +Mcl = 4 +3πR3 +clnHµmp +(22) +where µ ∼ 1.4 is the average atomic mass per proton and mp +is the proton mass. +We summarize the statistics for the derived FF, Rcl, Mcl +values in Table 2, where their values for individual galaxy +are shown in the last three columns in Table 3. For FF, which +is the only derived parameter dependent on rn, we have as- +sumed rn = rram. If we assume rn = rphot, the derived Rcl and +Mcl stay the same, while FF for each galaxy becomes larger +with a mean and median value of 1.5% and 13 pc, respec- +tively. In Section 5.3, we compare these measurements with +common outflow models, and discuss their implications. +5. DISCUSSION +5.1. Comparisons with Other Outflow Density +Measurements +While we are presenting the first examples of density mea- +surements for the warm ionized gas in outflows based on +absorption lines, measurements of densities for the optical +emission-line gas in outflows have been available in low- +redshift starbursts for over thirty years (Heckman et al. 1990). +Here we summarize what has been learned from the optical +emission-line gas and compare the results to our new data. +For low-redshift starbursts, ne is commonly directly mea- +sured using the density-sensitive ratio of the [S II] 6717 and +6731 emission lines. These data can be used to map out the +radial variation in ne, and show a steady radial decline from +∼ 500 to 1000 cm−3 in the starburst to ∼ 50 to 100 cm−3 at +distances several times larger than r∗ (e.g., Heckman et al. +1990; Lehnert & Heckman 1996; Yoshida et al. 2019; Perna +et al. 2020; Marasco et al. 2022). The [S II] flux ratio reaches +its low-density limit at ne ∼ 10 to 100 cm−3 (Osterbrock & +Ferland 2006), so direct measurements of ne at larger radii +(lower densities) are not possible. This would be consistent +with the lower values of ne that we typically get in our sam- +ple, if we are probing larger radial scales in this case. +We have shown that with the assumption that the absorb- +ing gas is in pressure balance with the ram pressure of the +wind fluid, we do in fact derive large outflow radius. Is this +a plausible assumption? We believe it is, because both the +emission- and absorption-lines trace the warm ionized gas +phase. +Since observations establish that the density (and +pressure) profiles measured in the emission-line gas are con- +sistent with the radial profile of the wind ram pressure (Heck- +man et al. 1990; Lehnert & Heckman 1996), this supports +adopting the assumption of pressure balance to calculate Rram +(Section 4.3.2). +5.2. Comparisons with Other Measurements of Outflow +Structures +The outflow radii we derive assuming ram-pressure con- +finement are in the range rram ∼ 1 to 10 kpc. These size +scales are consistent with those measured for the outflows +traced by optical emission lines for starbursts with a range +in SFR similar to our sample (e.g., Armus et al. 1995; Lehn- +ert & Heckman 1996; Martin 1998; Ho et al. 2014; Yoshida +et al. 2019). For the sample of dwarf SF galaxies in Marasco +et al. (2022), which has weaker SFR than ours (∼ 10 times +smaller), they get r ∼ 1 kpc. This is around the lower bound +of our galaxies as expected from their lower SFR. +Perhaps a more revealing comparison is to the size scales +measured using resonantly-scattered emission arising from +the same gas that produces the absorption lines seen directly +along the LOS to the starburst (Rubin et al. 2011; Martin et al. +2013). This has been done recently using IFU instruments, +including VLT/MUSE and Keck/KCWI, to map out Mg II +emission lines surrounding starburst galaxies at intermediate +redshifts (e.g., Rupke et al. 2019; Burchett et al. 2021; Zabl +et al. 2021; Shaban et al. 2022). These data detect emission +out to radii of ∼ 10 to 20 kpc, with half-light radii of 5 to 10 +kpc. The latter is quite similar to values we derived for rram +for our sample. +Given a typical aperture size of 1′′ – 2′′ in current (non- +IFU) spectrographs, these sizes imply that a significant frac- +tion of the resonantly scattered line emission could lie outside +the aperture. Thus, this missing light helps explain why the +scattered (or fluorescently-reprocessed) emission-lines from +the outflow are often quite weak (e.g., Erb et al. 2012; Steidel +et al. 2018; Wang et al. 2020; Xu et al. 2022a), as can be seen +in radiative transfer models of outflows (e.g., Prochaska et al. +2011; Scarlata & Panagia 2015; Carr et al. 2021; and Huberty +et al. in prep). We explore this idea further with the current +data. In Figure 6, we show histograms of the ratio rram/rCOS +and rphot/rCOS, where rCOS is the projected physical size of +the COS aperture for a given galaxy. We find that the ra- +tios of rram/rCOS are > 1, while rphot/rCOS are most often ≲ +1. Thus, the larger sizes measured for rram may be consistent +with the relatively weak emission-lines seen in these galaxies +(e.g., Wang et al. 2020; Xu et al. 2022a). +Another apt comparison is to maps of the outflows of neu- +tral gas traced by the Na I D optical absorption-line (1 – 10 +kpc, e.g., Martin 2006; Rupke & Veilleux 2013; Perna et al. +2019; Avery et al. 2021; 2022). Our data are complementary +to these studies since they pertain to the ionized phase of the +outflow and represent integrals over the line-of-sight directly +into the starburst. Our data also provide information on key +parameters like the densities, filling factors, radii and masses +of the outflowing clouds. +Besides the outflows discussed above, SF galaxies can +exhibit outflows features in many other wavelength bands + +12 +XU ET AL. +Figure 6. Histograms showing the comparisons between the measured outflow radius (rram or rphot) and the projected physical size of the +HST/COS aperture for each galaxy. The large ratio of rram/rCOS suggests that the scattered or fluorescent emission lines should be weak in our +galaxies, which is consistent with what has been found in the literature. See details in Section 5.2. +and line diagnostics, where outflow distances are measured +(see reivews in Heckman & Thompson 2017; Veilleux et al. +2020). These include very hot gas detected in X-ray (at ∼ 1 +– 10 kpc, e.g., Strickland & Heckman 2007; Li et al. 2011; +Zhang et al. 2014), and atomic and molecular outflows ob- +served in infrared to radio bands (e.g., from [C II] and CO, +out to radii of a few kpc, Walter et al. 2002; Chisholm & +Matsushita 2016; Stuber et al. 2021). +Again, we emphasize that these various outflow sizes are +defined in different ways. In our case, we are defining the +size to be the radius at which our measured densities occur. +For the emission-line data, the sizes are typically just defined +by the radius at which the emission becomes undetectably +faint. Additionally, different diagnostics of outflows in dif- +ferent galaxies can reach intrinsically distinct scales, and the +relationships between them are not entirely clear. Detailed +comparisons are beyond the scope of this paper, but we plan +to study the relationships between different diagnostics and +phases of galactic outflows in future papers. +5.3. Comparisons with Models and Simulations of Galactic +Outflows +Galactic winds are complex and difficult to model be- +cause one needs to simultaneously capture the large spatial +scales for the whole galaxy and the fundamentally small- +scale process happening between the galaxy’s ISM/CGM +and the wind (see Naab & Ostriker 2017; and references +therein). Currently, a compelling model (e.g., Fielding & +Bryan 2022) comprises 1) a hot, volume-filling wind com- +ponent driven by thermalized ejecta of massive stars (Cheva- +lier & Clegg 1985) 5 and 2) a cold to warm component in +the form of embedded clouds, which are entrained by the hot +wind. This component produces the observed outflows seen +in UV absorption lines (e.g., Xu et al. 2022a). The exchange +of mass/momentum/energy between these two components +is in the turbulent radiative mixing layer (e.g., Gronke & Oh +2020; Tan et al. 2021; Fielding & Bryan 2022). +Two time-scales control the fate of the outflow clouds +(Gronke & Oh 2020): 1) the clouds grow by cooling of the +hot wind in a time scale of tcool, which depends mainly on +the pressure and metallicity; and 2) the clouds are destroyed +by turbulent shredding in a time scale of tmix. We have tmix ∝ +Rcl/Vturb, where Rcl is the average radius of the outflow clouds +and Vturb is the turbulent velocity. For large outflow clouds, +tcool < tmix so that the clouds can grow. For smaller outflow +clouds, the clouds are shredded before they can grow. Thus, +parameters related to Rcl are important for galactic outflows +but have rarely been constrained from observations. +In Section 4.4, we have shown that, based on our mea- +surements of outflow density and distances, we can constrain +these parameters, including FF, Rcl, and Mcl. Here we attempt +to compare our measurements to common outflow models. +We can use the criterion derived by Gronke & Oh (2020) +for the critical (minimum) size for a cloud to survive/grow +when exposed to the ram pressure of the wind: +Rcrit ∼ +T 5/2 +cl,4 Mwind +P3Λmix,−21.4 +χ +100α−1pc +(23) +5 This hot gas is only detectable inside the starburst (Heckman & Thompson +2017), where its density is relatively high. + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +13 +Here, Tcl,4 is the cloud temperature in units of 104 K, +Mwind is the Mach number of the hot wind fluid, P3 is the +cloud pressure in units of 103 K cm−3, Λmix,−21.4 is the value +of the cooling function in the turbulent mixing layer (in units +of 10−21.4 cm3 erg s−1), χ is the ratio between the cloud and +wind density, and α is a ‘fudge factor’ of order unity. Under +this model, if a cloud exposed to the hot wind has smaller +sizes than Rcrit, it is destroyed/shredded before being accel- +erated. +We assume Tcl = 104 K and can then use our measured val- +ues of ne to compute P3. We further take α = 1 and the fidu- +cial value of Λmix. To measure χ we adopt the model above +for clouds in pressure balance with the wind ram pressure. +We use the Chevalier & Clegg (1985) wind solution to obtain +Mwind assuming rout = rram (see Section 4.3.2 and left panel +of Figure 4). For balancing between the wind ram pressure +and the cloud thermal pressure, we have +Pcl = 2nclkTcl = ρwv2 +w +(24) +Then, since ρcl = nclmp, we have: +χ = ρcl +ρw += v2 +wmp +2kTcl +(25) +Finally, we adopt the Chevalier & Clegg (1985) model and +assume a wind velocity of vw = 1800 km/s (Strickland & +Heckman 2009), leading to a value of χ ∼ 2×104. We show +the results in Figure 7, in which we compare our derived Rcl +with Rcrit for our sample. We find that the estimated values of +Rcl all lie close to Rcrit. Within the uncertainties, the growth +criterion is satisfied in all 11 cases with ne measurements. +For the other 11 galaxies with ne as upper limits, the derived +values of Rcl and Rcrit are both lower limits (gray-open sym- +bols). However, Equations (20) and (23) show that both sizes +are inversely proportional to the density. This means that the +ratio of Rcl/Rcrit is independent of density, and thus we can +evaluate the growth criterion even in these cases. Within the +uncertainties, 20 out of 21 cases6 satisfy this criterion, i.e., +Rcl are large enough for them to survive under the impact of +the hot wind. +The fact that the cloud sizes are similar to Rcrit could be un- +derstood if the pre-existing population of clouds initially had +a power-law distribution of sizes (Ncl ∝ R−γ +cl ) that declines +with increasing Rcl. Then only the clouds with Rcl ≳ Rcrit +survive the interaction with the wind, while clouds with sizes +≫ Rcrit are rare (i.e., having a small total covering factor). +Additionally, in our assessment of cloud survival, Equa- +tions (20) and (23) imply that the ratio of Rcl/Rcrit depends +6 Among the total sample of 22 galaxies, one (J1612+0817) does not have +NH reported in Xu et al. (2022a) since its Si IV doublet troughs are in a +detector gap of HST/COS. Thus, its Rcl/Rcrit ratio is unknown. +Figure 7. Comparisons of the derived outflowing cloud radii (Rcl) +with the critical (minimum) radius for a cloud to survive/grow when +exposed to the ram pressure of the hot wind (Gronke & Oh 2020). +The labels and symbols are the same as Figure 3. Galaxies that +have Rcl as measurement or lower limits are shown as the red-filled +or gray-open symbols, respectively. The blue line represents the 1:1 +relationship. Within the uncertainties, 20/21 outflows have enough +cloud sizes large enough to survive. See discussion in Section 5.3. +only on the ratio of the column density to covering factor +[since we adopted fixed values for all the terms in Equa- +tion (23)]. Empirically, there is relatively small variation in +CFsh. In this case, the relatively small spread in the values +of Rcl/Rcrit seen in Figure 7 could imply that the total column +densities of the absorbing clouds in the outflows are directly +connected to the cloud-survival requirement. Future simula- +tions of galactic winds may answer these implications. +6. CONCLUSION AND FUTURE WORK +We have reported here the first direct measurements of the +density (ne) in outflows from starburst galaxies traced by ul- +traviolet absorption lines. These measurements were made +using COS on HST to measure the ratio of the column den- +sity of fine structure excited transitions of Si II (i.e., Si II*) +to those of the Si II resonance transitions. The sample of 22 +galaxies was drawn from Berg et al. (2022) and Heckman +et al. (2015), and limited to cases with SNR > 5, galaxy FUV +radii < 1.5′′, and detected outflows. Our main results are as +follows: +• We were able to measure ne in 11 cases and set upper +limits in the other 11 galaxies. The median density was +23 cm−3. We found a strong correlation between ne and +the star-formation rate per unit area in the starburst. +• Since the value of ne is derived along a line-of-sight, +its meaning is only simple in the case of an expanding +shell with constant density. In the case of a continu- +ous outflow in which the density drops with radius, we + +14 +XU ET AL. +showed that for radial density profiles with power-law +indices of –2, –1.5, and –1.2, the measured densities +would pertain to gas at respective radii of 1.7, 2.5, and +5.1 times the radius at which the outflow begins (taken +to be the starburst radius). +• Using the measured values of ne, we made two indirect +estimates of the radius of outflows (rn) at which this +density applies. The first assumes that the gas is photo- +ionized by radiation from the starburst. This required +making estimates for the fraction of intrinsic ionizing +radiation leaking out of the starburst and into the out- +flow. Typical radii from this method are 1 to 2 times +the starburst radius. We then assumed that the absorb- +ing gas clouds are in pressure equilibrium with the hot +wind fluid. These radii are typically 4 to 5 times the +starburst radius. +• We used the values of ne and our measured values for +the total hydrogen column density and the covering +fraction of the outflow to estimate the radii and masses +of the absorbing clouds. We found median values of ∼ +5 pc and 200 M⊙ respectively. We also estimated the +volume filling factor of the population of these clouds, +with typical values of 10−3 to 10−2. +• We have compared the outflow clouds sizes to theoret- +ical models in which clouds interact with a supersonic +wind fluid. We find that in 20 out of 21 cases, the esti- +mated cloud sizes exceed the critical cloud size, mean- +ing that these clouds are predicted to survive and grow +as they interact with a hot supersonic wind. +This is the first time that various essential absorption-line +outflow parameters have been estimated from observations, +including outflow density, volume filling-factor, and cloud +sizes/masses. There are plenty of compelling future projects +to do in both observations and simulations. For example, +how do our derived ne and rn values compare to direct mea- +surements from spatially resolved observations? What are +the differences and/or connections between the ne and out- +flow sizes measured from emission and absorption line out- +flows? Given the detected warm outflows, can we provide +constraints on the hot wind properties? How can the mea- +sured radii and masses of the absorbing clouds help constrain +simulations of outflows? We have plans to tackle some of +these questions in our future work. + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +15 +X.X. and T.H. thank M. Gronke, D. Fielding, and G. Bryan +for interesting discussions. +1 +2 +The CLASSY team is grateful for the support for this pro- +gram, HST-GO-15840, that was provided by NASA through +a grant from the Space Telescope Science Institute, which +is operated by the Associations of Universities for Research +in Astronomy, Incorporated, under NASA contract NAS5- +26555. BLJ thanks support from the European Space Agency +(ESA). CLM gratefully acknowledges support from NSF +AST-1817125. The CLASSY collaboration extends special +gratitude to the Lorentz Center for useful discussions during +the "Characterizing Galaxies with Spectroscopy with a view +for JWST" 2017 workshop that led to the formation of the +CLASSY collaboration and survey. +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +Funding for SDSS-III has been provided by the Alfred +P. Sloan Foundation, the Participating Institutions, the Na- +tional Science Foundation, and the U.S. Department of +Energy Office of Science. +The SDSS-III web site is +http://www.sdss3.org/. +15 +16 +17 +18 +19 +SDSS-III is managed by the Astrophysical Research Con- +sortium for the Participating Institutions of the SDSS- +III Collaboration including the University of Arizona, the +Brazilian Participation Group, Brookhaven National Labora- +tory, Carnegie Mellon University, University of Florida, the +French Participation Group, the German Participation Group, +Harvard University, the Instituto de Astrofisica de Canarias, +the Michigan State/Notre Dame/JINA Participation Group, +Johns Hopkins University, Lawrence Berkeley National Lab- +oratory, Max Planck Institute for Astrophysics, Max Planck +Institute for Extraterrestrial Physics, New Mexico State Uni- +versity, New York University, Ohio State University, Penn- +sylvania State University, University of Portsmouth, Prince- +ton University, the Spanish Participation Group, University +of Tokyo, University of Utah, Vanderbilt University, Univer- +sity of Virginia, University of Washington, and Yale Univer- +sity. +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +This research has made use of the HSLA database, devel- +oped and maintained at STScI, Baltimore, USA. +37 +38 +CHIANTI is a collaborative project involving George Ma- +son University, the University of Michigan (USA), and the +University of Cambridge (UK). +39 +40 +41 +Facilities: HST (COS) +Software: +astropy (Astropy Collaboration et al. 2022) +CalCOS (STScI), jupyter (Kluyver et al. 2016), MPFIT +(Markwardt 2009) +REFERENCES +Akritas, M. 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ESTIMATIONS OF βsh +As discussed in Section 4.4, different outflow clouds can “shadow” each other and produce smaller area covering fractions +(CF) in the LOS (e.g., Sun et al. 2017; Xu et al. 2020). Here we conduct a Monte Carlo (MC) experiment in two-dimension to +estimate the shadowing parameter, βsh, for Equation (18). +Given Ncl outflow clouds with radius Rcl, we randomly distributed them within the area of AUV. We also assume the ratio of +AUV/πR2 +cl = 4000. We have tested that the variations of this ratio have little effect on our final results below. We then vary Ncl +from 1000 to 15,000 and calculate two quantities: 1) Ncl × πR2 +cl/AUV, which represents the CF value if outflow clouds do not +shadow each other at all; and 2) the true CF by checking each spot in AUV to see if they are covered by any of the outflow clouds. +We show these two quantities in the y- and x-axis in Figure 8, respectively. +We find when Ncl grows, the true CF initially increases fast but then slows down. This is because when Ncl is small, we do +not expect to have strong shadowing effects given the relatively large area of AUV compared to the projected size of each outflow +cloud (i.e., πR2 +cl). But when Ncl is large (≳ 4000), the shadowing effects become more significant and the growth of the true CF +is slower. +For galaxies in our sample, we have measured the true CF from “down-the-barrel” observations of UV absorption lines (Xu +et al. 2022a). Thus, we can estimate y = Ncl ×πR2 +cl/AUV for each galaxy based on the true CF and the curve in Figure 8. Then we +can calculate βsh from Equation (18). For the 22 galaxies analyzed in this paper, we get βsh in the range of ∼ 0.3 to 0.6. +Figure 8. Comparisons between CF assuming no shadowing versus the true CF from MC simulations. We also show the number of clouds +(Ncl) for several positions on the line. See discussion in Appendix A. +B. TABLES +Here we present the tables for the derived quantities for each galaxy. + +CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS +19 +Table 3. Measured Parameters for Galaxies in the Combined Sample(1) +Object +log(SFR) +log(r∗) +log(NH) +CF(Si II) +N(Si II) +N(Si II*) +log(ne) +Adust +log(Qeff) +log(rphot) +log(rram) +log(FF) +log(RCloud) +log(MCloud) +βsh +M⊙/yr +kpc +cm−2 +1012cm−2 +1012cm−2 +cm−3 +mags. +s−1 +kpc +kpc +kpc +M⊙ +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +(12) +(13) +(14) +(15) +(16) +J0021+0052 +1.07 +-0.05 +20.78 +0.30 +243.40 +<62.55 +<2.41 +1.81 +52.76 +>-1.09 +>-0.09 +>-2.93 +>-3.09 +>1.20 +0.56 +J0150+1308(H15) +1.50 +0.10 +20.67 +0.85 +720.63 +18.89+0.50 +−0.50 +1.36+0.04 +−0.05 +2.81 +53.80 +0.09+0.12 +−0.16 +0.65+0.20 +−0.09 +-2.73+0.10 +−0.21 +-2.54+0.24 +−0.24 +1.81+0.73 +−0.72 +0.38 +J0808+3948 +1.26 +-0.59 +19.99 +0.40 +741.47 +<45.28 +<1.73 +2.07 +54.10 +>0.04 +>0.35 +>-3.49 +>-3.45 +>-0.54 +0.46 +J0823+2806 +1.48 +-0.30 +20.56 +0.96 +187.19 +13.89+4.50 +−4.50 +1.82+0.13 +−0.18 +3.46 +53.09 +-0.60+0.14 +−0.16 +0.41+0.19 +−0.11 +-3.06+0.22 +−0.24 +-3.18+0.30 +−0.27 +0.35+0.92 +−0.83 +0.34 +J0926+4427 +1.03 +0.12 +20.90 +0.30 +999.09 +7.93+0.71 +−0.71 +0.84+0.13 +−0.18 +1.21 +53.98 +0.48+0.13 +−0.11 +0.68+0.09 +−0.10 +-2.01+0.22 +−0.18 +-1.41+0.26 +−0.23 +4.67+0.80 +−0.70 +0.56 +J0938+5428 +1.05 +0.01 +20.86 +0.34 +1282.70 +14.40+1.93 +−1.93 +0.99+0.15 +−0.23 +1.00 +53.59 +0.15+0.16 +−0.14 +0.61+0.12 +−0.14 +-2.14+0.28 +−0.20 +-1.41+0.32 +−0.26 +4.81+0.97 +−0.82 +0.61 +J1016+3754 +-1.17 +-0.61 +20.74 +0.25 +44.12 +<18.21 +<2.65 +0.68 +51.99 +>-1.54 +>-1.33 +>-1.97 +>-3.29 +>0.85 +0.59 +J1024+0524 +0.21 +-0.27 +21.00 +0.62 +384.52 +<59.64 +<2.16 +0.90 +53.09 +>-0.56 +>-0.39 +>-2.16 +>-2.80 +>1.81 +0.48 +J1025+3622 +1.04 +0.20 +20.84 +0.69 +612.97 +2.57+0.67 +−0.67 +0.56+0.11 +−0.14 +1.65 +53.83 +0.58+0.12 +−0.12 +0.82+0.11 +−0.10 +-1.93+0.17 +−0.16 +-1.56+0.21 +−0.19 +3.95+0.65 +−0.59 +0.38 +J1144+4012 +1.51 +0.26 +20.71 +0.92 +1482.90 +23.40+1.06 +−1.06 +1.14+0.04 +−0.05 +2.92 +53.68 +0.21+0.11 +−0.16 +0.77+0.17 +−0.09 +-2.59+0.11 +−0.18 +-2.30+0.22 +−0.21 +2.31+0.67 +−0.65 +0.37 +J1148+2546 +0.53 +0.37 +21.06 +0.87 +1615.60 +<21.59 +<1.06 +2.98 +52.76 +>-0.18 +>0.31 +>-1.71 +>-1.67 +>4.10 +0.47 +J1150+1501 +-1.33 +-0.87 +20.71 +0.71 +380.18 +<53.65 +<2.12 +2.07 +51.23 +>-1.54 +>-1.14 +>-1.65 +>-3.04 +>1.07 +0.49 +J1200+1343 +0.75 +-0.35 +20.85 +0.84 +789.16 +<47.91 +<1.73 +2.58 +53.17 +>-0.39 +>0.09 +>-2.37 +>-2.83 +>1.30 +0.33 +J1253–0312 +0.56 +-0.11 +21.00 +0.91 +404.83 +12.42+2.38 +−2.38 +1.43+0.12 +−0.16 +2.51 +52.96 +-0.36+0.20 +−0.20 +0.14+0.10 +−0.10 +-1.96+0.20 +−0.16 +-2.44+0.24 +−0.21 +2.17+0.73 +−0.64 +0.29 +J1359+5726 +0.42 +0.17 +21.05 +0.81 +979.65 +<39.70 +<1.55 +1.35 +53.31 +>-0.17 +>0.02 +>-1.91 +>-2.26 +>2.85 +0.43 +J1416+1223 +1.57 +-0.26 +20.27 +0.54 +175.12 +8.24+0.71 +−0.71 +1.62+0.07 +−0.09 +2.20 +53.63 +-0.18+0.16 +−0.15 +0.56+0.15 +−0.11 +-3.29+0.14 +−0.17 +-3.15+0.22 +−0.21 +0.22+0.66 +−0.63 +0.40 +J1428+1653 +1.22 +0.32 +20.70 +0.62 +436.32 +2.32+0.77 +−0.77 +0.66+0.14 +−0.20 +1.04 +54.26 +0.63+0.17 +−0.13 +0.86+0.13 +−0.15 +-2.21+0.26 +−0.19 +-1.72+0.30 +−0.25 +3.58+0.90 +−0.79 +0.43 +J1429+0643 +1.42 +-0.06 +20.81 +0.67 +675.03 +17.69+0.78 +−0.78 +1.36+0.06 +−0.07 +1.58 +53.80 +0.05+0.09 +−0.11 +0.61+0.10 +−0.06 +-2.55+0.11 +−0.13 +-2.29+0.17 +−0.16 +2.55+0.53 +−0.49 +0.44 +J1448–0110 +0.39 +-0.60 +20.50 +0.83 +106.60 +3.22+0.92 +−0.92 +1.42+0.12 +−0.17 +2.76 +52.45 +-0.68+0.13 +−0.11 +0.06+0.10 +−0.10 +-2.38+0.20 +−0.16 +-2.84+0.24 +−0.20 +0.96+0.73 +−0.64 +0.34 +J1545+0858 +0.37 +-0.31 +21.38 +0.61 +1435.50 +<45.45 +<1.44 +1.55 +53.12 +>-0.29 +>0.05 +>-1.50 +>-1.66 +>4.53 +0.51 +J1612+0817 +1.58 +0.01 +. . . +0.80 +2665.30 +<230.03 +<1.89 +2.75 +53.83 +... +>0.42 +... +... +... +... +J2103–0728(H15) +1.29 +-0.32 +20.47 +0.84 +1526.10 +<209.23 +<2.10 +4.18 +53.43 +>-0.45 +>0.17 +>-3.20 +>-3.44 +>-0.16 +0.40 +Notes: (1) Measured parameters for 22 galaxies in our combined sample that have high SNR spectra of Si II and Si II* (see Section 3.2). Galaxies from Heckman et al. (2015) is +marked as (H15). Descriptions for each column: (2) – (6) are adopted from Xu et al. (2022a) (see Section 3.1 for a summary); (2) The log of star-formation rate (SFR) of the galaxy; +(3) The log of starburst radius of the galaxy, where we take r∗ = 2 ×r50 from Xu et al. (2022a); (4) The log of total hydrogen column density of the outflow; (5) The mean covering +fraction derived from Si II outflow absorption lines; (6) The column density of Si II in the outflows; (7) The column density of excited states of Si II (i.e., Si II*) measured in the +outflow absorption troughs for each galaxy (Section 3.2); (8) The log of electron number density of the outflows (Section 3.3); (9) Dust extinction derived from SED fittings (Section +4.3.1); (10): The effective ionizing photon rate per second for the outflows (Section 4.3.1); (11) The radius of the observed outflows derived from photoionization models (Section +4.3.1); (12) The radius of the observed outflows derived from assuming pressure equilibrium (Section 4.3.2); (13) The volume filling factor for outflows (Section 4.4); (14) and (15) +The average radius and mass of the outflowing clouds (Section 4.4) adopting rram (Section 4.4). 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' AlbaNova University Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' SE-10691,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Sweden 8Minnesota Institute for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' University of Minnesota,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 116 Church Street SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Minneapolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' MN 55455,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' USA 9National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2-21-1 Osawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Tokyo 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Japan 10Waseda Research Institute for Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Faculty of Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Waseda University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3-4-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Okubo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Shinjuku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Tokyo 169-8555,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Japan Submitted to AASJournal ApJ ABSTRACT Galaxy formation and evolution are regulated by the feedback from galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Absorption lines provide the most widely available probe of winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' However, since most data only provide information integrated along the line-of-sight, they do not directly constrain the radial structure of the outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this paper, we present a method to directly measure the gas electron density in outflows (ne), which in turn yields estimates of outflow cloud properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', density, volume filling-factor, and sizes/masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also estimate the distance (rn) from the starburst at which the observed densities are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We focus on 22 local star-forming galaxies primarily from the COS Legacy Archive Spectroscopic SurveY (CLASSY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In half of them, we detect absorption lines from fine structure excited transitions of Si II (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We determine ne from relative column densities of Si II and Si II*, given Si II* originates from collisional excitation by free electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find that the derived ne correlates well with the galaxy’s star-formation rate per unit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' From photoionization models or assuming the outflow is in pressure equilibrium with the wind fluid, we get rn ∼ 1 to 2r∗ or ∼ 5r∗, respectively, where r∗ is the starburst radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Based on comparisons to theoretical models of multi-phase outflows, nearly all of the outflows have cloud sizes large enough for the clouds to survive their interaction with the hot wind fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Most of these measurements are the first-ever for galactic winds detected in absorption lines and, thus, will provide important constraints for future models of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Keywords: Galactic Winds (572), Galaxy evolution (1052), Galaxy kinematics and dynamics(602), Starburst galaxies (1570), Ultraviolet astronomy (1736), Galaxy spectroscopy (2171) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' INTRODUCTION Corresponding author: Xinfeng Xu xinfeng@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='edu ∗ Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in As- tronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Galactic winds are essential to the evolution of galaxies and the intergalactic medium (IGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In star-forming galax- ies (without accreting black holes), these winds are driven by mass, energy, and momentum supplied by star-formation, in the form of radiation, stellar winds, and supernovae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The latter two result in the cre- ation of a tenuous and energetic wind fluid that flows out and accelerates existing gas clouds, which are observable as warm to cold outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Galac- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='11498v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='GA] 27 Jan 2023 2 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' tic winds and the outflows they drive are able to transport mass/energy/momentum against the gravitational potential of the hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, they have been proposed to explain vari- ous feedback effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', regulating the star formation rate (SFR) of the host galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Martin 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Rupke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Cazzoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Heckman & Borthakur 2016), chemically enriching the circum-galactic medium (CGM) and IGM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Dalcanton 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2018), and explaining the “overcooling problem" in cosmological simulations by reducing the baryon fractions in galactic discs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Outflows can also clear neutral gas away from young star- bursts and regulate the escape of Lyman continuum photons, which is responsible for the cosmic reionization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Heck- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Hogarth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In the last few decades, galactic winds and outflows have been intensely studied in the literature, especially in star- forming and starburst galaxies, which commonly host pow- erful outflows (see reviews in Heckman & Thompson 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Rupke 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Outflows are multi-phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Fluetsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022), but the most abundant data probe the warm ionized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This material is believed to be accelerated by the combined momentum of a hot wind fluid created by stellar ejecta and radiation pressure (Heckman & Thompson 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The two major ways to detect the warm ionized gas are from rest-frame UV absorption lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', O I, Si II, Si IV, and C IV), and optical emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', [O III] and Hα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Even though both emission and absorption lines can show kinematic features that represent the outflows, they are thought to arise from different environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Emission lines are weighted towards the denser environments (brightness scales with outflow elec- tron density (ne) squared), while ne have been found to be ∼ 100 – 1000 cm−3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' On the contrary, absorption lines trace lower density environments (optical depth scales with ne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2015) suggests that UV absorption lines can trace larger-scale galactic outflows and are more re- liable tracers of warm gas in starburst-driven outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' There exist well-developed ways to constrain various im- portant outflow parameters from the absorption lines, includ- ing outflow velocity (Vout), ionization, and column density (NH) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Martin 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Rupke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Scarlata & Panagia 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Heck- man & Borthakur 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The strength of the absorption outflows and their potential feedback effects can then be quantified by their mass, mo- mentum, and energy rates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', ˙Mout ∝ NHroutVout, ˙pout ∝ NHroutV 2 out, and ˙Eout ∝ 1 2NHroutV 3 out, respectively, where rout is the assumed radius of the outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The major uncertainty for these outflow rates is from rout (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is because these surveys of galactic outflows in absorption lines only have integrated spectra in a single aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' It is not possible to measure rout directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Most previous studies either assume a fiducial radius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', 1 – 5 kpc in Martin 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Rupke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012), or assume rout starts at a few times the starburst radius (r∗, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Chevalier & Clegg 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Recently, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2020) showed that the non- resonant emission lines are much weaker and narrower than the corresponding absorption lines in a sample of starburst galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' They suggest that observed absorbing material for outflows could be located at radii significantly larger than r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Moreover, the meaning of rout is only well-defined for the idealized case in which the outflow is a thin bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In the more general case where the outflow is continuous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', rout is a distribution), the appropriate value of rout for calculating outflow rates will depend upon the radial variation of den- sity and velocity in the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Without knowing the radial structure of the outflow, outflow rates are uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In addition to uncertainties in the radial structure of the outflows probed by absorption lines, there is the long- standing theoretical problem about the nature of outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' How can the absorbing material survive long enough to be accelerated to hundreds of km/s without being shredded by the hydro-dynamical interaction with the wind fluid (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2023)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Recent work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Gronke & Oh 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Fielding & Bryan 2022) imply that clouds exposed to an out- flowing hot wind can either grow by accreting gas at the cloud’s interface with the hot phase (for large clouds), or be destroyed (for small clouds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To date, there are no good em- pirical constraints on the cloud masses (Mcl) or radii (Rcl) in outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this paper, we aim to shed light on a method to mea- sure ne, the radius at which these densities apply (rn), Mcl, and Rcl from outflow absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We focus on 22 local star-forming galaxies selected from the COS Legacy Archive Spectroscopy SurveY (CLASSY) atlas (Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022) and Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These galax- ies have high signal-to-noise ratio (SNR) HST/COS spectra which cover their rest-frame UV bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In half of the galax- ies, we can securely detect absorption lines from the fine structure excited transitions of Si II, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' From it, we determine various important physical parameters of the out- flows, including ne, rn, outflow volume filling factor, outflow cloud sizes, and cloud masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Since the majority of these measurements are the first-ever for galactic winds detected in absorption lines, we discuss their implications and what CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 3 observational constraints they provide for future models of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Section 2, we introduce the data and observations that are used in this pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We then describe how to measure the column density from Si II and Si II* in Section 3 and how to derive ne from these two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Section 4, we present results for the ne and rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We compare them with empirical estimates that are commonly adopted in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also describe how to derive several other important outflow parameters (the cloud masses, radii, and volume filling factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Finally, in Section 5, we discuss and compare our results with other outflow den- sity and radius measurements in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also con- trast our results with current outflow models in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We conclude the paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We adopt a cosmology with H0 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='6 km s−1 Mpc−1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='286, and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='714 (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2014), and we use Ned Wright’s Javascript Cosmology Calculator website (Wright 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this paper, we adopt the notation r for distances from the starburst, and use R to represent outflow cloud radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTIONS In this paper, we select galaxies from the parent sample of the CLASSY dataset (Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022), which includes 45 lo- cal star-forming galaxies (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='002 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='182).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These galax- ies are observed by the G130M+G160M+G185M/G225M gratings on Hubble Space Telescope (HST)/Cosmic Ori- gins Spectrograph (COS) for their rest-frame far-ultraviolet (FUV) spectral regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To enlarge the dynamic range of the sample at the highest star-formation rates (SFR), we also include five similar galaxies from the Lyman Break Analog (LBA) sample in Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These galaxies have similar quality HST/COS observations as CLASSY ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We then apply three selection criteria: 1) the SNR per resolution element (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='18 Å) in the continuum near 1260 Å in the rest- frame is ≥ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2) the UV half-light radius of the starburst is < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5′′ (so that the COS spectrum represents the majority of the starburst);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3) an outflow has been detected (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These criteria result in a final sample of 22 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' All data were reduced locally using the COS data- reduction package CalCOS v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='101, including spectra ex- traction and wavelength calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We refer readers to Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022) and James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022) for more details about these data reductions and spectral coaddition procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also apply the same reductions to the LBA galaxies from Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Therefore, the whole sample was reduced and processed in a self-consistent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have re- sampled the spectra into bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='18 Å (spectral resolution ∼ 6000 – 10000 from the blue to red end) (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='com/spacetelescope/calcos/releases These galaxies’ redshift are derived from fitting the optical emission lines discussed in Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' ANALYSES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Summary of Previous Outflow Analyses For our sample, the detailed analyses of outflow proper- ties and their relationship to the host galaxy properties are reported in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We briefly summarize the key steps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given the reduced data from CalCOS, we start with fitting the stellar continuum of galaxies using stellar models from Starburst99 (Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We follow the methodology discussed in Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We then normalize the spectra by the best-fit stellar continuum for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For each galaxy, the final reduced HST/COS spectra cover ∼ 1200 Å – 2000 Å in the observed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this region, various lines from galactic outflows are de- tected as absorption troughs, from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', O I λ1302, C II λ1334, Si II multiplet (λ1190, 1193, 1260, 1304, and 1526), Si III λ1206, and Si IV λλ1393, 1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To isolate the outflowing gas component from the static ISM, we fit a double-Gaussians model to each absorption trough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The first Gaussian has a fixed ve- locity center at v = 0 km s−1, which represents the static ISM component, and the second Gaussian has a veloc- ity center < 0 km s−1, which stands for the blueshifted outflow component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Since the line-spread-functions (LSF) from HST website2 is only suitable for point sources, we have constructed non-point source line- spread-functions (LSF) for each galaxy and convolved them with standard Gaussian profiles in the fitting pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To robustly measure the ionic column density (Nion) of outflows, we apply partial coverage (PC) models to Si II multiplet and Si IV doublet absorption troughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' From the PC models, we have determined the optical depths, covering fraction (CF), and Nion for Si II and Si IV as functions of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We then compare the measured Nion to grids of pho- toionization models from CLOUDY [version c17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='01, (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2017)] to determine the total silicon and hydrogen column densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', N(Si) and NH, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='edu/hst/instrumentation/cos/performance/ spectral-resolution 4 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We derive the mass/momentum/energy outflow rates given the derived NH and Vout, while we assume rout = r∗, which we take to be the radius enclosing 90% of the starburst FUV emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Measurements of Column Density from Si II* Galactic outflows not only show absorption lines from res- onance transitions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II λ1260, but also from fine struc- ture excited transitions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II* λ1265 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Jaskot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The combination of both can be adopted to derive the electron number density (ne) of the outflows (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this subsection, we focus on measuring N(Si II*) for galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' There are a total of six Si II* lines observable in the rest- frame FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We list their important atomic information in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find the observed absorption lines from Si II* are commonly weak in our galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is consistent with the assumed low ne (∼ 10 cm−3) for typical starburst galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This low ne has both pros and cons for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' On the one hand, the weaker Si II* troughs are generally optically thin (τ ≪ 1), and we can safely mea- sure N(Si II*) by adopting CF = 1, given the apparent optical depth (AOD) assumption (Savage & Sembach 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' On the other hand, the shallow Si II* troughs are sometimes difficult to measure, even in our high SNR HST/COS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' An- other complexity is that the emission lines from Si II* (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', so-called fluorescent lines) can contaminate the blue-shifted absorption troughs of Si II*, especially when the outflow ve- locity (Vout) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The steps for our fitting process of Si II* lines and measurements of N(Si II*) are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We fit the fluorescent emission and fine-structure ab- sorption lines from Si II* λ1197, 1265, 1309, and 1533 for each galaxy simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We exclude Si II* λ1194 because it is commonly blended with the ab- sorption trough from Si II λ1193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For each galaxy, we also exclude Si II* lines that fall into a chip gap or are contaminated by Galactic lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II λ1197 can be affected by Galactic Lyα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For each Si II* absorption line, we assume it has a Gaussian optical depth profile: Ik(v) = e−τk(v) τk(v) = bk σ √ 2π ×exp((v−vc)2 2σ2 ) (1) where k stands for the kth Si II* absorption line, Ik(v) is the normalized intensity, τk(v) is the optical depth at each velocity of the absorption trough, v is the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given the AOD assumption, the optical depths of dif- ferent Si II* absorption lines (scaled by coefficient bk) are linked by their oscillator strength ( f) ratios (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The velocity center (vc) and dispersion (σ) of the Si II* absorption lines are fixed among all Si II* lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These fixed values are chosen to be the same as the median values from all Si II resonance absorption lines (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This assumes that the same outflow clouds have produced Si II and Si II* absorption lines, which is true since both lines have close energy levels (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For each Si II* emission line, we model it using only one Gaussian profile in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is because Si II* in our sample show weak and narrow fluorescent emission-lines, and are inconsistent with arising from the outflowing gas seen in absorption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', broad lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This implies that most of the emission from the outflow arises on scales larger than the projected COS aperture (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For all Si II* emission lines, we fix vc at the systematic velocity, and σ is set to be in the range between 0 and the median FWHM of the static ISM component of Si II resonance lines (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Their amplitudes are free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Then, we conduct χ2 minimization to fit all 2×N pro- files simultaneously to the spectral regions of Si II*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here the 2 stands for the emission and absorption line for each Si II*, and N equals the number of Si II* lines that are clean and used in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We adopt the fitting routine mpfit (Markwardt 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Finally, assuming AOD, N(Si II*) can be derived from the best-fitted τk(v) as follows (Savage & Sembach 1991): Nion(v) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='8× 1014 fk ·λk τk(v) Nion = � Nion(v)dv (2) where λk is the wavelength for the kth Si II* line that has τk(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Note that under the AOD assumption, choices of different Si II* lines in Equation (2) lead to the exact same Nion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' There are two close transitions of Si II* at ∼ 1265 Å, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II* λ1264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='73 and λ1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Both are from the same lower energy level at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 eV (= 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='24 cm−1), but have slightly different upper energy levels due to fine structure splitting (δE ∼ 4×10−4 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Since the velocity offset between these two lines is only 69 km s−1, we can barely resolve their ab- sorption lines in the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, we adopt the combined f value in the calculations of N(Si II*) (Borguet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Since Si II* λ1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='02 has ∼ 10 times smaller f value than that of Si II* λ1264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='73 (Table 1), the absorption trough is al- ways dominated by the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Example of fitting to the absorption and emission lines for Si II* spectral regions for galaxy J0150+1308 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='14668).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The black and gray histograms are the data and errors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In each panel, the blueshifted outflow component for the Si II resonance line is shown in blue (adopted from Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The fitted absorption and emission models for Si II* are shown in green and orange, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The summed model for Si II* is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The green and orange dashed lines mark the velocity centers for the fitted absorption and emission lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' See detailed fitting methods in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Atomic Data for the Resonance and Excited Transitions of Si II (a) Ions Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' flk Akl Elow – Eup (1) (2) (3) (4) (5) Si II 1190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='77 × 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='53 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='41 Si II 1193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='75 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='69 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='39 Si II 1260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='57 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='84 Si II 1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='37 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='28 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='64 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='50 Si II 1526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='33 × 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='81 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='12 Si II* 1194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='37 × 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='45 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='41 Si II* 1197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='50 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='40 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='39 Si II* 1264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='73(b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='04 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='84 Si II* 1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='02(b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='13× 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='73 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='84 Si II* 1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='28 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='00 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='23 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='50 Si II* 1533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='33 × 10−1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='52 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='12 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' – (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Data are obtained from National Institute of Standards and Technology (NIST) atomic database (Kramida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Vacuum wavelengths in units of Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Oscillator strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Einstein A coefficients in units of s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Energies from lower to upper levels in units of eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Si II* has two close transitions at ∼ 1265 Å, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II* λ1264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='73 and λ1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Both are from the same lower energy level at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 eV, but have slightly different upper energy levels due to fine structure splitting (δE ∼ 4×10−4 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' See discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' An example of the fitted Si II* spectral regions is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The outflow components for the Si II resonance lines are shown in blue (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1), while the fitted absorp- tion and emission models for Si II* are shown in green and orange, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The overall model for Si II* by sum- ming both the absorption and emission is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' There is a clear absorption trough from Si II* λ1265, and it is well- fitted, while there is no trough seen in Si II* λ1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is as expected since f1265/f1309 = 15, which leads to τ1265/τ1309 = 15 under AOD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Overall, we have measured N(Si II*) se- curely in 11 out of 22 galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These galaxies that have N(Si II*) measured yield a mean N(Si II*)/N(Si II) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Table 3, we report the measured N(Si II) and N(Si II*) in column 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Mechanisms for Generating Si II*: Collisions v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Radiative Pumping As shown in Table 1, the observed fine-structure transitions of Si II* in FUV have lower energy levels as Elow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='036 eV (= 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='24 cm−1), which is the first excited energy level of Si II (hereafter, Si II* specifically stands for this level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Two mechanisms can populate Si II*: 1) Collisional excitation of the ground state of Si II by free electrons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Silva & Vie- gas 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Osterbrock & Ferland 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Borguet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, a higher ne would yield a higher n(Si II*)/n(Si II) ratio, where n(Si II*) and n(Si II) stand for the level popu- lation of the first excited and ground state of Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2) In- direct UV pumping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', the Si II ground state is excited by absorption of a UV photon to an upper energy level, followed by a spontaneous decay to the excited level at Si II* 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='24 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, a stronger radiation field leads to higher n(Si II*)/n(Si II) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To check if indirect UV pumping can be the dominant mechanism, we estimate the radiation intensity G (in units of ergs cm−2 s−1) suffered by outflows for galaxies in our sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We first measure each galaxy’s continuum flux (λFλ) around Si II λ1260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Then we convert it to luminosity as: λLλ = 4πD2 L × λFλ, where DL is the luminosity distance of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We conservatively assume the location of ob- 6 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' served outflowing gas is at or beyond the starburst radius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', rout > r∗), which we will show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 to be a fair assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Finally, we derive G for each galaxy as G = λLλ/(4πr2 out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find the majority of galaxies in our sam- ple have G/G0 < 103 (with two exceptions), while the mean G/G0 is only ∼ 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here, G0 represents the interstellar FUV intensity of our Milky Way (Habing 1968), which is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='6 × 10−3 ergs cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' As shown in Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2006), n(Si II*)/n(Si II) < 10−4 when G/G0 < 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given our observed mean N(Si II*)/N(Si II) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='013, we conclude that indirect UV pumping commonly contribute < 10−4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='01 = 1% of the observed population of Si II*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is different from the fine structure absorption lines detected in γ-ray bursts in Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2006), where indirect UV pumping dom- inates because the radiation field is much stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, collisional excitation is the dominant mechanism for populating n(Si II*) in our galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We show the relation- ship between level population ratio and ne in Figure 2 for Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The modelled curves are calculated using the CHIANTI database (v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='7, Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2015), assuming collisional excitation under three different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The relation is only weakly dependent on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The critical den- sity (ncr) for Si II* is defined at the position where n(Si II*) = n(Si II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For T = 10,000 K, we get ncr ∼ 2000 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' From Figure 2, we can derive ne from the observed column den- sity ratio of N(Si II*)/N(Si II) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Borguet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The errors of ne are propagated from the errors of N(Si II*)/N(Si II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given galaxies in our sample have N(Si II*)/N(Si II) in the range between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='001 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1, we get ne from a few to ∼ 100 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The derived ne for each galaxy is listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For galaxies that show no absorption on N(Si II*), we present upper limits on ne based on their upper limits of N(Si II*) (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' RESULTS We summarize the main notations and measured quanti- ties in this paper at Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We illustrate their details in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Outflow Density Distribution and Correlations In Table 2, we show the statistics of the derived ne for 11 galaxies which have secure measurements of their N(Si II*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find outflows in the galaxies have the mean and median value ne ∼ 23 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These values are consistent with what has been estimated before from absorption-line data for star- burst galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', ne= 19 – 34 cm−3 in Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Figure 3, we show a strong positive correlation 3 In the LOS, the observed N(Si II*)/N(Si II) = n(Si II*)/n(Si II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Population ratio of Si II’s fine structure level (Elow = 287 cm−3) to the ground state (Elow = 0 cm−3) versus the electron number density (ne) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Osterbrock & Ferland 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The mod- elled curves are calculated using the CHIANTI database (v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='7, Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2015), assuming collisional excitation under three different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The green vertical line represents the me- dian value of ne measured from galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 for more discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Strong orrelations between outflow electron number den- sity (ne) and SFR surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Galaxies that have ne measure- ment (Mea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=') or upper limits (UL) are shown as the red-filled or gray-open symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Kendall’s τ correlation coeffi- cients are shown at the bottom-right corner of each panel, where we have considered the upper limits following Akritas & Siebert (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The best linear-fit to all measurements is shown as the or- ange dashed line, and the fitted slope and intercept are shown in the top-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The blue line represents the model from Cheva- lier & Clegg (1985) assuming the outflow gas is in pressure balance with the wind fluid at the radius of the starburst [Equation (12) and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 7 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Summary of the Notations and Measured Quantities Notation Definition Reference Mean Median STDDEV(a) (1) (2) (3) (4) (5) (6) rout Actual outflow radius distribution Section 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' r∗ Starburst radius of the galaxy Section 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' rn Outflow radius at the derived ne from Si II* Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' ne Outflow electron number density Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 cm−3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='8 cm−3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='7 cm−3 rphot Outflow radius assuming photoionization Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='6 kpc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 kpc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 kpc rram Outflow radius assuming pressure equilibrium Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 kpc 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1 kpc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 kpc FF Outflow volume filling factor Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4% Rcl Outflow cloud size Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 13 pc 5 pc 15 pc Mcl Outflow cloud mass Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1 × 104 M⊙ 202 M⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='0 × 104 M⊙ Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' – (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The first part of this table are the important notations adopted throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The second part (beginning with ne) shows the measured quantities (galaxies with lower and upper limits are excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We show the mean, median values, and standard deviations for these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' rout is a range or distribution, which can only be measured from spatially resolved detections of outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' rout = rn only when the outflow is a thin bubble (see discussions in Sections 1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Based on different assumptions, we can measure rn as rphot or rram specifically (illustrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' between ne with the SFR surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We will discuss the implications of this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In all Figures in this Section, galaxies in our sample with ne measurement or upper limits are shown as red-filled or gray-open symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Kendall’s τ correlation co- efficient (rk) and the probability of the null hypothesis (pk) are shown at the bottom-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have taken account of these upper limits in the Kendall τ test following Akritas & Siebert (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Interpretations of Outflow Density in Models Since our HST/COS spectra are integrated over the whole line-of-sight (LOS), the derived ne values for outflows also represent mean values over the velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To better interpret the measured ne discussed above, we consider two common outflow models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The simplest case for outflow is an expanding thin shell model given a mean electron number density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', ne = ns) and shell thickness (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, we have: N(Si II*) = n(Si II*)×s N(Si II) = n(Si II)×s (3) where Nion and nion represent the column and number den- sity for a certain ion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For galactic outflows, ne varies between ∼ 10 cm−3 and ∼ 2000 cm−3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Chevalier & Clegg 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this range, the curve in Figure 2 is approximately linear, so we have: n(Si II*) n(Si II) ≈ ne ncr (4) Combining Equation (3) and (4), we get N(Si II*)/N(Si II) ≈ ne/ncr = ns/ncr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Therefore, for a thin shell outflow model, the derived ne from N(Si II*)/N(Si II) discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 is just the mean density in the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In the second case, we consider a mass-conserving galac- tic wind with constant velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, the outflow has a density profile n(r) = n0(r/r0)−2, where r0 is the radius at which the outflow begins, and n0 is the density at this radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, we have: N(Si II) = � ∞ r0 n(Si II)dr = C0 ×n0r0 (5) where the integration is from r0 to infinity (note n(∞) = 0) and C0 = n(Si II)0/n0 is the conversion factor from gas num- ber density to Si II number density at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' C0 depends on gas metallicity and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Similarly, for Si II*, we get: N(Si II*) = � ∞ r0 n(Si II*)dr ≈ � ∞ r0 n(r)2 ncr ×C(r)dr = C0n2 0r0 3ncr (6) where in the second row we have adopted Equation (4) to replace n(Si II*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, this mass-conserving outflow model yields N(Si II*)/N(Si II) = n0/(3ncr), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', the derived ne from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 is a third of n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Equivalently, our derived ne cor- responds to the gas density at rn = √ 3r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Hereafter, we define rn as the radius of the outflows at which the mean ne derived from fine-structure absorption lines above would occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 8 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Similarly, if we take a general form of n(r) = n0(r/r0)−γ, we get: N(Si II) = C0 ×n0r0 (γ −1) N(Si II*) = C0n2 0r0 (2γ −1)ncr N(Si II*)/N(Si II) = γ −1 2γ −1 n0 ncr (7) Thus, in the general form, the derived ne from integrated spectra corresponds to γ−1 2γ−1n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Equivalently, the derived ne is the gas density at a radius of ( 2γ−1 γ−1 ) 1 γ × r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given the evidence for relatively shallow radial density profiles found in outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021), we consider the additional cases γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2, and get rn = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='52 r0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='06 r0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 We will compare these sizes with those estimated from our measured values of ne below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We note that if the outflows are more complex than a gen- eral form of n(r) = n0(r/r0)−γ, or if there is a range in density at a given radius, the exact interpretation of our measured ne and rout will be different and dependent on the actual form of n(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We do not dive in this direction, which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Derivations of Outflow Distances from ne For galactic outflows, their radial extent (also referred as the “outflow distance”) can not be determined given only LOS integrated spectra (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In fact, for a continuous outflow, there is no unique way to define distances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', one could define minimum or maximum val- ues, or a half-mass radius, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Moreover, while we have measured a single value for ne for a continuous outflow, this value will only apply at some specific radius in the outflow (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here we will estimate rn based on two methods as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Outflow Distances Assuming Photoionization In star-forming galaxies, the ultraviolet outflow absorption lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', from O I, Si II, Si III, Si IV) have been found to be well-described by photoionization models instead of shock- heating models (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, we have: UH = QH 4πr2 photonHc −→ rphot = � QH 4πUHnHc (8) where UH is the ionization parameter, QH is the source emis- sion rate of ionizing hydrogen photons, c is the speed of light, and nH is the hydrogen number density of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' On the 4 Note that these expressions diverge for γ ≤ 1, so we do not consider these shallower profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' right side of the Equation (8), we show the solved formula for outflow distance r assuming photoionization (hereafter, rphot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For QH, we adopt the values from spectral energy distribu- tion (SED) fitting with UV and optical photometry described in Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We note that the resulting QH value is the intrinsic value, but only a portion of the ionizing photons can reach the observed outflows due to attenuation by neutral hydrogen and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, we estimate the escaped ionizing photon rate (QH,esc) as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022b): QH,esc = QH,tot ×(1−CF)×10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4E(B−V)k(912) (9) where, for each galaxy, CF represents the covering fraction of the static ISM component derived from the absorption line profiles in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a), E(B−V) is the internal dust ex- tinction (derived in Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022), and k(912) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='87 is the extinction curve at the Lyman limit by assuming the ex- tinction law from Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The second term on the right of Equation (9) represents the attenuation by neu- tral hydrogen, where a fraction of CF around the galaxy is covered by ISM and is generally optically thick to QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The third term stands for the attenuation by dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We note that this assumes all the extinction arises inside the starburst and that the outflow is at least as large as the starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For UH, we adopt the values determined from outflow ab- sorption lines of Si II and Si IV as described in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For nH, we approximate it as ∼ ne/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2, which is applicable for ionized gas, assuming ∼ 90% hydrogen and ∼ 9% helium and some metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Overall, we can solve rphot from Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The derived results are shown in Table 3, which are in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 – 5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In the left panel of Figure 4, we compare the rphot values with r∗, which is the commonly assumed outflow radius in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We see a strong cor- relation with rphot∼ 1 to 2r∗ as we move from the smallest to largest galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Outflow Distances Assuming Pressure Equilibrium In this section, we compare our data to the simple ana- lytic model for a starburst-driven wind by Chevalier & Clegg (1985) (CC85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To review, CC85 model assumes that mas- sive stars return mass and kinetic energy to the starburst through supernova explosions and stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These ejecta are thermalized through shocks to form a very hot region of gas inside the starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This gas expands through a sonic radius (the starburst radius r∗) and becomes a high-velocity supersonic wind that can accelerate clouds in its path, pro- ducing the blue-shifted absorption lines we see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This latter gas is much denser and cooler than the wind fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The CC85 model requires a density of the wind fluid at its sonic point that depends on SFR/r2 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We see this dependence for the absorption-line gas in our data (see Figure 3), so it is CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons of the derived outflow distances with the starburst radius (r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The labels and symbols are the same as Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Galaxies that have r as a measurement or a lower limit are shown as the red-filled or gray-open symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Left: Derived rphot from photoionization outflow models (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Right: Derived rram by assuming pressure equilibrium (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We show the 1:1 correlation as the blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In both cases, the derived r correlate strongly with r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' worth exploring the connection between the wind fluid and the gas we measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We begin by comparing the pressures implied by the densities we measure via Si II to the pressure of the hot wind fluid predicted by CC85 at r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This pressure takes two forms, the thermal pressure of the wind fluid and its ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In convenient units, at r∗ the total (summed) pressure is given as: Ptot/k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='79×105SFR×r−2 ∗ [K cm−3] (10) where SFR is in units of M⊙/yr, r∗ is in units of kpc, and k is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' If we assume that the gas we mea- sure with Si II is in pressure equilibrium with the hot wind fluid, we have: Ptot = 2nekT (11) where ne is the density of the outflows, the factor 2 of is due to the gas is highly ionized, and we assume T = 10,000 K, for the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, for pressure balance with the wind fluid, the gas densities traced by Si II at r∗ are proportional to SFR/r2 ∗ as: ne ≃ 9×SFR×r−2 ∗ [cm−3] (12) This allows us to compare the relationship predicted by the model to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We show this model as the blue line in the right panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We see that the densities we measure are about an order-of-magnitude lower than the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This suggests that we are measuring densities at radii significantly larger than r∗ where pressures are lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this region, the CC85 model shows that ram pressure is dominant over thermal pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Direct measurements of the radial density profiles for the optical emission-line gas show that this material is in pressure balance with the wind ram pressure (Lehnert & Heckman 1996), so this is plausible for the ionized absorption-line gas as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We can therefore compute the location (rram) at which the observed absorption- line gas is in pressure balance with the hot wind’s ram pres- sure: Pram = ˙pSFR/4πr2 ram (13) where Pram is the wind ram pressure and ˙pSFR is the total momentum flux of the wind, which equals the input momen- tum from the starburst [reported in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a) for our galaxies].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Combining Equations (11) and (13), we can solve rram as: rram = � ˙pSFR/(8πne kT) (14) The derived results are shown in Table 3, and are in the range ∼ 1 to 8 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In the right panel of Figure 4, we compare rram with r∗, which shows a strong correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Figure 5, we also compare rphot (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1) with rram for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find a linear relation with the pressure-based sizes being typically ∼ 5 times larger than the starburst radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Outflow Sizes: Summary We have discussed three estimates of the radius of the outflow at the location at which the measured density oc- curs,based on different assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The first is based on an outflow with a power-law radial density profile n(r) = n0(r/r0)−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' It predicts our mea- sured densities occur at the characteristic radius rn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='73, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='52, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='06r∗ for γ = 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 respec- tively (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here r∗ is the radius of the star- burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The second assumes the gas is photoionized by a frac- tion of the starburst ionizing flux that reaches the out- 10 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons between outflow distances (r) derived from assuming photoionization (y-axis) and pressure equilibrium (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The labels and symbols are the same as Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find the pressure-based sizes are typically ∼ 2 to 5 times larger than the values derived from photoionization models, with the ratio decreas- ing from the small to large cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This estimates that the measured densities occur at a typical distance ∼ 1 to 2r∗ (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Finally, we have assumed that the gas we measure is in pressure balance with the ram pressure of the hot wind fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This estimates that the measured densities occur at a typical distance ∼ 4 to 5r∗ (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given the systematic uncertainties in these estimates, we regard this level of agreement as satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In all cases, we are tracing the region of the outflow where densities are high enough to measure with our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' As explained at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3, the maximum extent of the outflow could be considerably larger than rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Other Important Outflow Parameters Besides the density and structure of outflows, there are var- ious essential parameters of outflows that have rarely been measured from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this sub-section, we con- strain these parameters for outflows in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We com- pare them with simulations of outflows and discuss the im- plications in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We start with the volume filling factor (FF) of the ob- served outflow clouds, where we treat the absorbing material as an ensemble of clouds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Fielding & Bryan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This yields: FF = Ncl ×4/3πR3 cl AUVrn (15) where Ncl is the number of outflow clouds entrained in the hot wind at the outflow distance rn, and AUV is the cross- sectional area of the starburst UV continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also have the definition of outflow column density (NH) as: NH = Ncl ×4/3πR3 cl ×nH AUV (16) One can estimate FF from Equations (15) and (16) as: FF = NH nHrn (17) where, in this equation, all variables on the right side can be measured for at least part of the galaxies in our sample (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a), we also derived the area covering frac- tion of the outflow (CF) from the Si II and Si IV absorption lines (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We can rewrite CF as: CF = βsh × Ncl ×πR2 cl AUV (18) where βsh is a coefficient between 0 and 1 to account for the shadowing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is because the projected areas by dif- ferent outflow clouds in the LOS can overlap each other so that their total covered area drops by the factor of βsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This factor depends on 1) the overall spatial distribution of out- flow clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and 2) the second term of Equation (18), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', the number and relative size of each cloud to AUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Ap- pendix A, we show how to estimate βsh from Monte Carlo simulations and the measured CF in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For the 22 galaxies analyzed in this paper, we get βsh in the range of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For simplicity of symbols, we define CFsh = CF/βsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' From Equations (15) and (18), we can solve the size of the outflow clouds as: Rcl = 3 4 FF CFsh rn (19) Using the above expression for FF in Equation (17), this can be rewritten as: Rcl = 3 4 NH nHCFsh (20) This shows that Rcl does not depend on rn and can be com- puted from directly measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Once we have Rcl constrained, we can combine Equations (18) and (20) to get Ncl as: Ncl = CFsh × AUV πR2 cl = CFsh × R2 UV R2 cl (21) where RUV is the UV size of a galaxy and we approximate it as r∗ that we measured from the HST/COS acquisition im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Note that Ncl is also independent of rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find the mean and median values of Ncl are 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5 and 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='9, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 11 Finally, we can estimate the average mass of the individual outflow clouds (Mcl) by: Mcl = 4 3πR3 clnHµmp (22) where µ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 is the average atomic mass per proton and mp is the proton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We summarize the statistics for the derived FF, Rcl, Mcl values in Table 2, where their values for individual galaxy are shown in the last three columns in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For FF, which is the only derived parameter dependent on rn, we have as- sumed rn = rram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' If we assume rn = rphot, the derived Rcl and Mcl stay the same, while FF for each galaxy becomes larger with a mean and median value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5% and 13 pc, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3, we compare these measurements with common outflow models, and discuss their implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons with Other Outflow Density Measurements While we are presenting the first examples of density mea- surements for the warm ionized gas in outflows based on absorption lines, measurements of densities for the optical emission-line gas in outflows have been available in low- redshift starbursts for over thirty years (Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here we summarize what has been learned from the optical emission-line gas and compare the results to our new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For low-redshift starbursts, ne is commonly directly mea- sured using the density-sensitive ratio of the [S II] 6717 and 6731 emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These data can be used to map out the radial variation in ne, and show a steady radial decline from ∼ 500 to 1000 cm−3 in the starburst to ∼ 50 to 100 cm−3 at distances several times larger than r∗ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Lehnert & Heckman 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The [S II] flux ratio reaches its low-density limit at ne ∼ 10 to 100 cm−3 (Osterbrock & Ferland 2006), so direct measurements of ne at larger radii (lower densities) are not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This would be consistent with the lower values of ne that we typically get in our sam- ple, if we are probing larger radial scales in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have shown that with the assumption that the absorb- ing gas is in pressure balance with the ram pressure of the wind fluid, we do in fact derive large outflow radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Is this a plausible assumption?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We believe it is, because both the emission- and absorption-lines trace the warm ionized gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Since observations establish that the density (and pressure) profiles measured in the emission-line gas are con- sistent with the radial profile of the wind ram pressure (Heck- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Lehnert & Heckman 1996), this supports adopting the assumption of pressure balance to calculate Rram (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons with Other Measurements of Outflow Structures The outflow radii we derive assuming ram-pressure con- finement are in the range rram ∼ 1 to 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These size scales are consistent with those measured for the outflows traced by optical emission lines for starbursts with a range in SFR similar to our sample (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Lehn- ert & Heckman 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Martin 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For the sample of dwarf SF galaxies in Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022), which has weaker SFR than ours (∼ 10 times smaller), they get r ∼ 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is around the lower bound of our galaxies as expected from their lower SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Perhaps a more revealing comparison is to the size scales measured using resonantly-scattered emission arising from the same gas that produces the absorption lines seen directly along the LOS to the starburst (Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This has been done recently using IFU instruments, including VLT/MUSE and Keck/KCWI, to map out Mg II emission lines surrounding starburst galaxies at intermediate redshifts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Rupke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Zabl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Shaban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These data detect emission out to radii of ∼ 10 to 20 kpc, with half-light radii of 5 to 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The latter is quite similar to values we derived for rram for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given a typical aperture size of 1′′ – 2′′ in current (non- IFU) spectrographs, these sizes imply that a significant frac- tion of the resonantly scattered line emission could lie outside the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, this missing light helps explain why the scattered (or fluorescently-reprocessed) emission-lines from the outflow are often quite weak (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a), as can be seen in radiative transfer models of outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Scarlata & Panagia 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and Huberty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We explore this idea further with the current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Figure 6, we show histograms of the ratio rram/rCOS and rphot/rCOS, where rCOS is the projected physical size of the COS aperture for a given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find that the ra- tios of rram/rCOS are > 1, while rphot/rCOS are most often ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, the larger sizes measured for rram may be consistent with the relatively weak emission-lines seen in these galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Another apt comparison is to maps of the outflows of neu- tral gas traced by the Na I D optical absorption-line (1 – 10 kpc, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Martin 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Rupke & Veilleux 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Avery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Our data are complementary to these studies since they pertain to the ionized phase of the outflow and represent integrals over the line-of-sight directly into the starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Our data also provide information on key parameters like the densities, filling factors, radii and masses of the outflowing clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Besides the outflows discussed above, SF galaxies can exhibit outflows features in many other wavelength bands 12 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Histograms showing the comparisons between the measured outflow radius (rram or rphot) and the projected physical size of the HST/COS aperture for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The large ratio of rram/rCOS suggests that the scattered or fluorescent emission lines should be weak in our galaxies, which is consistent with what has been found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' See details in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and line diagnostics, where outflow distances are measured (see reivews in Heckman & Thompson 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These include very hot gas detected in X-ray (at ∼ 1 – 10 kpc, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Strickland & Heckman 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2014), and atomic and molecular outflows ob- served in infrared to radio bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', from [C II] and CO, out to radii of a few kpc, Walter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Chisholm & Matsushita 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Stuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Again, we emphasize that these various outflow sizes are defined in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In our case, we are defining the size to be the radius at which our measured densities occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For the emission-line data, the sizes are typically just defined by the radius at which the emission becomes undetectably faint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Additionally, different diagnostics of outflows in dif- ferent galaxies can reach intrinsically distinct scales, and the relationships between them are not entirely clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Detailed comparisons are beyond the scope of this paper, but we plan to study the relationships between different diagnostics and phases of galactic outflows in future papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons with Models and Simulations of Galactic Outflows Galactic winds are complex and difficult to model be- cause one needs to simultaneously capture the large spatial scales for the whole galaxy and the fundamentally small- scale process happening between the galaxy’s ISM/CGM and the wind (see Naab & Ostriker 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Currently, a compelling model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Fielding & Bryan 2022) comprises 1) a hot, volume-filling wind com- ponent driven by thermalized ejecta of massive stars (Cheva- lier & Clegg 1985) 5 and 2) a cold to warm component in the form of embedded clouds, which are entrained by the hot wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This component produces the observed outflows seen in UV absorption lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The exchange of mass/momentum/energy between these two components is in the turbulent radiative mixing layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Gronke & Oh 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Fielding & Bryan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Two time-scales control the fate of the outflow clouds (Gronke & Oh 2020): 1) the clouds grow by cooling of the hot wind in a time scale of tcool, which depends mainly on the pressure and metallicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and 2) the clouds are destroyed by turbulent shredding in a time scale of tmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have tmix ∝ Rcl/Vturb, where Rcl is the average radius of the outflow clouds and Vturb is the turbulent velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For large outflow clouds, tcool < tmix so that the clouds can grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For smaller outflow clouds, the clouds are shredded before they can grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, parameters related to Rcl are important for galactic outflows but have rarely been constrained from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4, we have shown that, based on our mea- surements of outflow density and distances, we can constrain these parameters, including FF, Rcl, and Mcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here we attempt to compare our measurements to common outflow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We can use the criterion derived by Gronke & Oh (2020) for the critical (minimum) size for a cloud to survive/grow when exposed to the ram pressure of the wind: Rcrit ∼ T 5/2 cl,4 Mwind P3Λmix,−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 χ 100α−1pc (23) 5 This hot gas is only detectable inside the starburst (Heckman & Thompson 2017), where its density is relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 13 Here, Tcl,4 is the cloud temperature in units of 104 K, Mwind is the Mach number of the hot wind fluid, P3 is the cloud pressure in units of 103 K cm−3, Λmix,−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 is the value of the cooling function in the turbulent mixing layer (in units of 10−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 cm3 erg s−1), χ is the ratio between the cloud and wind density, and α is a ‘fudge factor’ of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Under this model, if a cloud exposed to the hot wind has smaller sizes than Rcrit, it is destroyed/shredded before being accel- erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We assume Tcl = 104 K and can then use our measured val- ues of ne to compute P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We further take α = 1 and the fidu- cial value of Λmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' To measure χ we adopt the model above for clouds in pressure balance with the wind ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We use the Chevalier & Clegg (1985) wind solution to obtain Mwind assuming rout = rram (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2 and left panel of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For balancing between the wind ram pressure and the cloud thermal pressure, we have Pcl = 2nclkTcl = ρwv2 w (24) Then, since ρcl = nclmp, we have: χ = ρcl ρw = v2 wmp 2kTcl (25) Finally, we adopt the Chevalier & Clegg (1985) model and assume a wind velocity of vw = 1800 km/s (Strickland & Heckman 2009), leading to a value of χ ∼ 2×104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We show the results in Figure 7, in which we compare our derived Rcl with Rcrit for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find that the estimated values of Rcl all lie close to Rcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Within the uncertainties, the growth criterion is satisfied in all 11 cases with ne measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For the other 11 galaxies with ne as upper limits, the derived values of Rcl and Rcrit are both lower limits (gray-open sym- bols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' However, Equations (20) and (23) show that both sizes are inversely proportional to the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This means that the ratio of Rcl/Rcrit is independent of density, and thus we can evaluate the growth criterion even in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Within the uncertainties, 20 out of 21 cases6 satisfy this criterion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Rcl are large enough for them to survive under the impact of the hot wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The fact that the cloud sizes are similar to Rcrit could be un- derstood if the pre-existing population of clouds initially had a power-law distribution of sizes (Ncl ∝ R−γ cl ) that declines with increasing Rcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Then only the clouds with Rcl ≳ Rcrit survive the interaction with the wind, while clouds with sizes ≫ Rcrit are rare (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', having a small total covering factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Additionally, in our assessment of cloud survival, Equa- tions (20) and (23) imply that the ratio of Rcl/Rcrit depends 6 Among the total sample of 22 galaxies, one (J1612+0817) does not have NH reported in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a) since its Si IV doublet troughs are in a detector gap of HST/COS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, its Rcl/Rcrit ratio is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons of the derived outflowing cloud radii (Rcl) with the critical (minimum) radius for a cloud to survive/grow when exposed to the ram pressure of the hot wind (Gronke & Oh 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The labels and symbols are the same as Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Galaxies that have Rcl as measurement or lower limits are shown as the red-filled or gray-open symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The blue line represents the 1:1 relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Within the uncertainties, 20/21 outflows have enough cloud sizes large enough to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' See discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' only on the ratio of the column density to covering factor [since we adopted fixed values for all the terms in Equa- tion (23)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Empirically, there is relatively small variation in CFsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In this case, the relatively small spread in the values of Rcl/Rcrit seen in Figure 7 could imply that the total column densities of the absorbing clouds in the outflows are directly connected to the cloud-survival requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Future simula- tions of galactic winds may answer these implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK We have reported here the first direct measurements of the density (ne) in outflows from starburst galaxies traced by ul- traviolet absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These measurements were made using COS on HST to measure the ratio of the column den- sity of fine structure excited transitions of Si II (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II*) to those of the Si II resonance transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The sample of 22 galaxies was drawn from Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022) and Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2015), and limited to cases with SNR > 5, galaxy FUV radii < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5′′, and detected outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Our main results are as follows: We were able to measure ne in 11 cases and set upper limits in the other 11 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The median density was 23 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We found a strong correlation between ne and the star-formation rate per unit area in the starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Since the value of ne is derived along a line-of-sight, its meaning is only simple in the case of an expanding shell with constant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' In the case of a continu- ous outflow in which the density drops with radius, we 14 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' showed that for radial density profiles with power-law indices of –2, –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5, and –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2, the measured densities would pertain to gas at respective radii of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='5, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1 times the radius at which the outflow begins (taken to be the starburst radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Using the measured values of ne, we made two indirect estimates of the radius of outflows (rn) at which this density applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The first assumes that the gas is photo- ionized by radiation from the starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This required making estimates for the fraction of intrinsic ionizing radiation leaking out of the starburst and into the out- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Typical radii from this method are 1 to 2 times the starburst radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We then assumed that the absorb- ing gas clouds are in pressure equilibrium with the hot wind fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' These radii are typically 4 to 5 times the starburst radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We used the values of ne and our measured values for the total hydrogen column density and the covering fraction of the outflow to estimate the radii and masses of the absorbing clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We found median values of ∼ 5 pc and 200 M⊙ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also estimated the volume filling factor of the population of these clouds, with typical values of 10−3 to 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have compared the outflow clouds sizes to theoret- ical models in which clouds interact with a supersonic wind fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find that in 20 out of 21 cases, the esti- mated cloud sizes exceed the critical cloud size, mean- ing that these clouds are predicted to survive and grow as they interact with a hot supersonic wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is the first time that various essential absorption-line outflow parameters have been estimated from observations, including outflow density, volume filling-factor, and cloud sizes/masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' There are plenty of compelling future projects to do in both observations and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For example, how do our derived ne and rn values compare to direct mea- surements from spatially resolved observations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' What are the differences and/or connections between the ne and out- flow sizes measured from emission and absorption line out- flows?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given the detected warm outflows, can we provide constraints on the hot wind properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' How can the mea- sured radii and masses of the absorbing clouds help constrain simulations of outflows?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have plans to tackle some of these questions in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 15 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Gronke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Fielding, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Bryan for interesting discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 1 2 The CLASSY team is grateful for the support for this pro- gram, HST-GO-15840, that was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Associations of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5- 26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' BLJ thanks support from the European Space Agency (ESA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLM gratefully acknowledges support from NSF AST-1817125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The CLASSY collaboration extends special gratitude to the Lorentz Center for useful discussions during the "Characterizing Galaxies with Spectroscopy with a view for JWST" 2017 workshop that led to the formation of the CLASSY collaboration and survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 3 4 5 6 7 8 9 10 11 12 13 14 Funding for SDSS-III has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Sloan Foundation, the Participating Institutions, the Na- tional Science Foundation, and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Department of Energy Office of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' The SDSS-III web site is http://www.' 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Mexico State Uni- versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Penn- sylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Prince- ton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' the Spanish Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Univer- sity of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and Yale Univer- sity.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Itoh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', & Hattori, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2019, PASJ, 71, 87, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1093/pasj/psz069 Zabl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Bouché, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Wisotzki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2021, MNRAS, 507, 4294, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1093/mnras/stab2165 Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Thompson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Murray, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', & Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2014, ApJ, 784, 93, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1088/0004-637X/784/2/93 18 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' ESTIMATIONS OF βsh As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4, different outflow clouds can “shadow” each other and produce smaller area covering fractions (CF) in the LOS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Here we conduct a Monte Carlo (MC) experiment in two-dimension to estimate the shadowing parameter, βsh, for Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Given Ncl outflow clouds with radius Rcl, we randomly distributed them within the area of AUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also assume the ratio of AUV/πR2 cl = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We have tested that the variations of this ratio have little effect on our final results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We then vary Ncl from 1000 to 15,000 and calculate two quantities: 1) Ncl × πR2 cl/AUV, which represents the CF value if outflow clouds do not shadow each other at all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' and 2) the true CF by checking each spot in AUV to see if they are covered by any of the outflow clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We show these two quantities in the y- and x-axis in Figure 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We find when Ncl grows, the true CF initially increases fast but then slows down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' This is because when Ncl is small, we do not expect to have strong shadowing effects given the relatively large area of AUV compared to the projected size of each outflow cloud (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', πR2 cl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' But when Ncl is large (≳ 4000), the shadowing effects become more significant and the growth of the true CF is slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For galaxies in our sample, we have measured the true CF from “down-the-barrel” observations of UV absorption lines (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Thus, we can estimate y = Ncl ×πR2 cl/AUV for each galaxy based on the true CF and the curve in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Then we can calculate βsh from Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' For the 22 galaxies analyzed in this paper, we get βsh in the range of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Comparisons between CF assuming no shadowing versus the true CF from MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' We also show the number of clouds (Ncl) for several positions on the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' See discussion in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' TABLES Here we present the tables for the derived quantities for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' CLASSY VI: DENSITY, STRUCTURE AND SIZE OF GALACTIC OUTFLOWS 19 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Measured Parameters for Galaxies in the Combined Sample(1) Object log(SFR) log(r∗) log(NH) CF(Si II) N(Si II) N(Si II*) log(ne) Adust log(Qeff) log(rphot) log(rram) log(FF) log(RCloud) log(MCloud) βsh M⊙/yr kpc cm−2 1012cm−2 1012cm−2 cm−3 mags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' s−1 kpc kpc kpc M⊙ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) J0021+0052 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='40 Notes: (1) Measured parameters for 22 galaxies in our combined sample that have high SNR spectra of Si II and Si II* (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Galaxies from Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2015) is marked as (H15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' Descriptions for each column: (2) – (6) are adopted from Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1 for a summary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2) The log of star-formation rate (SFR) of the galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (3) The log of starburst radius of the galaxy, where we take r∗ = 2 ×r50 from Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (2022a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (4) The log of total hydrogen column density of the outflow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (5) The mean covering fraction derived from Si II outflow absorption lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (6) The column density of Si II in the outflows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (7) The column density of excited states of Si II (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=', Si II*) measured in the outflow absorption troughs for each galaxy (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (8) The log of electron number density of the outflows (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (9) Dust extinction derived from SED fittings (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (10): The effective ionizing photon rate per second for the outflows (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (11) The radius of the observed outflows derived from photoionization models (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (12) The radius of the observed outflows derived from assuming pressure equilibrium (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (13) The volume filling factor for outflows (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (14) and (15) The average radius and mass of the outflowing clouds (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4) adopting rram (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content=' (16) The coeffcient between 0 –1 to account for the shadowing effects of outflow clouds (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} +page_content='4 and Appendix A)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFJT4oBgHgl3EQfSCyU/content/2301.11498v1.pdf'} diff --git a/dtE4T4oBgHgl3EQfpg2t/content/tmp_files/2301.05193v1.pdf.txt b/dtE4T4oBgHgl3EQfpg2t/content/tmp_files/2301.05193v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..35c60c3b58293a73586d8d5f3fc953498d92ef44 --- /dev/null +++ b/dtE4T4oBgHgl3EQfpg2t/content/tmp_files/2301.05193v1.pdf.txt @@ -0,0 +1,1991 @@ +LEARNING DYNAMICAL SYSTEMS FROM INVARIANT +MEASURES +Jonah Botvinick-Greenhouse∗ +Center for Applied Mathematics +Cornell University +Ithaca, NY 14850 +jrb482@cornell.edu +Robert Martin +DEVCOM Army Research Laboratory +Research Triangle Park +Durham, NC 27709 +robert.s.martin163.civ@army.mil +Yunan Yang +Institute for Theoretical Studies +ETH Z¨urich +Z¨urich, Switzerland 8092 +yunan.yang@eth-its.ethz.ch +January 13, 2023 +ABSTRACT +We extend the methodology in [65] to learn autonomous continuous-time dynamical systems from +invariant measures. We assume that our data accurately describes the dynamics’ asymptotic statistics +but that the available time history of observations is insufficient for approximating the Lagrangian +velocity. Therefore, invariant measures are treated as the inference data and velocity learning is +reformulated as a data-fitting, PDE-constrained optimization problem in which the stationary distribu- +tional solution to the Fokker–Planck equation is used as a differentiable surrogate forward model. +We consider velocity parameterizations based upon global polynomials, piecewise polynomials, and +fully connected neural networks, as well as various objective functions to compare synthetic and +reference invariant measures. We utilize the adjoint-state method together with the backpropagation +technique to efficiently perform gradient-based parameter identification. Numerical results for the +Van der Pol oscillator and Lorenz-63 system, together with real-world applications to Hall-effect +thruster dynamics and temperature prediction, are presented to demonstrate the effectiveness of the +proposed approach. +Keywords Dynamical systems · Invariant measure · Inverse Frobenius–Perron problem · Parameter identification · +Fokker–Planck equation · Neural networks · Time-delay embedding · Computational ergodic theory +1 +Introduction +Data-driven models have proven to be instrumental across numerous scientific disciplines for their ability to predict +and control the behavior of complex physical systems [47]. The method by which models are constructed is highly +dependent on the available data, as well as prior knowledge of the physical process in question. Popular approaches +such as the shooting methods [2, 46], neural differential equations [8, 29, 49], and SINDy [6] adopt a Lagrangian +perspective and seek to directly minimize the error between modeled trajectories and observed data. However, the +following scenarios can render these standard techniques unreliable. The observational data may be sparsely and +irregularly sampled in time, such that derivatives cannot be accurately approximated. In a worst-case scenario, one +may not even know the times at which individual samples were drawn. Moreover, measurements may be contaminated +with large amounts of noise and the system in question could be highly sensitive to initial conditions. +In the specialized setting of traffic simulation, the issue of sparse sampling was approached by merging the tasks of +imitation learning, and interpolation under a single generative adversarial network [63]. Moreover, the Ensemble-SINDy +method learns noisy chaotic dynamics by averaging over a family of models formed using bootstrapped samples [17]. +However, a general modeling framework that is robust to each of the aforementioned difficulties remains elusive. +In [21, 65], an Eulerian perspective was adopted to handle such difficulties, and velocity models were constructed to +yield the same asymptotic statistics as the observed measurements, rather than seeking a pointwise match with time +∗Corresponding author +arXiv:2301.05193v1 [math.DS] 12 Jan 2023 + +trajectories or Lagrangian velocities. More specifically, instead of directly treating the noisy observations {xη(ti)}n +i=1 +of an autonomous flow ˙x = v∗(x) as inference data, the approaches in [21, 65] considered the occupation measure ρ∗, +where for each measurable set B, +ρ∗(B) := 1 +n +n +� +i=1 +χB (xη(ti)) , +χB(x) = +�1, +x ∈ B, +0, +x ̸∈ B. +(1) +When the occupation measures generated by a set of initial conditions all weakly converge to the same invariant +measure and the set of initial conditions has nonzero Lebesgue measure, such an invariant measure is said to be +physical [66]. These convergence properties can be delicate, and an overview of the relevant theory is provided in +Section 2.1. In this work, we consider the class of autonomous systems for which the occupation measures of almost +all initial conditions converge to a unique physical measure. Notably, this encompasses chaotic attractors such as the +Lorenz-63 system [39, 60]. +Going forward, we write v = v(θ) = v(x; θ) to denote the dependence of the reconstructed velocity fields on a set of +parameters θ ∈ Θ where Θ ⊂ Rm is the admissible set of all parameter values. The concrete form of θ depends on the +hypothesis space of v, which will be discussed in Section 4.2. The task is now to find the best-parameterized model +v(x; θ) approximating the true velocity v∗ by minimizing the mismatch +inf +θ∈Θ J (θ), +J (θ) := D(ρε(v(θ)), ρ∗). +(2) +The formulation (2) represents an inverse data-matching optimization problem, in which D denotes a metric or +divergence on the space of probability measures and ρε(v(θ)) is a regularized approximation to the physical measure of +the dynamical system, given some regularization parameter ε > 0 and the current velocity v(θ). +Although one could approximate ρ(v(θ)) by numerically integrating a trajectory and binning the observed states to +a histogram [21], this approach does not permit simple differentiation of the resulting measure with respect to the +parameters θ. When the size of θ is large, it is practical to use gradient-based optimization methods for solving the +optimization problem (2), and one has to compute the essential gradient ∂θJ . In [65], this was handled by viewing +ρε(v(θ)) as the dominant eigenvector of a regularized Markov matrix originating from an upwind finite volume +discretization of the continuity equation. The derivative ∂θJ was then seamlessly computed via the adjoint-state +method [65]. The computation time of the adjoint-state method is independent of the size of θ, making the framework +presented in [65] well-suited for large-scale computational inverse problems. +Different from [65], we consider the Fokker–Planck equation as the partial differential equation (PDE) forward model +for ρε(v(θ)), rather than the continuity equation. This is motivated by the Fokker–Planck equation’s larger modeling +capacity. Indeed, the Fokker–Planck equation reduces to the continuity equation when its diffusion term is zero, and it +can fit intrinsic noise present in trajectories which reduces over-fitting the parameterized velocity v(θ). Moreover, the +Fokker–Planck equation can be seen as an alternative to the teleportation regularization used for the continuity equation +in [65], in order to guarantee the uniqueness of the computed stationary solution ρε(v(θ)). +In this work, we build upon the framework proposed in [65] and investigate dynamical system velocity learning +with a large-scale parameter space applied to real data. There are three essential new contributions. First, we use +an isotropic diffusion to regularize the forward continuity equation, which differs from the so-called teleportation +technique [20] proposed in [65]. Second, in contrast to only learning three coefficients as done in [65], we parameterize +the velocity v(θ) using piecewise polynomial, global polynomial, and neural network discretizations, which can all +yield large parameter spaces with thousands of dimensions. We compare the reconstructed velocity in each case and +further discuss how the choice of parameterization affects the inverse problem’s well-posedness and the reconstructed +velocity’s regularity. We also consider various metrics/divergences as the choice of the objective function. Third, we +investigate velocity learning in time-delay coordinates, which can characterize the full dynamics even with partial state +measurements [59]. It is worth noting that there is no analytic form for the velocity in time-delay coordinates, even for +well-studied dynamical systems. The proposed differentiable forward model and gradient-based methods are powerful +and scalable tools for modeling time-delayed dynamics, and they permit larger-scale modeling than the approaches +proposed in [21]. +The paper is organized as follows. In Section 2, we review essential background on dynamical systems, invariant +measures, the Fokker–Planck equation, and time-delay embedding. In Section 3, we introduce the forward surrogate +model ρε(v(θ)) and analyze its modeling errors. In Section 4, we present an efficient gradient calculation for the +objective function J (θ) by treating (2) as a PDE-constrained optimization problem and utilizing the adjoint-state +method. We then adapt the gradient calculation to various velocity parameterizations, including neural network +discretizations in which the gradient is computed along with the backpropagation technique [35]. Finally, in Section 5, +we present velocity reconstructions for the Van der Pol oscillator and the Lorenz-63 system. We also model dynamics in +time-delay coordinates based on real-world data from a Hall-effect thruster, and we provide an example of temperature +prediction with uncertainty quantification. Conclusions and further discussions follow in Section 6. +2 + +2 +Background +This section reviews the essential background on invariant measures, stochastic dynamics, the Fokker–Planck equation, +and time-delay coordinates. Finally, we review the Eulerian approach for parameter identification proposed in [21, 65], +as well as prior work on the discrete inverse Frobenius–Perron problem [45]. +2.1 +Physical Measures and SRB Measures +Physical measures characterize the long-term statistical behavior of a physically significant subset of dynamical +trajectories. When a dynamical system is chaotic and exhibits sensitive dependence on initial conditions, the existence +of a physical measure unifies the statistical properties of trajectories which are pointwise dissimilar. While ergodic +measures also describe the long-term statistical behavior of dynamical trajectories, they may have very small support or, +in fact, be singular, which makes them difficult to observe computationally. On the other hand, if a dynamical system +admits a physical measure, it holds that the dynamical trajectories corresponding to a positive Lebesgue measure subset +of initial conditions will all share the same statistical behavior. We will next formalize these ideas using the language of +ergodic theory. For a more thorough treatment of physical measures and the foundations of ergodic theory, the reader is +encouraged to consult [9, 19, 66] and [14, 30, 43, 48], respectively. +Following [66], we assume that M is a compact Riemannian manifold and that T : M → M is a diffeomorphism. A +probability measure µ is said to be invariant with respect to the map T if µ(T −1(B)) = µ(B) for all B ∈ B, where B +denotes the Borel σ-algebra [14, Definition 2.1]. Hereafter, we will assume that µ is an invariant measure. A point +x ∈ M is said to be generic [66, Section 2.2] if for all g ∈ C(M), it holds that +lim +N→∞ +1 +N +N−1 +� +k=0 +g(T k(x)) = +� +M +g dµ. +(3) +The left-hand side of (3) is known as the time-average of a function g ∈ C(M) whereas the right-hand side of (3) is +known as the space average. It follows Birkhoff’s pointwise ergodic theorem [14, Theorem 2.30] that the time-average +of any g ∈ C(M) necessarily exists on a set of full µ-measure. To formally discuss the statistical properties of +dynamical trajectories, we now define the N-step occupation measure given the initial condition x ∈ M as +µx,N(B) := 1 +N +N−1 +� +k=0 +χB(T k(x)), +∀B ∈ B. +(4) +As noted in [14, Definition 4.19], the condition that a point x ∈ M is generic is equivalent to the condition +lim +N→∞ µx,N = µ , +(5) +where convergence takes place in the weak-* topology. Since the quantity µx,N(B) approximates the average amount +of time for which the orbit {T k(x)}∞ +k=0 initiated at x ∈ M resides in a measurable set B ∈ B, this convergence +indicates that the collection of generic points all share the same asymptotic statistical behavior. +When the measure µ is ergodic (see [14, Definition 2.13]), it holds that µ-almost every x ∈ M is a generic point [14, +Corollary 4.20]. However, if µ is an ergodic measure that is singular with respect to the Lebesgue measure, the resulting +collection of generic points may be physically insignificant and difficult to observe computationally. Motivated by this +perspective, an invariant measure µ is said to be physical if there exists a collection of generic points with positive +Lebesgue measure [66, Definition 2.3]. Physical measures are closely related to so-called Sinai–Ruelle–Bowen (SRB) +measures [9, 66], and in the case of Axiom A attractors where T ∈ C2(M, M), the two have been shown to coincide +[66, Theorem 1]. Importantly, the existence of a physical measure cannot provide insight into the complexity of the +dynamics for the set of generic points. However, as noted in [66], an SRB measure with nonzero Lyapunov exponents +[19, Chapter 1] is a particular type of physical measure that describes chaotic attractors. +We will next discuss the ways in which a physical invariant measure µ can be computationally approximated. If one +collects the measurements {T k(x)}N +k=1, the weak-* convergence in (5) suggests that the physical measure µ will +describe the statistics of our measurements provided that N is sufficiently large. Motivated by this perspective, we +can discretize the domain M and directly compute the occupation measure (4) for each cell in the discretization to +approximate the physical measure. This procedure is illustrated in Figure 1 and has been previously used to approximate +physical measures in [1, 21, 65]. +Other approaches have been proposed to compute the invariant measure as the stationary vector of the finite-dimensional +approximation of the continuous Frobenius–Perron operator [34], including Ulam’s method [19] and Galerkin-type +3 + +methods [10, 33]. More precisely, these discretizations are used to construct a Markov matrix that represents a +random dynamical system approximating the deterministic map T : M → M. An invariant measure for the discrete +approximation is then recovered as a stationary vector of the resulting Markov matrix. As the discretization is refined, +certain assumptions guarantee that the desired SRB measure will be recovered in the weak-* limit [10, Theorem 4.14]. +The process of identifying SRB measures as so-called zero-noise limits has also led to theoretical results which prove +their existence for certain classes of dynamical systems [9]. Though we have reviewed the theory of physical measures +and SRB measures in the context of arbitrary discrete-time dynamical systems, we will hereafter focus on applications +in which the dynamics are given by a time-∆t flow map for some ∆t > 0. +Figure 1: To computationally approximate a physical measure without knowledge of the dynamical system’s functional +form or the sampling times of trajectory data, the following procedure is used. We first obtain trajectory measurements +(left), then form a finite mesh over the state space (middle), and lastly bin the sampled trajectory to a histogram (right) +to represent the probability of occupation for each cell in the discretization of the state space. +2.2 +Stochastic Dynamics and the Fokker–Planck Equation +Consider an Itˆo stochastic differential equation (SDE) of the form +dXt = v(Xt)dt + σ(Xt)dWt, +X0 = x. +(6) +Above, Wt is a Brownian motion, v is the velocity, and σ determines the diffusion matrix Σ(x) = 1 +2σ(x)σ(x)⊤. For +simplicity, we will consider the case of a constant diffusion. Similar to the deterministic setting, there are analogous +notions of invariant measuress, ergodicity, and physical measures in the stochastic setting [4, 24]. One may use the +Euler–Maruyama method to obtain the numerical solution to (6) on the time interval [0, T], +Xj+1 = Xj + v(Xj)∆t + σ(Xj)ξj +√ +∆t, +∆t := T/N, +j ∈ {0, . . . , N − 1}, +where {ξj} are independently and identically distributed (i.i.d.) from N(0, I), the standard normal distribution on Rd. +The Fokker–Planck equation provides a PDE description of the probability density ρ(x, t) of the random variable Xt. +By [53, Page 88], the density evolves as +∂ρ(x, t) +∂t += −∇ · (ρ(x, t)v(x)) + ∇ · +� +∇ · (Σ(x)ρ(x, t)) +� +. +(7) +By assuming a constant diffusion, we may write Σ(x) = DI, where I denotes the identity and D > 0 is a constant +representing the scale of the diffusion. Equation (7) can then be simplified to read +∂ρ(x, t) +∂t += −∇ · (ρ(x, t)v(x)) + D∇2ρ(x, t). +(8) +We leave the study of a non-constant or anisotropic diffusion for later work. We remark that if D = 0, (8) reduces to +the so-called continuity equation, which instead models the probability flow of the ODE given by ˙x = v(x). Under +certain conditions [27], the steady-state solution ρ(x) of (8) exists and satisfies +∇ · (ρ(x)v(x)) = D∇2ρ(x). +(9) +Since (9) describes a limiting distribution limt→∞ ρ(x, t), it has been previously used to provide approximations of +invariant measures for stochastically-forced dynamical systems [1]. +4 + +Simulate Trajectory +Create Mesh +Bin to Histogram +4. +4 +3.7 +3 - +3 +0.006 +2.7 +2 - +2 +1.7 +0.005 +1 - +1 +0.7 +0.004 +y +0 +y +-0.3 +0 +0.003 +-1 - +-1 +1.3 +0.002 +-2 - +-2 - +-2.3 +-3 - +-3 - +0.001 +3.3 +-4 +4 +0.000 +-4 +-2 +-1 +0 +1 +2 +3.3-2.3-1.3-0.3 +0.7 +1.7 +2.7 +3.7 +-4 +-2 +0 +2 +4 +X +X2.3 +Delay Coordinates and Takens’ Theorem +The technique of time-delay embedding is a popular approach for reconstructing chaotic dynamical systems from limited +observations and has found numerous applications in the physical sciences [5, 21, 32, 58]. The procedure involves +embedding time series measurements ψ(t) = ψ(x(t)) of a full state x(t) into d-dimensional Euclidean space by +considering the vector of time-lagged measurements Ψd,τ(t) = (ψ(t), ψ(t − τ), . . . , ψ(t − (d − 1)τ)), for some τ > 0. +Takens’ theorem [59] provides suitable assumptions under which Ψd,τ(t) and x(t) are related via a diffeomorphism, +implying that the time-lagged vector of partial observations Ψd,τ(t) is sufficient for reconstructing the full state x(t). +Notably, the embedding dimension provided in [59] is d = 2m + 1 where m is the dimension of a compact manifold +M on which the flow map ft for the original dynamics is defined. In cases when trajectories are attracted to a compact +subset A with box-counting dimension (see [56, Page 586]) dA strictly less than m, it turns out that lower-dimensional +reconstructions can be obtained. +Indeed, it was shown in [56, Theorem 2.5] that for an open subset U ⊂ Rk, a flow ft : U → Rk, and a compact subset +A ⊂ U with box-counting dimension dA, suitable assumptions on the equillibria and periodic points of ft guarantee that +for almost all C1 observation functions ψ : U → R, the mapping Ψd,τ(x) = (ψ(x), ψ(f−τ(x)), . . . , ψ(f−(d−1)τ(x))) +is one-to-one on A, where d is an integer strictly larger than 2dA. Moreover, the mapping Ψd,τ(·) preserves any +manifold structure on A, i.e., it is an immersion for all compact subsets C of a smooth manifold which is contained +within A. This result is a helpful generalization of Takens’ theorem, as the box counting dimension for chaotic attractors +can be much smaller than the dimension of Euclidean space on which their flows are defined [55, Example 3.2.4]. +When a time-series projection ψ(t) of an unknown system ˙x = v(x) is observed, one can try to numerically determine +a suitable embedding dimension d and time delay τ; see for example [7, 40, 42, 62]. Choosing a proper embedding +dimension and time delay is important for obtaining a reliable surrogate model of the original dynamics in time-delayed +coordinates. Notably, in Section 5.2, we demonstrate that models for the velocity in time-delayed coordinates can incur +excess uncertainties when the embedding dimension is not sufficiently large. +2.4 +Prior Work on Learning Dynamics from Invariant Measures +For chaotic systems, trajectories are sensitive to initial conditions and estimation parameters. Sometimes, the approx- +imated reference velocity field {ˆv(x(ti))} cannot be accurately estimated from trajectory {x(ti)} due to the lack of +observational data, slow sampling, discontinuous or inconsistent time trajectories, and noisy measurements. To tackle +such difficulties, instead of working with the Lagrangian trajectories, [21, 65] propose an Eulerian approach by treating +the occupation measure (4) as the data. When enough samples are available, the occupation measure can be treated as +an approximation to the invariant measure; see Section 2.1. Finding the optimal parameter θ is then translated into the +optimization problem (2). The reference measure ρ∗ is the occupation measure converted from the observed trajectories +{ˆx(ti)}; see Figure 1. In [21], the approximated synthetic ρε(v(θ)) is generated by first simulating the synthetic +trajectories {x(ti; θ)} based on the dynamical system and then computing its histogram following (4). Since this +approach requires lengthy trajectory simulation, each evaluation of ρε(v(θ)) for a given θ is relatively costly. Moreover, +it is difficult to compute the derivative of ρε(v(θ)) with respect to θ, due to the histogram approximation of nonlinear +trajectories. As an improvement to the original idea in [21], [65] proposes a surrogate model to approximate ρε(v(θ)) +that is differentiable in θ and sometimes faster to compute. The key idea is to solve for ρε(v(θ)) as the distributional +steady-state solution to the continuity equation (i.e., (9) with D = 0) using a finite volume upwind scheme together +with the teleportation regularization. The gradient of the objective function J in (2) with respect to the parameter θ can +be efficiently computed based on the adjoint-state method [65, Sec. 5]. +The task of learning a dynamical system from an invariant measure has also been studied in the discrete-time setting +under the inverse Frobenius–Perron problem [45, 51, 54, 64]. The Frobenius–Perron operator, also known as the transfer +operator, characterizes the time-evolution of an initial measure µ0 according to some prespecified dynamical system. +Given a probability measure µ, the inverse Frobenius–Perron problem seeks to construct a dynamical system for which +µ is a fixed point of the associated transfer operator. The most widely studied case involves recovering an ergodic +map T on [0, 1] for which a prescribed absolutely continuous measure is the unique fixed point of the discrete transfer +operator. In this particular setting, various approaches such as topological conjugation [22] and matrix methods [50] +have been introduced to solve the inverse problem. The multivariate inverse Frobenius–Perron problem was also studied +in [18], where ergodic maps were constructed to adhere to the statistics of two-dimensional densities. Moreover, due to +inherent non-uniqueness in the inverse problem, recent approaches further restrict the solution space of the discrete +ergodic maps to those with a prescribed power spectrum [44]. To the best of our knowledge, [65] and our contributions +here are the first works that numerically solve the inverse Frobenius–Perron problem in the continuous-time setting. +Moreover, we do not assume that µ is absolutely continuous, as we use a finite-volume discretization to approximate +the Frobenius–Perron operator. +5 + +3 +The Forward Model and Modeling Errors +A central contribution of this work is to consider a different regularized forward model than the one in [65], especially +for trajectory measurements containing intrinsic noise which can be interpreted as sample paths of stochastic dynamical +systems (6). In those cases, the Fokker–Planck equation (7) is a better candidate as the PDE surrogate model, as it +contains a diffusion term which can fit noise present in the data. Based on the relationship between (6) and (7), one can +learn both the velocity field v(x) and the diffusion tensor Σ(x) in the optimization framework (2). For simplicity, we +only consider a fixed diffusion constant and leave the investigation of multi-parameter inversion to future work. +We will use (9) as the forward model to fit invariant measures generated by trajectories with intrinsic noise. While, +the diffusion term allows the model to fit the intrinsic noise and prevent over-fitting the noise into the target velocity +component, it also controls the scaling of the reconstructed velocity v(x; θ). Indeed, when D = 0 and ˜v(x) = a v(x), +we have ∇ · (ρ(x)˜v(x)) = 0 as long as ∇ · (ρ(x)v(x)) = 0, for any a > 0. However, for most cases, �v and v will not +solve the stationary Fokker-Planck equation (9) for D > 0. +Figure 2: As the mesh size of the forward model discretization is refined, we visually observe the convergence of +the computed steady-state solution to (9) (top row) to the approximate physical measure obtained by binning a time +trajectory based on the SDE (6) (bottom row). The Van der Pol oscillator (18) with c = 1 is used in this example, and +the histograms indicate mass-per cell. +3.1 +Finite Volume Discretization +We assume that our system evolves on the d-dimensional rectangular state space Ω = [a1, b1] × · · · × [ad, bd] ⊂ Rd, +with a spatially dependent velocity v : Ω → Rd. We define ni ∈ Z+, 1 ≤ i ≤ d, to be the number of equally-spaced +points along the i-th spatial dimension at which we wish to approximate the solution of (8), as well as the mesh spacing +∆xi := bi−ai +ni−1 . We are interested in obtaining a solution to the forward problem at points of the form +xk1,...,kd := +� +a1 + k1∆x1, . . . , ad + kd∆xd +� +∈ Ω, +ki ∈ {1, . . . , ni}. +We will index our coordinates using column-major order and write xk1,...,kd = xj where +j = k1 + +d +� +i=2 +(ki − 1)Si , +Si := +i−1 +� +j=1 +nj . +(10) +We will regard xj as the center of the cell Cj where +Cj = +d +� +i=1 +� +ai + +� +ki − 1 +2 +� +∆xi, ai + +� +ki + 1 +2 +� +∆xi +� +⊂ Ω . +Following the approach in [3], we implement a first-order upwind finite volume discretization of the continuity +equation, adding a diffusion term using the central difference scheme and enforcing a zero-flux boundary condition [36]. +6 + +Forward Model vs. Trajectory Histogram with Diffusion = O.oo1 +Forward dx = 0.2 +Forward dx = 0.1 +Forward dx = 0.05 +Forward dx = 0.005 + 0.0200 +0.00200 +0.0175 +0.006 +0.00175 +0.0150 +0.005 +0.00150 +0.0125 +0.004 +1 - +0.00125 +0 + 0.0100 +F0 +0.00100 +0.003 +0.0075 +1 +0.00075 +0.002 +- 0.0050 +0.00050 +-2 +0.001 +0.0025 +0.00025 +0.0000 +0.000 +0.00000 +Trajectory dx = 0.2 +Trajectory dx = 0.1 +Trajectory dx = 0.05 +Trajectory dx = 0.005 +0.0200 +F 0.00200 + 0.0175 +0.006 +0.00175 +0.0150 +2 - +0.005 +21 +0.00150 +0.0125 +0.004 +1 + 0.00125 +- 0.0100 +0 +0 +0.00100 +0 + 0.003 +0.0075 +-1 +0.00075 +-1 +0.002 +0.0050 +-2 +0.00050 +2 +0.001 +0.0025 +0.00025 +3 +0.0000 +0.000 +0.00000This allows us to obtain an explicit time-evolution of the probability vector ρ = [ρ1 +ρ2 +. . . +ρN]⊤ ∈ RN, where +N = �d +i=1 ni. While ρ is a discrete probability measure over the cells Cj, it also corresponds to a piecewise-constant +probability density function on Ω. With an abuse of notation, we will refer to both the piecewise-constant density +and the discrete probability measure as ρ. We discretize the time domain with a step size ∆t chosen to satisfy the +Courant–Friedrichs–Lewy (CFL) stability condition [36, Chapter 20]. Based on (8), the probability vector at the l-th +time step evolves as +ρ(l+1) = ρ(l) + Kρ(l), +K = +d +� +i=1 +∆t +∆xKi, +where each Ki is a tridiagonal matrix given by +Ki := +... +−vi,− +j−1 + D +∆xi +... +... +vi,− +j−1 − wi,+ +j−1 − 2D +∆xi +−vi,− +j ++ D +∆xi +... +... +... +wi,+ +j−1 + D +∆xi +vi,− +j +− wi,+ +j +− 2D +∆xi +−vi,− +j+1 + D +∆xi +... +... +... +wi,+ +j ++ D +∆xi +vi,− +j+1 − wi,+ +j+1 − 2 D +∆xi +... +... +wi,+ +j+1 + D +∆xi +... +� +��������������������������������� +� +��������������������������������� +Si +∈ RN×N. +Above, we have defined for each j ∈ {1, . . . , N} the upwind velocities +vi,− +j +:= min +� +0, vi +j +� +, +vi,+ +j +:= max +� +0, vi +j +� +, +wi,− +j +:= min +� +0, wi +j +� +, +wi,+ +j +:= max +� +0, wi +j +� +, +where vi +j := v +� +xj − ei∆xi/2 +� +· ei and wi +j := v +� +xj + ei∆xi/2 +� +· ei denote the i-th components of the velocity field at +the center of cell faces, and {ei} is the standard basis in Rd. We remark that if xj is away from ∂Ω, then wi,± +j += vi,± +j+1. +To enforce the zero-flux boundary condition, we set both the velocity v and diffusion D to be zero on ∂Ω. As a result, +the columns of K each sum to zero, and the total probability +ρ(l) · 1 = 1, +1 := [1 +. . . +1]⊤ ∈ RN, +is conserved under time-evolution. Since numerical artifacts cause the flux accumulation along the boundary, we also +enforce ρ = 0 on ∂Ω. When the boundary ∂Ω is sufficiently far from the trajectory data, this artifact is insignificant. +Hereafter, we assume the uniform spatial discretization ∆xi = ∆x for all i = 1, . . . , d. For a complete description of +the finite volume scheme, we refer to [36]. We remark that there are many higher-order structure-preserving schemes to +solve (8); see [26] for example. A more accurate numerical scheme can further reduce the forward modeling error. +3.2 +Teleportation and Diffusion Regularization +We use the finite volume discretization of the Fokker–Planck equation in Section 3.1 to approximate its steady-state +solution. After discretization, finding such stationary distributions to (9) is equivalent to solving the linear system +ρ = Kρ + ρ ⇐⇒ (I + K)ρ = ρ. +Since the columns of K sum to zero, we have that M := I + K is a column-stochastic Markov matrix. When D ̸= 0, +M is a transition matrix for an ergodic Markov chain, which has a unique equilibrium. When D = 0, to guarantee the +uniqueness of the equilibrium, [65] applies the so-called teleportation regularization [20] and considers +Mε := (1 − ε)M + ε U, +U = N −111⊤ ∈ RN×N. +7 + +There is now a unique solution to the linear system +Mερ = ρ, +ρ · 1 = 1, +ρ > 0. +(11) +From a computational aspect, it is useful to take advantage of the fact that M − I is sparse where I ∈ RN×N is the +identity matrix, and to instead solve +(1 − ε)(M − I)ρ = −N −1ε1, +where we have simply rearranged terms in (11) and used the fact that ρ · 1 = 1. +Since U is also a column stochastic Markov matrix with the uniform probability of visiting any point of the mesh, +using Mε amounts to stopping the dynamics based on M at a random time and restarting it from a uniformly randomly +chosen initial point. The size of ε represents the restarting frequency–the smaller ε, the rarer we restart [65]. +On the other hand, adding the diffusion component D to the tridiagonal matrix K can be seen as another way of +regularizing the noise-free Markov matrix by adding a scaled Brownian motion after each discrete evolution of the +deterministic dynamics. For deterministic dynamics with D = 0, the solution to (9) might not be unique if there is more +than one attractor. The use of teleportation connects all attractors through the “random restart”, and the solution ρε to +the linear system (11) has support that connects all the disjoint attractors. Similarly, when D ̸= 0, the Brownian motion +connects all disjoint attractors of the deterministic dynamics, giving a unique steady-state solution. In this scenario, the +use of teleportation for the diffusive case is simply a numerical treatment to improve the conditioning of matrix M +rather than to guarantee the uniqueness of ρ. +It is worth noting that both the teleportation regularization and an incorrect diffusion coefficient could be sources of +modeling error when we perform parameter identification. Although these regularizations enable faster evaluation of +ρε(v(θ)) and better posedness of the forward problem, they may reduce the accuracy of the inverse problem solution. +3.3 +Numerical Diffusion +In Figure 2, we illustrate the ρε computed as the steady-state solution to the Fokker–Planck equation in the top row and +the approximation to physical invariant measures of the corresponding SDE in the bottom row. From Figure 2, we see +that on a coarse mesh, the first-order finite volume scheme incurs a lot of numerical error, which gives a computed +solution with an artificial diffusion effect and thus is often referred to as the numerical diffusion [3]. The amount of +numerical diffusion is reduced as the mesh is refined since it is incurred by the first-order scheme, which is expected +to decay as O(maxi ∆xi) in the L∞ norm as we refine the mesh [36]. Besides the teleportation and the modeling +diffusion D, the presence of numerical diffusion is another modeling error incurred from solving the forward problem. +4 +Gradient Calculation & Velocity Parameterization +Another main contribution of this paper is to reconstruct the velocity field v(x) using large-scale parameterizations +v(x; θ), which turns an infinite-dimensional problem of searching for v(x) in a function space to a finite-dimensional +optimization problem of finding θ ∈ Θ ⊂ Rm. Here, we introduce parameterizations based on piecewise-constant, +neural network, and global polynomial functions. We also investigate various data-fitting objective functions J +that compare the mismatch between the observed and simulated invariant measures, ρ∗ and ρε(v(θ)). We compute +the gradient of such functions with respect to the coefficients θ in the parameterized velocity model v(x; θ) based +on the adjoint-state method for the PDE-constrained part and the backpropagation technique [35] for the neural +network part. Thanks to these techniques, we can then efficiently evaluate the gradients of J with respect to θ and +thus conveniently use gradient-based optimization algorithms to iteratively update θ, e.g., steepest descent, L-BFGS, +conjugate gradient descent methods as well as stochastic methods such as Adam [31]. For notational simplicity, we will +write ρ(v(θ)) = ρε(v(θ)) throughout this section. +4.1 +Gradient Calculation Through the Adjoint-State Method +Recall the finite volume scheme in Section 3.1 for solving (9). The forward model yields a discrete measure ρ(v(θ)) = +ρ(θ) = [ρ1(θ) . . . ρj(θ) . . . ρN(θ)]⊤ over the cells {Cj}, which converges to the solution to (9) in the weak sense as +we refine the discretization parameters. For the explicit form of ρ(v(θ)), we refer to [65, Eqn. (5.1)]. Note that we have +highlighted the dependence of our approximate steady-state distributional solution to the Fokker–Planck equation (9) +on the velocity v(x; θ). Our goal is to solve the optimization problem (2): +inf +θ∈Θ J (ρ(v(θ)), ρ∗) +by using gradient-based methods, where J is the cost function, and ρ∗ represents our inference data. The adjoint-state +method is an efficient technique by which we can evaluate the derivative ∂θJ , as the computation time is largely +8 + +independent of the size of θ. One can derive the adjoint-state method for gradient computations by differentiating the +discrete constraint [52], which in our case is the eigenvector problem +g(ρ(θ), θ) = Mε(θ)ρ(θ) − ρ(θ) = 0, +where ρ(θ) · 1 = 1. Specifically, we will compute ∂θJ = λ⊤∂θg where λ solves (∂ρg)⊤ λ = − (∂ρJ )⊤. In our case, +this linear system is the adjoint equation [65, Eqn. (5.8)] +(M ⊤ +ε − I)λ = − (∂ρJ )⊤ + (∂ρJ )⊤ ρ 1, +(12) +and the derivative +∂θJ = λ⊤� +∂θMε +� +ρ. +(13) +As a result, we only need to compute the derivatives ∂ρJ and ∂θMε to determine the gradient ∇θJ = (∂θJ )⊤. The +former depends on the choice of the objective function, while the latter is based on a specific parameterization of the +velocity field v(x; θ) determined by its hypothesis space. +4.1.1 +The Computation of ∂ρJ +For the objective function J , we consider the quadratic Wasserstein distance, the squared L2 norm, the Kullback–Leibler +(KL) Divergence, and the Jensen–Shannon (JS) Divergence. +Quadratic Wasserstein Distance: For probability measures ρ and ρ∗ on Ω, with finite second-order moments, the +squared quadratic Wasserstein distance is defined by +W 2 +2 (ρ, ρ∗) := +inf +Tρ,ρ∗∈T +� +Ω +|x − Tρ,ρ∗(x)|2dρ(x), +where T := {T : Ω → Ω : ρ(T −1(B)) = ρ∗(B), for all measurable B} is the set of maps that push ρ forward into ρ∗ +[61]. With an abuse of notation, we also use ρ(x) and ρ∗(x) to denote the densities of ρ and ρ∗ respectively. For efficient +computation of the W2 distance, we utilize the back-and-forth method [28], which instead uses the dual Kantorovich +formulation [61] +W 2 +2 (ρ, ρ∗) = sup +φ,ψ +�� +Ω +φ(x)ρ∗(x)dx + +� +Ω +ψ(x)ρ(x)dx +� +, +where φ ∈ L1 +ρ∗(Ω) and ψ ∈ L1 +ρ(Ω) are required to satisfy φ(x) + ψ(y) ≤ |x − y|2. In this case, the Fr´echet derivative +of J = W 2 +2 (ρ, ρ∗) with respect to ρ is given by +∂J +∂ρ = ψ. +Squared L2 Norm: The squared L2 distance as the objective function and its Fr´echet derivative are given by +J = 1 +2 +� +Ω +|ρ(x) − ρ∗(x)|2dx, +∂J +∂ρ = ρ − ρ∗. +KL-Divergence: The KL-divergence and its Fr´echet derivative are given by +J = DKL(ρ, ρ∗) := +� +Ω +ρ∗(x) log +�ρ∗(x) +ρ(x) +� +dx, +∂DKL +∂ρ += −ρ∗(x) +ρ(x) . +We remark that our definition of the KL-divergence differs from many applications in which it is commonly computed +as J = DKL(ρ∗, ρ). +JS-Divergence: Defining ρ′ := (ρ + ρ∗)/2, the JS-divergence and its Fr´echet derivative are given by +J = DJS(ρ, ρ∗) = 1 +2DKL(ρ, ρ′) + 1 +2DKL(ρ∗, ρ′), +∂DJS +∂ρ += 1 +2 log +� +2ρ +ρ + ρ∗ +� +. +Based on definitions of the KL and JS divergence, it is clear that we may encounter numerical instability issues if either +ρ or ρ∗ is not supported on the entire domain Ω. Thus, we remark that for the computation of both the KL and JS +divergences, we restrict the domain Ω to regions where both ρ and ρ∗ are strictly positive. This is equivalent to the +definition of the KL and JS divergence based upon the so-called Csiszar divergence [57, Eqn. (1)]. +9 + +4.1.2 +The Computation of ∂θJ +We have presented a few cases of ∂ρJ for different choices of J . Next, we show how to obtain ∂θMε, which is +the other necessary component in the adjoint-state method for gradient calculation; see (12)-(13). To begin with, we +consider θ = {vi +j} for all i = 1, . . . , d and j = 1, . . . , N, which corresponds to one variant of piecewise-constant +velocity parameterization. +Since we are only interested in computing the gradient away from ∂Ω, we can utilize the property that wi,± +j += vi,± +j+1. +First, observe that +∂Mε +∂vi +j += (1 − ε) +d +� +ℓ=1 +∆t +∆x +∂Kℓ +∂vi +j += (1 − ε) ∆t +∆x +∂Ki +∂vi +j +, +as well as that +∂Ki +∂vi +j += +... +0 +... +... +−H(vi,+ +j +) +−(1 − H(vi,− +j +)) +... +... +... +H(vi,+ +j +) +(1 − H(vi,− +j +) +0 +... +... +... +0 +0 +... +... +0 +... +� +������������������������ +� +������������������������ +Si +H(x) := +�1, +x > 0 +0, +x ≤ 0 +. +Above, H(·) is the Heaviside function. We remark that ∂vi +jKi can only be nonzero in the (j, j), (j, j − Si), (j − Si, j), +and (j − Si, j − Si)-th entries where Si is defined in (10). After solving (12) for λ and applying (13), we deduce that +∂J +∂vi +j += λ · ∂Mε +∂vi +j +ρ = (1 − ε) ∆t +∆x +� +λ · ∂Ki +∂vi +j +ρ +� += (1 − ε) ∆t +∆x +� +H(vi,+ +j +)ρj−Siλj + (1 − H(vi,− +j +))ρjλj − H(vi,+ +j +)ρj−Siλj−Si − (1 − H(vi,− +j +))ρjλj−Si +� += (1 − ε) ∆t +∆x (λj − λj−Si) +� +H(vi,+ +j +)ρj−Si + (1 − H(vi,− +j +))ρj +� +. +(14) +Equation (14) provides an efficient way for computing the gradient of the objective function with respect to the +piecewise-constant velocity based on cells {Cj} from our finite-volume discretization. +Alternatively, if the velocity v = v(x; θ) is smoothly parameterized by the vector θ = [θ1, . . . , θk, . . . , θm]⊤ ∈ Rm, for +each θk, we can then evaluate +∂J +∂θk += +N +� +j=1 +d +� +i=1 +∂J +∂vi +j +∂vi +j +∂θk +, +∂vi +j +∂θk += ei · ∂v +∂θk +���� +(xj−ei∆xi/2;θ) +, +(15) +to determine the derivative ∂θJ . By using a similar indexing convention to Section 3.1, we can collect the terms ∂vi +jJ +and ∂θkvi +j into the vectors ∂vJ and ∂θkv, respectively. Therefore, the double summation in (15) is achieved by the +inner-product ∂vJ · ∂θkv. Note that for different θk, we only need to change ∂θkv as ∂vJ does not depend on θk. +4.2 +Velocity Parameterization +In this subsection, we apply Equations (14) and (15) to evaluate the gradients of several parameterized velocity models. +Specifically, we consider piecewise constant, global polynomial, and neural network parameterizations of the velocity. +10 + +4.2.1 +Piecewise-Constant Parameterization +In the case of the piecewise-constant parameterization, we model the velocity as +v(x; θ) = +d +� +i=1 +N +� +i=j +vi +j χCj(x) ei, +θ = {vi +j}. +(16) +Here, we again use the column-major ordering from Section 3.1 to accumulate vectors of cells Cj with centers xj, and +velocity components, +vi +j = v(xj − ei∆xi/2) · ei, +along the i-th direction of the cell face located at xj − ei∆x/2. The parameter space of the model presented in (16) is +given by {vi +j}, which has size N · d, and the gradient of the parameters {vi +j} can be directly evaluated by (14). +We remark that (16) is only one variant of piecewise-constant parameterization since the parameterization mesh is +the same as the discretization mesh in the finite-volume method; see Section 3.1. These two meshes do not have to +be coupled together. To reduce the numerical error from the first-order scheme, it is preferable to reduce the spacing +{∆xi}, but we can keep the parameterization mesh fixed, so the size of the optimization problem does not change. In +this case, we need to apply the chain rule (15) to obtain the final gradient after evaluating (14). +The model defined by (16) can be learned by gradient-based optimization methods. The regularity of the piecewise- +constant model defined by (16) can be improved to a C0 function by interpolating between the values vi +j using either +piecewise linear or higher-order piecewise polynomial functions, as in [38]. +4.2.2 +Global Polynomial Parameterization +Though the regularity of the piecewise-constant model given by (16) can be improved by interpolation, the inverted +velocity v(x; θ) may still be highly oscillatory if the mesh size ∆x is small. Modeling approaches, such as SINDy [6], +learn the velocity fields of dynamical systems from a polynomial basis together with sparse regression. Here, we show +how the gradient derivation in (15) can be adapted to such polynomial basis parameterization of the velocity field +v(x; θ) = [v1(x; θ), . . . , vd(x; θ)]⊤ = +d +� +i=1 +vi(x; θ) ei. +The i-th component of the velocity field vi(x; θ) parameterized by a linear combination of the monomial basis of degree +at most K can be written as +vi(x; θ) = +M +� +ℓ=1 +ai +ℓ(x⊤e1) +1ki +ℓ . . . (x⊤ed) +dki +ℓ, +M = +�d + K +K +� +, +θ = {ai +ℓ}, +(17) +where the powers are represented by multi-indices ki +ℓ = (1ki +ℓ, . . . , dki +ℓ), 1 ≤ ℓ ≤ M, and |ki +ℓ| ≤ K. The size of θ in +this case is d · M. +To learn the model parameterized by (17), we can use (15) to compute the gradient ∂ai +ℓJ . Without loss of generality, +we assume ∆xi = ∆x, for all 1 ≤ i ≤ d. The only term in (15) which explicitly depends upon the velocity +paramaterization is +∂vi +j +∂ai +ℓ += +� +(xj − ei∆x/2)⊤e1 +�1ki +ℓ . . . +� +(xj − ei∆x/2)⊤ed +�dki +ℓ , +1 ≤ j ≤ N, +where i, ai +ℓ and the multi-index ki +ℓ are fixed. Note that ∂ai +ℓvi′ +j = 0 if i′ ̸= i. Thus, we can again use gradient-based +methods to infer proper polynomial coefficients {ai +ℓ}. +Although a global polynomial parameterization guarantees ideal C∞ regularity of the parameterized velocity v(x; θ), +the Runge phenomenon could be a potential downside of this approach. Specifically, as we increase the maximum +degree K of the polynomial basis, we may encounter substantial interpolation errors near the boundary ∂Ω. +4.2.3 +Neural Network Parameterization +Motivated by the universal approximation theory of neural networks [25], we may also choose to model each component +of the velocity vi(x; θ) as a feed-forward neural network, where the tunable parameters θ make up the network’s weights +11 + +Figure 3: In two dimensions, both velocity components v1(x; θ1) and v2(x; θ2) are parameterized by fully connected +neural networks with smooth activation functions. Due to the upwind finite volume method, the adjoint-state gradient +calculation for v1(x; θ1) is performed on the mesh {xj − e1∆x/2}, whereas the gradient calculation for v2(x; θ2) is +performed on the mesh {xj − e2∆x/2}. Both gradients are used in conjunction with backproppogation to train the +networks, which ultimately yields a smooth map x �→ (v1(x; θ1), v2(x; θ)). +and biases. We follow [37] to combine the adjoint-state method for the PDE constraints and the backpropagation +technique to update the weights and biases of the neural network. +The term ∂vi +jJ in the gradient calculation (15) can be computed by first evaluating the neural network on the mesh of +cell face centers oriented in the direction of ei to obtain {vi +j}, which is then plugged into (14) to obtain ∂vi +jJ . Figure 3 +illustrates where the neural network is evaluated for gradient computation. The remaining term ∂θv in (15) is then +computed via the backpropagation technique [35]. +For simplicity, we restrict ourselves to single-layer feed-forward networks. Moreover, by using a smooth activation +function, such as the hyperbolic tangent or the sigmoid function, we can guarantee C∞ regularity of the reconstructed +velocity v(x; θ) on the domain Ω. To enforce the zero-flux boundary condition, we manually set v = 0 on ∂Ω. +Consequently, the neural network parameterization may lack regularity near ∂Ω. However, if the domain is sufficiently +large, the support of the physical measure will be very far from ∂Ω, in which case we will not observe any discontinuities +originating from the boundary condition while simulating the trajectories based on (6). As we increase the number of +nodes in the hidden layer of the neural network, both the approximation power and the potential difficulty of training +the neural network are expected to increase. +5 +Numerical Results +In this section, we present several numerical examples to demonstrate the utility of the proposed approach for learning +dynamical systems from invariant measures with intrinsic noise1. In Section 5.1, we study the inverse problem for the +Van der Pol oscillator with a neural network parameterization of the velocity. In Section 5.2, we time-delay embed a +signal sampled from a Hall-effect thruster and proceed to model the dynamics in delay-coordinates based upon the +time-delayed invariant measure. We then illustrate that a very low-dimensional embedding will increase the uncertainty +of the learned model and that the choice of parameterization largely affects the regularity of the reconstructed velocity. +In Section 5.3, we study rolling averages of weekly temperature data and perform uncertainty quantification using the +learned Fokker–Planck PDE in time-delayed coordinates. We conclude in Section 5.4 by inverting a component of the +Lorenz-63 system’s velocity using a neural network parameterization. +5.1 +Van der Pol Oscillator +We begin by considering the autonomous Van der Pol oscillator [23], given by +� +˙x = y, +˙y = c(1 − x2)y − x. +(18) +Our results for learning a dynamical system with prescribed statistical properties given by the stochastically-forced +Van der Pol oscillator are shown in Figure 4. In the top row, the first panel features the velocity of (18) for the choice +of c = 0.5, the second panel shows the approximate invariant measure from a diffuse trajectory simulated via the +1For those interested, we also include an example Google Colaboratory file demonstrating the velocity inversion for a polynomial +parameterization: https://tinyurl.com/PDEinv. +12 + +j-ei△/2] +v1(x; 01) +(x; 02) +[αj -e2△α/2] + > (*(x; 01), v(x; 02)) < (Euler–Maruyama method, the third panel shows the diffuse dynamics which were used to approximate the invariant +measure, and the fourth panel shows the dynamics of the oscillator without stochastic forcing. Throughout, we color +trajectories by their histogrammed density to illustrate the connection between the Lagrangian and Eulerian perspectives. +Figure 4: Learning velocity fields to reproduce the statistics of the stochastically-forced Van der Pol oscillator. The first +row features the ground truth velocity, mass-per cell, and dynamics of the Van der Pol oscillator (18) with c = .5 and +diffusion D = .05. The subsequent rows show the resulting optimization based on the observed density using each of the +four objective functions studied in Section 4.1.1 along with a neural network paramaterization utilizing a single hidden +layer of 100 nodes and a hyperbolic tangent activation function. During the optimization, each objective function was +reduced to .05% of its initial value. We emphasize that the density in the first row is approximated via the procedure in +Figure 1, whereas the densities in the remaining rows are solutions of the PDE forward model from Section 3.1 +. +In the following rows of Figure 4, we use a neural network parameterization to solve the inverse problem using the +optimization framework from Sections 3.1 and 4. In the first column of Figure 4, we see that each objective function in +Section 4.1.1 is reduced below .05% of its initial value. Moreover, the reconstructed velocity for each objective function +is shown to vary significantly from the true velocity shown in the first row of Figure 4. This is largely due to the lack +of data away from the main attracting limit cycle. In regions of the state space with no available data, we can only +expect that the modeled velocity v(x; θ) will direct trajectories towards the attracting limit cycle on which the invariant +measure is supported. Indeed, this is what we observe. Moreover, both the PDE forward model (9) and the statistics of +the SDE simulation using the learned velocity match the ground truth invariant measure from the first row of Figure 4. +These observations indicate that the learned velocity need not agree with the ground truth velocity to yield a dynamical +system with the same invariant measure. Finally, in the last column of Figure 4, we approximate the noise-free limit +cycle by removing the diffusion term in (6), which corresponds to the simulation of an ODE rather than an SDE. +13 + +Mass per Cell +Velocity +Diffuse DynamicsOptimization1L3KLIterationsISIW2To reduce the computational requirements of the inversion in the final row of Figure 4, we compute J = W 2 +2 on a +coarsened mesh. Among the four objective functions in Figure 4, it is worth noting that the W2 metric does not compare +the two densities pointwisely and is well-defined for comparing singular measures, which is different from the other +three. The distance reflects both the local intensity differences and the global geometry mismatches [16]. It has also +been shown that the Wasserstein metric is robust to noise [12, 15]. Thanks to the geometric nature of the optimal +transportation problem, the Wasserstein metric is primarily sensitive to global changes such as translation and dilation +and is robust to small local perturbations such as noisy measurements of ρ∗. The better stability also brings a downside +as the optimization landscape can be relatively flat around the ground truth, which may lead to compromised accuracy +in the velocity reconstruction. +5.2 +Hall-Effect Thruster +We now turn to the more realistic setting of experimentally sampled time-series data. Specifically, we study the +Cathode–Pearson signal sampled from a Hall-effect thruster (HET) in its breathing mode. Hall-effect thrusters are +in-space propulsion devices that exhibit dynamics resembling stable limit cycles while in breathing mode. For details +about the experimental setup used to collect the data, the reader is encouraged to consult [13, 41]. In Section 5.2.1, we +utilize Takens’ theorem [59] to reformulate the large-scale optimization framework presented in Sections 3 and 4 to be +compatible with scalar time-series observations, and in Section 5.2.2 we demonstrate numerical results based upon this +reformulation. +5.2.1 +Methods +Intrinsic physical fluctuations present in the Cathode–Pearson signal indicate that the HET’s dynamics may be modeled +well by a Fokker–Planck equation. Motivated by this insight, we first time-delay embed the Cathode–Pearson signal +C(t) in d-dimensions to form the trajectory Cd,τ(t) := (C(t), C(t − τ) . . . , C(t − (d − 1)τ)). We then use the +procedure outlined in Figure 1 to compute the occupation measure ρ∗ of Cd,τ(t). By viewing each dimension of the +coordinate system on which the measure ρ∗ is supported as the independent variables C−kτ(t) := C(t − kτ) where +0 ≤ k ≤ d − 1, we then seek a solution to the optimization problem (2) for a velocity v = v(Cd,τ; θ). Such a velocity +can then provide us with a model of the asymptotic statistics of the embedded trajectory Cd,τ(t), provided that a +suitable diffusion coefficient can be found. +We note that forming the time-delay coordinates Cd,τ(t) does require a knowledge of measurements at uniform +increments in time. However, the available data may still be sampled slowly enough such that it is impractical to +seek a direct approximation of the Lagrangian velocity through the standard approaches described in Section 1. This +perspective motivates our use of the approach developed in Sections 3 and 4 to learn dynamical systems from invariant +measures in time-delay coordinates. +Moreover, there are a few additional considerations which arise when adapting the modeling framework presented in +Sections 3 and 4 to real-world data. Namely, we do not know the proper diffusion coefficient a priori (as was the case in +Section 5.1). Moreover, the invariant measure that the model is based on does not contain any information about the +time-scale at which the system evolves. Towards this, we utilize the following three-step procedure as a computationally +efficient means to mitigating these difficulties. +1. Bin the trajectory Cd,τ(t) onto a d-dimensional mesh with spacing ∆x along each axis to form the occupation +measure ρ∗, assume a constant diffusion coefficient D > 0, and learn the velocity v = v(Cd,τ; θ), using the +framework from Sections 3 and 4. +2. Bin the trajectory Cd,τ(t) onto another d-dimensional mesh with spacing ∆ˆx ≤ ∆x to create a new occupation +measure ˆρ∗ and adjust the diffusion coefficient by solving the optimization problem +˜D = arg min +ˆ +D∈R +J (ρε(v; ˆD), ˆρ∗), +(19) +where the term ρε(v; ˆD) in (19) denotes the forward model evaluation with the diffusion coefficient ˆD. +3. Rescale both the velocity and diffusion by solving the optimization problem +˜a = arg min +a∈R +N +� +i=1 +��� ˆC(ti; a) − Cd,τ(ti) +��� +2 +2 , +(20) +where ˆC(ti; a) denotes the time-ti solution of the ODE initial value problem with velocity av(·; θ) and initial +condition Cd,τ(t0) . The final velocity and diffusion are then given by ˜av(·; θ) and ˜a ˜D, respectively. +14 + +The three-step approach makes repeated use of the fact that ρε(v; D) = ρε(av; aD), for any scalar multiple a > 0. +Indeed, if the true diffusion coefficient D∗ > 0 is unknown a priori, but we instead seek a solution v(·, θ) with a +different diffusion D > 0, it is guaranteed that the velocity v = (D/D∗)v∗ will still provide a solution to the inverse +problem. This observation motivates step one, in which an arbitrary diffusion coefficient is used to find a solution v(·; θ) +to the inverse problem. As the dimensionality d is increased, solving the large scale optimization problem in step 1 on a +fine mesh becomes infeasible. As such, step one is typically performed on a coarse mesh where additional Gaussian +filtering is applied to the inference measure ρ∗ to make the large-scale optimization more feasible. +The diffusion coefficient is then adjusted in step two on a finer mesh via (19) to mitigate the errors due to the Gaussian +filtering, numerical diffusion, and histogram errors incurred during step one (see Figure 2). Finally, in step three the +scale of both the velocity and diffusion are adjusted via (20) such that the time evolution of simulated trajectories is +consistent with the inference trajectory Cd,τ(t) in delay coordinates. Since diffusion plays a relatively small role over +short time-scales for the quasi-periodic HET data, we use the trajectory to calibrate a reasonable time-scaling between +our model and the available data. However, as the magnitude of the diffusion increases, the least squares fit in (20) +will become less reliable and it may be preferable to instead minimize a transport cost between a collection of model +samples and a collection of data samples at each time-step. +5.2.2 +Results +The results of the three-step procedure in Section 5.2.1 for learning the HET dynamics are shown in Figure 5 for +an embedding dimension of d = 3 and time-delay of τ = 1.4 · 10−5 seconds, or rather τ = .23 when normalizing +the time-scale to the HET breathing mode frequency (16.6kHz). The modeled trajectory accurately reconstructs +the shape of the embedded Cathode–Pearson signal but cannot capture the variable diffusion present throughout the +time-delayed signal. We do not expect to capture such details, as we assume a constant diffusion coefficient in our +model. Nevertheless, we regard the reconstruction of the 3D globally attracting limit cycle as a success and leave +extending the model to account for the case of a non-constant diffusion tensor to future work. +Figure 5: Learning the velocity from embedded Cathode-Pearson signal’s invariant measure. We present the time- +delay embedded signal C3,τ(t) (left), the reconstructed velocity field from the embedded signal’s occupation measure +(middle), where blue indicates slow speed and red indicates fast. Finally, we show a trajectory simulated with the +Euler–Maruyama method from the learned velocity and diffusion (right). The velocity was parameterized by a neural +network with 500 nodes in a single hidden layer and learned using the KL divergence loss function. The three-step +procedure in Section 5.2.1 is used to learn the model, and in step one, additional Gaussian filtering is applied to the +occupation measure ρ∗ to simplify the resulting optimization. +The dimensionality of the original HET dynamics is unknown, and as such, a sufficient embedding dimension for the +Cathode–Pearson signal is unclear, though likely very high. Interestingly, we can compare the model learned in Figure 5 +with a 2D analog to demonstrate that when the number of time-delays is not sufficiently large, there is more uncertainty +in modeling the time-delayed dynamics. This phenomenon is most evident when inspecting regions of the delayed +Cathode–Pearson signal for which the 2D embedding lacks structure readily observed in 3D. +Specifically, consider a collection of nearby samples {C3,τ(ti)}n +i=1 in the 3D time-delay coordinate system +(C0, C−τ, C−2τ). The corresponding 2D samples {C2,τ(ti)}n +i=1 will also be nearby one another in the 2D time- +delay coordinate system (C0, C−τ). In Figure 6, we initiate uniform distributions centered about these samples in +both 2D and 3D time-delay coordinate systems. We then evolve both the samples and initial uniform distributions +forward in time. The evolution of the ground truth samples is simply determined by the time-delayed Cathode Pearson +signal Cd,τ(t), and the evolution of the uniform distributions is given by Fokker–Planck models constructed from +the time-delayed Cathode Pearson signal’s invariant measure. As the modeled probability densities and ground truth +15 + +DelayedCathode-PearsonSignal +Modeled Velocity +Modeled Trajectory +0.6 +0.6 +0.6 +0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +C +C +C +0.0 +0.0 +0.0 +2 +2 +-0.2 +-0.2 +0.2 +-0.4 +0.4 +0.4 +0.6 +-0.6 +-0.6 +-0.6 +0.4 +-0.6 +-0.4 +0.6 +-0.4 +-0.2 +-0.2 +0.2 +-0.6 +0.0 +0.6 +0.4 +0.0 +-0.6 +-0.4 +0.0 +0.2 +Co +-0.2 +0.2 +Co +0.2 +Co +0.0 +0.0 +0.2 +0.4 +0.2 +0.4 +0.4 +C +0.2 +0.4 +0.4 +0.4 +0.60.6 +0.60.6time = 0 +time = .17 +time = .33 +time = .5 +time (1/HET Breathing Mode Frequency) +Figure 6: Comparing the model accuracy and uncertainty for the embedded Cathode–Pearson signal with 2D and 3D +time delays. The time-evolution of the models is compared to a collection {Cd,τ(ti)}n +i=1 of samples (plotted in black) +from the time-delayed Cathode–Pearson signal. The first two rows feature four snapshots from the 2D and 3D models +from the time interval [0, .5]. The bottom row compares the uncertainty of the models to the ground truth data. The +time units are normalized to the inverse of a HET breathing mode frequency (16.6kHz). Both models utilized a neural +network velocity parameterization with 500 nodes in a single hidden layer and reduced the KL divergence objective +function to .1% of its initial value during training. As in Figure 5, the three-step procedure in Section 5.2.1 is used to +learn the models, and in step one, additional Gaussian filtering is applied to the occupation measure ρ∗ to simplify the +resulting optimization. The 3D visualization was plotted using [55]. +samples evolve in time, we observe in Figure 6 that the mean of the 3D model matches the true sample mean more +closely than the 2D model, and that it has less uncertainty. +In Figure 7, we study the three parameterizations from Section 4.2 for learning the time-delayed Cathode–Pearson +signal’s velocity, now with an embedding dimension of two to allow for clearer visualizations. It can be seen that the +density associated with each velocity parameterization indeed matches the ground truth density in Figure 7, but that +the velocity fields differ significantly from one another. The piecewise-constant velocity in Figure 7 suffers from poor +regularity with discontinuities on the attracting limit cycle. As a result, we lose the connection between the Eulerian +16 + +0.6 +2D model mean +3D model mean +True sample mean +0.4 +2D model one-o Interval +3D model one- Interval +王 +0.2 +True one-o Interval0.0 +-0.2 +0.0 +0.1 +0.2 +0.3 +0.4 +0.50.008 +0.005 +0.003 +0.0000.020 +0.013 +0.007 +0.000Figure 7: Comparison between the three parameterizations detailed in Section 4.2 for learning a velocity field +from the time-delayed Cathode–Pearson signal’s invariant measure, using a diffusion coefficient D = 0.01. The +learned velocities and densities for the piecewise constant (PC), global polynomial (GP), and neural network (NN) +discretizations are shown in the three columns, respectively. We show the velocity field on the full state space (top +row), a close-up of the velocity field’s direction near the attracting limit cycle (middle row), and the forward model +output ρε(v(θ)) for each parameterization (bottom row). The resulting parameter spaces of these discretizations have +dimensionality of 9800 (PC), 56 (GP), and 400 (NN). The L2 loss is reduced below .1% of its initial cost for the PC +and NN discretizations and reduced below .7% of its initial value for the GP case when we stopped the optimization. +and Lagrangian dynamics and cannot reconstruct zero-diffusion trajectories which form a stable limit cycle. On the +other hand, the velocities parameterized by the global polynomial and the neural network are both C∞. The differences +among these three can clearly be seen via the zoomed-in velocity plots in the second row of Figure 7. The global +polynomial and neural network discretizations are both global parameterizations of the velocity, and as such, their +values near the domain’s boundary are dictated by the available data in the center of the domain. This causes the +polynomial velocity to rapidly increase near the boundary, and a similar effect can also be seen for the neural network. +It is worth noting that the initial condition for the optimization in Figure 7 can play a large role in the reconstructed +velocity, which is related to the optimization landscape of the nonconvex optimization problem (2) we tackle. In +the case of the piecewise-constant discretization, we initialize all velocities to be significantly less than the diffusion +coefficient D = 0.1. Thus, diffusion initially dominates in the finite volume solver, and all non-boundary cells will +contain nonzero mass, which allows for accurate gradient updates everywhere. This phenomenon can also help neural +network training, though it is not always necessary due to the global nature of parameterization. Moreover, we initialize +our polynomial basis to form the velocity ( ˙x, ˙y) = (−y + x(0.1 − x2 − y2), x + y(0.1 − x2 − y2)), which describes a +globally attracting limit cycle. Thus, to converge to the ground truth limit cycle of the time-delayed Cathode–Pearson +signal, this initial velocity only needs to be translated and deformed. +17 + +Piecewise Constant Velocity +Global Polynomial Velocity +NeuralNetworkVelocity +1.0 +1.0 +1.0 +1.4 +0.8 +0.7 +0.8 +0.8 +1.2 +0.6 +0.6 +0.6 +0.6 +0.4 +1.0 +0.4 +0.5 +0.4 +办： +0.2 + 0.4 +0.2 +0.2 +0.8 +0.0 +0.0 +0.0 +0.3 +0.6 +-0.2 +0.2 +-0.2 +0.2 +0.4 +-0.4 +-0.4 +1 +-0.4 +0.1 + 0.2 +-0.6 +-0.6 +0.6 +-0.6-0.4-0.20.0 +0.20.40.60.81.0 +-0.6 -0.4 -0.20.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.6-0.40.20.00.20.40.6 +0.81.0 +Co +Co +CoPiecewise Constant Velocity +Global Polynomial Velocity +Neural Network Velocity +0.6 +0.6 +0.6 - +0.4 - +0.4 +0.4 +0.2 +0.2 +0.2 +c +0.0 +0.0 +0.0 +0.2 +0.2 +0.2 +-0.4 - +-0.4 +0.4 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +Co +Co +CoPiecewise Constant Mass per Cell +Global Polynomial Mass per Cell +Neural NetworkMassperCell +1.0 +0.005 +1.0 +0.005 +1.0 +T0.005 +0.8 +0.8 +0.8 +0.004 +0.004 +0.004 +0.6 +0.6 +0.6 +0.4 +0.4 +0.4 +0.003 +0.003 +0.003 +0.2 +0.2 +0.2 +0.0 +0.0 +0.0 +0.002 +0.002 +0.002 +-0.2 +-0.2 +0.2 +-0.4 +0.001 +-0.4 +0.001 +-0.4 +0.001 +-0.6 +0.6 +-0.6 +0.000 +0.000 +0.000 +-0.6-0.4-0.20.00.20.40.60.81.0 +-0.6-0.4 -0.20.00.20.40.60.81.0 +-0.6 -0.4 -0.20.0 +0.20.40.60.81.0 +Co +Co +Co5.3 +Temperature Uncertainty Quantification +We now study 2D time-delay embedded data of weekly rolling averages of the temperature in Ithaca, NY, between +2006 and 2020 [11]. We view temperature fluctuations over short time scales as an intrinsic diffusion process and the +approximately periodic oscillation of seasonal temperatures driven by some nonzero velocity. Thus, we model the 2D +data in delay coordinates as a diffuse limit cycle. We again follow the procedure in Section 5.2.1 to learn a velocity +v(x; θ) and diffusion coefficient D, which closely matches the occupation measure. +Figure 8: Performing prediction and uncertainty quantification for Ithaca, NY’s temperature in 2019. The top left +plot shows the ground truth occupation measure accumulated from 13 years of weekly rolling averaged temperature +observations, normalized by an affine transformation to [−1, 1]. The top middle and right plots show the learned model +and velocity vector field, obtained using the three-step procedure in Section 5.2.1. In the bottom plot, the PDE model +with a uniform initialization in the box from the top middle plot is evolved in time and used to quantify the uncertainty in +the measurements of C0. Observed trajectories of the temperature in delay coordinates with initial conditions displayed +in the top left plot are also shown to demonstrate the effectiveness of the learned model. A time delay of τ = 280 days +is used, and the model is trained using a neural network parameterization and the KL-divergence objective function. +As in Section 5.2 we can use the trained model v(x; θ) to quantify measurement uncertainties through the Fokker– +Planck equation (9), whose solution is a probability density in the time-delay coordinates (C0, C−τ). Specifically, if +we know some initial probability distribution that captures the current state of the temperature system well, we can +consider the time-evolution of the distribution using our trained model to quantify the uncertainty of future temperature +measurements. The process of evolving both the Fokker–Planck PDE from a uniform distribution and the ground +truth sample paths from past temperature measurements is shown in Figure 8. The uncertainty bounds from the model +accurately capture fluctuations in the training data used to form the occupation measure (plotted in black), as well as a +testing sample path previously unseen by the model (plotted in red). +It is also worth noting that the confidence intervals we construct may be larger than the actual range due to several +factors, including additional extrinsic noise from filtering the data, modeling errors accumulated from the hypothesis +space, numerical diffusion in the forward model, and a sub-optimal embedding dimension. Reducing such errors may +result in tighter confidence intervals. Considering time-delays in higher dimensions could yield better predictions of the +temperature’s transient behaviors. +18 + +Occupation Measure +1.0 +0.00200 +X +Data IC +0.8 - +0.00175 +0.6 - +0.00150 +0.4 - +0.00125 +0.2 - +0.00100 +C +0.0 - +0.00075 +-0.2 +0.00050 +-0.4 +0.00025 +-0.6 +0.00000 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CoLearned Model +1.0 +Model IC +0.0014 +0.8 +0.0012 +0.6 - +0.0010 +0.4 - +0.0008 +0.2 - +C +0.0006 +0.0 +0.0004 +-0.2 +-0.4 - +0.0002 +-0.6 +0.0000 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CoLearned Velocity +1.0 +0.25 +0.8 +0.6 +0.20 +0.4 +Z +0.15 +0.2 +0.0 +0.10 +-0.2 +0.05 +-0.4 +-0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CoTemperature Forecast +0.8 +0.6 +0.4 +0.20.0 +Two-o interval +One-o interval +-0.2 +Mean +Training +Testing +-0.4 +0 +50 +100 +150 +200 +250 +300 +350 +Days5.4 +Lorenz-63 System +We conclude this section by studying the Lorenz-63 system [39], defined by +� +� +� +˙x += c1(y − x) +˙y += x(c2 − z) − y +˙z += xy − c3z +, +(21) +where we consider (c1, c2, c3) = (10, 28, 8/3). For these choices of parameters, the Lorenz-63 system exhibits chaotic +behavior and admits a unique physical measure [60]. In Figure 9, we assume that the quantities ˙y and ˙z are known, and +we learn a model for the velocity in the x-direction, using the stochastically-forced Lorenz-63 system’s occupation +measure. We emphasize that the samples used to form the occupation measure are not correlated in time and were, +in fact, randomly sampled from a trajectory as in [65, Figure 6]. This demonstrates the applicability of the proposed +approach for learning the dynamics of physical processes for which measurement times are unknown. From the +occupation measure of these samples, we invert the first component ˙x of the Lorenz-63 system’s velocity via a neural +network parameterization and successfully simulate dynamical trajectories which recover the original Lorenz attractor. +We remark that when ˙x, ˙y, and ˙z are all simultaneously inverted, the optimization is unsuccessful at reconstructing +the true velocity (21). While we may be able to learn a velocity that approximately recovers the stationary state of the +Lorenz-63 system in the sense of (9), the corresponding dynamical trajectories do not form a chaotic attractor, i.e., the +physical property (3) does not hold. Whether the difficulties of inverting all velocity components of the Lorenz-63 +system are due to inherent non-uniqueness in the inverse problem or simply inconvenient local minima during training +is worth further investigation in future work. +Figure 9: Neural network parameterization of ˙x using the Lorenz system’s stochastically perturbed invariant measure +with D = 10 and ∆x = 2. The top row displays the learned velocity and a single trajectory plotted both with and +without diffusion. The bottom row displays the true velocity and true trajectories both with and without diffusion. For +visualization of the occupation measure used to learn the model displayed in the top row, we refer to [65]. +6 +Conclusion +In this paper, we performed large-scale parameter identification to model the velocity v(x; θ) of intrinsically noisy +autonomous dynamical systems. We first adapted the invariant measure surrogate model ρε(v(θ)) in [65] based upon +the continuity equation to the Fokker–Planck equation. This increased our modeling capacity and prevented overfitting +the reconstructed velocity v(θ) while modeling intrinsically noisy trajectories. We next extended the three-coefficient +learning performed in [65] to thousands of coefficients by modeling the velocity v(x; θ) via global polynomials, +piecewise polynomials, and fully connected neural networks. The efficient gradient computation presented in Section 4 +made these large-scale parameterizations of the velocity computationally tractable. We finally studied velocity inversion +for invariant measures of time-delay embedded observables. The method of time-delay embedding is useful for +analyzing real-world data, where in many cases, only limited observations of complex systems are available. As such, +we proceeded to learn the velocity in time-delay coordinates for a Hall-effect thruster system and weekly temperature +measurements. Both systems exhibit periodic behavior with intrinsic noise, which led us to model the time-delayed +19 + +Zero-Diffusion Trajectory +50 +40 +30 +Z +20 +10 +30 +10 +20 +-20 +-10 +0 +0 +-10 +10 +-20 +Y +X +20 +-30Lorenz-63 Samples +50 +40 +30 +Z +20 +10 +0 +30 +10 +20 +-20 +-10 +0 +0 +-10 +10 +-20 +Y +X +20 +-30Initial Velocity +-20 +-15 +-10 +-5 +X +0 +5 +10 +15 +20 +-20 +-10 +0 +10 +20 +Y +ZModeled Velocity +20 +-15 +-10 +-5 +X +5 +10 +15 +20 +-20 +-10 +0 +10 +20 +Y +ZTrue Velocity +-20 +-15 +-10 +-5 +X +0 +5 +10 +15 +20 +-20 +-10 +0 +10 +20 +Y +Zdynamics as globally attracting limit cycles. Using these models, we predicted future states of the systems and quantified +uncertainty in forecasts by evolving the Fokker–Planck equation forward in time, similar to [3]. +In future work, there are several computational directions to explore. The first-order upwind finite volume discretization +incurs numerical diffusion for large values of ∆x and is costly to implement as a forward model for small values of ∆x. +Higher-order methods would be both more accurate and efficient. Moreover, most globally attracting systems admit +invariant measures which are singular with respect to the Lebesgue measure, and consequently, most of the cells in the +finite-volume discretization are unused, even if intrinsic noise is present. Therefore, it would be more efficient to solve +the Fokker–Planck equation on an unstructured mesh that adapts to the density of the available data. +Given the complex interactions of a large number of variables in real-world systems, the low-dimensional embedding +presented in Section 5 may not be sufficient to resolve the full dynamics of the HET and temperature systems. In many +cases, time-delay embeddings of dimensionality greater than three are required, and as such, it would be useful to +adapt the proposed approach to higher dimensions. The current framework suffers from the curse of dimensionality, +so further investigation into mesh-free approaches for approximating invariant measures and solving the resulting +PDE-constrained optimization (2) would be fruitful. Finally, the forward model ρε(v(θ)) should be adapted so that it +can describe the invariant measures of multiple attractors. +On the theoretical side, numerous questions about the well-posedness of the inverse problem remain open. Perhaps +the most important ones are the following questions, which are directly related to the uniqueness of the reconstructed +velocity v(x; θ). Given a physical measure µ of the autonomous dynamics ˙x = v(x), when does there exist another +dynamical process ˙y = w(y) which admits the same physical measure µ but satisfies v(x) ̸= w(x) on supp(µ)? Does +the presence of intrinsic noise affect the answer to this question? If the velocity v(x) is not unique on supp(µ), is there +a restriction to a reasonable function space where uniqueness can be obtained? A deeper investigation into the stability +of the inverse problem is also needed. Specifically, it is desirable to understand how perturbations to a velocity v(θ) +affect the associated physical measure and our surrogate approximation via the upwind finite volume method. Finally, if +the uniqueness of the velocity can be guaranteed on supp(µ), then the stability of the reconstructed velocity v(x; θ) +with respect to perturbations of the inference data µ would be the next interesting question. +Acknowledgements +J. Botvinick-Greenhouse was supported by the Department of Defense (DoD) through the National Defense Science & +Engineering Graduate (NDSEG) Fellowship Program. R. Martin was partially supported by AFOSR Grants FA9550- +20RQCOR098 (PO: Leve) and FA9550-20RQCOR100 (PO: Fahroo). This work was done in part while Y. Yang was +visiting the Simons Institute for the Theory of Computing in Fall 2021. Y. Yang acknowledges support from Dr. Max +R¨ossler, the Walter Haefner Foundation and the ETH Z¨urich Foundation. This material is based upon work supported +by the National Science Foundation under Award Number DMS-1913129. +We thank Dr. Chen Li for his helpful suggestions and generosity in sharing code for the approach of Section 4.2.3. +References +[1] Altan Allawala and J. B. Marston. Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck +equation and cumulant expansions. Physical Review E, 94(5), nov 2016. +[2] Ellen Baake, Michael Baake, HG Bock, and KM Briggs. Fitting ordinary differential equations to chaotic data. +Physical Review A, 45(8):5524, 1992. +[3] Thomas R Bewley and Atul S Sharma. 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Journal of statistical +physics, 108(5):733–754, 2002. +23 + diff --git a/dtE4T4oBgHgl3EQfpg2t/content/tmp_files/load_file.txt b/dtE4T4oBgHgl3EQfpg2t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a40c5235b7072aa75570125d3f9bb6115064c612 --- /dev/null +++ b/dtE4T4oBgHgl3EQfpg2t/content/tmp_files/load_file.txt @@ -0,0 +1,1502 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf,len=1501 +page_content='LEARNING DYNAMICAL SYSTEMS FROM INVARIANT MEASURES Jonah Botvinick-Greenhouse∗ Center for Applied Mathematics Cornell University Ithaca, NY 14850 jrb482@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='edu Robert Martin DEVCOM Army Research Laboratory Research Triangle Park Durham, NC 27709 robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='martin163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='civ@army.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='mil Yunan Yang Institute for Theoretical Studies ETH Z¨urich Z¨urich, Switzerland 8092 yunan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='yang@eth-its.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='ch January 13, 2023 ABSTRACT We extend the methodology in [65] to learn autonomous continuous-time dynamical systems from invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We assume that our data accurately describes the dynamics’ asymptotic statistics but that the available time history of observations is insufficient for approximating the Lagrangian velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Therefore, invariant measures are treated as the inference data and velocity learning is reformulated as a data-fitting, PDE-constrained optimization problem in which the stationary distribu- tional solution to the Fokker–Planck equation is used as a differentiable surrogate forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We consider velocity parameterizations based upon global polynomials, piecewise polynomials, and fully connected neural networks, as well as various objective functions to compare synthetic and reference invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We utilize the adjoint-state method together with the backpropagation technique to efficiently perform gradient-based parameter identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Numerical results for the Van der Pol oscillator and Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, are presented to demonstrate the effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Keywords Dynamical systems · Invariant measure · Inverse Frobenius–Perron problem · Parameter identification · Fokker–Planck equation · Neural networks · Time-delay embedding · Computational ergodic theory 1 Introduction Data-driven models have proven to be instrumental across numerous scientific disciplines for their ability to predict and control the behavior of complex physical systems [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The method by which models are constructed is highly dependent on the available data, as well as prior knowledge of the physical process in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Popular approaches such as the shooting methods [2, 46], neural differential equations [8, 29, 49], and SINDy [6] adopt a Lagrangian perspective and seek to directly minimize the error between modeled trajectories and observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, the following scenarios can render these standard techniques unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The observational data may be sparsely and irregularly sampled in time, such that derivatives cannot be accurately approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In a worst-case scenario, one may not even know the times at which individual samples were drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, measurements may be contaminated with large amounts of noise and the system in question could be highly sensitive to initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In the specialized setting of traffic simulation, the issue of sparse sampling was approached by merging the tasks of imitation learning, and interpolation under a single generative adversarial network [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, the Ensemble-SINDy method learns noisy chaotic dynamics by averaging over a family of models formed using bootstrapped samples [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, a general modeling framework that is robust to each of the aforementioned difficulties remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In [21, 65], an Eulerian perspective was adopted to handle such difficulties, and velocity models were constructed to yield the same asymptotic statistics as the observed measurements, rather than seeking a pointwise match with time ∗Corresponding author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='05193v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='DS] 12 Jan 2023 trajectories or Lagrangian velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' More specifically, instead of directly treating the noisy observations {xη(ti)}n i=1 of an autonomous flow ˙x = v∗(x) as inference data, the approaches in [21, 65] considered the occupation measure ρ∗, where for each measurable set B, ρ∗(B) := 1 n n � i=1 χB (xη(ti)) , χB(x) = �1, x ∈ B, 0, x ̸∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (1) When the occupation measures generated by a set of initial conditions all weakly converge to the same invariant measure and the set of initial conditions has nonzero Lebesgue measure, such an invariant measure is said to be physical [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' These convergence properties can be delicate, and an overview of the relevant theory is provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In this work, we consider the class of autonomous systems for which the occupation measures of almost all initial conditions converge to a unique physical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Notably, this encompasses chaotic attractors such as the Lorenz-63 system [39, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Going forward, we write v = v(θ) = v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) to denote the dependence of the reconstructed velocity fields on a set of parameters θ ∈ Θ where Θ ⊂ Rm is the admissible set of all parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The concrete form of θ depends on the hypothesis space of v, which will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The task is now to find the best-parameterized model v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) approximating the true velocity v∗ by minimizing the mismatch inf θ∈Θ J (θ), J (θ) := D(ρε(v(θ)), ρ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (2) The formulation (2) represents an inverse data-matching optimization problem, in which D denotes a metric or divergence on the space of probability measures and ρε(v(θ)) is a regularized approximation to the physical measure of the dynamical system, given some regularization parameter ε > 0 and the current velocity v(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Although one could approximate ρ(v(θ)) by numerically integrating a trajectory and binning the observed states to a histogram [21], this approach does not permit simple differentiation of the resulting measure with respect to the parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When the size of θ is large, it is practical to use gradient-based optimization methods for solving the optimization problem (2), and one has to compute the essential gradient ∂θJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In [65], this was handled by viewing ρε(v(θ)) as the dominant eigenvector of a regularized Markov matrix originating from an upwind finite volume discretization of the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The derivative ∂θJ was then seamlessly computed via the adjoint-state method [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The computation time of the adjoint-state method is independent of the size of θ, making the framework presented in [65] well-suited for large-scale computational inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Different from [65], we consider the Fokker–Planck equation as the partial differential equation (PDE) forward model for ρε(v(θ)), rather than the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This is motivated by the Fokker–Planck equation’s larger modeling capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Indeed, the Fokker–Planck equation reduces to the continuity equation when its diffusion term is zero, and it can fit intrinsic noise present in trajectories which reduces over-fitting the parameterized velocity v(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, the Fokker–Planck equation can be seen as an alternative to the teleportation regularization used for the continuity equation in [65], in order to guarantee the uniqueness of the computed stationary solution ρε(v(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In this work, we build upon the framework proposed in [65] and investigate dynamical system velocity learning with a large-scale parameter space applied to real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' There are three essential new contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' First, we use an isotropic diffusion to regularize the forward continuity equation, which differs from the so-called teleportation technique [20] proposed in [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Second, in contrast to only learning three coefficients as done in [65], we parameterize the velocity v(θ) using piecewise polynomial, global polynomial, and neural network discretizations, which can all yield large parameter spaces with thousands of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We compare the reconstructed velocity in each case and further discuss how the choice of parameterization affects the inverse problem’s well-posedness and the reconstructed velocity’s regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We also consider various metrics/divergences as the choice of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Third, we investigate velocity learning in time-delay coordinates, which can characterize the full dynamics even with partial state measurements [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It is worth noting that there is no analytic form for the velocity in time-delay coordinates, even for well-studied dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The proposed differentiable forward model and gradient-based methods are powerful and scalable tools for modeling time-delayed dynamics, and they permit larger-scale modeling than the approaches proposed in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 2, we review essential background on dynamical systems, invariant measures, the Fokker–Planck equation, and time-delay embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 3, we introduce the forward surrogate model ρε(v(θ)) and analyze its modeling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 4, we present an efficient gradient calculation for the objective function J (θ) by treating (2) as a PDE-constrained optimization problem and utilizing the adjoint-state method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We then adapt the gradient calculation to various velocity parameterizations, including neural network discretizations in which the gradient is computed along with the backpropagation technique [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, in Section 5, we present velocity reconstructions for the Van der Pol oscillator and the Lorenz-63 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We also model dynamics in time-delay coordinates based on real-world data from a Hall-effect thruster, and we provide an example of temperature prediction with uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Conclusions and further discussions follow in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 2 2 Background This section reviews the essential background on invariant measures, stochastic dynamics, the Fokker–Planck equation, and time-delay coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, we review the Eulerian approach for parameter identification proposed in [21, 65], as well as prior work on the discrete inverse Frobenius–Perron problem [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Physical Measures and SRB Measures Physical measures characterize the long-term statistical behavior of a physically significant subset of dynamical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When a dynamical system is chaotic and exhibits sensitive dependence on initial conditions, the existence of a physical measure unifies the statistical properties of trajectories which are pointwise dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' While ergodic measures also describe the long-term statistical behavior of dynamical trajectories, they may have very small support or, in fact, be singular, which makes them difficult to observe computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' On the other hand, if a dynamical system admits a physical measure, it holds that the dynamical trajectories corresponding to a positive Lebesgue measure subset of initial conditions will all share the same statistical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We will next formalize these ideas using the language of ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For a more thorough treatment of physical measures and the foundations of ergodic theory, the reader is encouraged to consult [9, 19, 66] and [14, 30, 43, 48], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Following [66], we assume that M is a compact Riemannian manifold and that T : M → M is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' A probability measure µ is said to be invariant with respect to the map T if µ(T −1(B)) = µ(B) for all B ∈ B, where B denotes the Borel σ-algebra [14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Hereafter, we will assume that µ is an invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' A point x ∈ M is said to be generic [66, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2] if for all g ∈ C(M), it holds that lim N→∞ 1 N N−1 � k=0 g(T k(x)) = � M g dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (3) The left-hand side of (3) is known as the time-average of a function g ∈ C(M) whereas the right-hand side of (3) is known as the space average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It follows Birkhoff’s pointwise ergodic theorem [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='30] that the time-average of any g ∈ C(M) necessarily exists on a set of full µ-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To formally discuss the statistical properties of dynamical trajectories, we now define the N-step occupation measure given the initial condition x ∈ M as µx,N(B) := 1 N N−1 � k=0 χB(T k(x)), ∀B ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (4) As noted in [14, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='19], the condition that a point x ∈ M is generic is equivalent to the condition lim N→∞ µx,N = µ , (5) where convergence takes place in the weak-* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since the quantity µx,N(B) approximates the average amount of time for which the orbit {T k(x)}∞ k=0 initiated at x ∈ M resides in a measurable set B ∈ B, this convergence indicates that the collection of generic points all share the same asymptotic statistical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When the measure µ is ergodic (see [14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='13]), it holds that µ-almost every x ∈ M is a generic point [14, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, if µ is an ergodic measure that is singular with respect to the Lebesgue measure, the resulting collection of generic points may be physically insignificant and difficult to observe computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Motivated by this perspective, an invariant measure µ is said to be physical if there exists a collection of generic points with positive Lebesgue measure [66, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Physical measures are closely related to so-called Sinai–Ruelle–Bowen (SRB) measures [9, 66], and in the case of Axiom A attractors where T ∈ C2(M, M), the two have been shown to coincide [66, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Importantly, the existence of a physical measure cannot provide insight into the complexity of the dynamics for the set of generic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, as noted in [66], an SRB measure with nonzero Lyapunov exponents [19, Chapter 1] is a particular type of physical measure that describes chaotic attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We will next discuss the ways in which a physical invariant measure µ can be computationally approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' If one collects the measurements {T k(x)}N k=1, the weak-* convergence in (5) suggests that the physical measure µ will describe the statistics of our measurements provided that N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Motivated by this perspective, we can discretize the domain M and directly compute the occupation measure (4) for each cell in the discretization to approximate the physical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This procedure is illustrated in Figure 1 and has been previously used to approximate physical measures in [1, 21, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Other approaches have been proposed to compute the invariant measure as the stationary vector of the finite-dimensional approximation of the continuous Frobenius–Perron operator [34], including Ulam’s method [19] and Galerkin-type 3 methods [10, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' More precisely, these discretizations are used to construct a Markov matrix that represents a random dynamical system approximating the deterministic map T : M → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' An invariant measure for the discrete approximation is then recovered as a stationary vector of the resulting Markov matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As the discretization is refined, certain assumptions guarantee that the desired SRB measure will be recovered in the weak-* limit [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The process of identifying SRB measures as so-called zero-noise limits has also led to theoretical results which prove their existence for certain classes of dynamical systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Though we have reviewed the theory of physical measures and SRB measures in the context of arbitrary discrete-time dynamical systems, we will hereafter focus on applications in which the dynamics are given by a time-∆t flow map for some ∆t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 1: To computationally approximate a physical measure without knowledge of the dynamical system’s functional form or the sampling times of trajectory data, the following procedure is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We first obtain trajectory measurements (left), then form a finite mesh over the state space (middle), and lastly bin the sampled trajectory to a histogram (right) to represent the probability of occupation for each cell in the discretization of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Stochastic Dynamics and the Fokker–Planck Equation Consider an Itˆo stochastic differential equation (SDE) of the form dXt = v(Xt)dt + σ(Xt)dWt, X0 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (6) Above, Wt is a Brownian motion, v is the velocity, and σ determines the diffusion matrix Σ(x) = 1 2σ(x)σ(x)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For simplicity, we will consider the case of a constant diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Similar to the deterministic setting, there are analogous notions of invariant measuress, ergodicity, and physical measures in the stochastic setting [4, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' One may use the Euler–Maruyama method to obtain the numerical solution to (6) on the time interval [0, T], Xj+1 = Xj + v(Xj)∆t + σ(Xj)ξj √ ∆t, ∆t := T/N, j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , N − 1}, where {ξj} are independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=') from N(0, I), the standard normal distribution on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The Fokker–Planck equation provides a PDE description of the probability density ρ(x, t) of the random variable Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' By [53, Page 88], the density evolves as ∂ρ(x, t) ∂t = −∇ · (ρ(x, t)v(x)) + ∇ · � ∇ · (Σ(x)ρ(x, t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (7) By assuming a constant diffusion, we may write Σ(x) = DI, where I denotes the identity and D > 0 is a constant representing the scale of the diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Equation (7) can then be simplified to read ∂ρ(x, t) ∂t = −∇ · (ρ(x, t)v(x)) + D∇2ρ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (8) We leave the study of a non-constant or anisotropic diffusion for later work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that if D = 0, (8) reduces to the so-called continuity equation, which instead models the probability flow of the ODE given by ˙x = v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Under certain conditions [27], the steady-state solution ρ(x) of (8) exists and satisfies ∇ · (ρ(x)v(x)) = D∇2ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (9) Since (9) describes a limiting distribution limt→∞ ρ(x, t), it has been previously used to provide approximations of invariant measures for stochastically-forced dynamical systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4 Simulate Trajectory Create Mesh Bin to Histogram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 3 - 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='006 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 2 - 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='005 1 - 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='004 y 0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='003 1 - 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='002 2 - 2 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 3 - 3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='000 4 2 1 0 1 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7 4 2 0 2 4 X X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 Delay Coordinates and Takens’ Theorem The technique of time-delay embedding is a popular approach for reconstructing chaotic dynamical systems from limited observations and has found numerous applications in the physical sciences [5, 21, 32, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The procedure involves embedding time series measurements ψ(t) = ψ(x(t)) of a full state x(t) into d-dimensional Euclidean space by considering the vector of time-lagged measurements Ψd,τ(t) = (ψ(t), ψ(t − τ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , ψ(t − (d − 1)τ)), for some τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Takens’ theorem [59] provides suitable assumptions under which Ψd,τ(t) and x(t) are related via a diffeomorphism, implying that the time-lagged vector of partial observations Ψd,τ(t) is sufficient for reconstructing the full state x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Notably, the embedding dimension provided in [59] is d = 2m + 1 where m is the dimension of a compact manifold M on which the flow map ft for the original dynamics is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In cases when trajectories are attracted to a compact subset A with box-counting dimension (see [56, Page 586]) dA strictly less than m, it turns out that lower-dimensional reconstructions can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Indeed, it was shown in [56, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='5] that for an open subset U ⊂ Rk, a flow ft : U → Rk, and a compact subset A ⊂ U with box-counting dimension dA, suitable assumptions on the equillibria and periodic points of ft guarantee that for almost all C1 observation functions ψ : U → R, the mapping Ψd,τ(x) = (ψ(x), ψ(f−τ(x)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , ψ(f−(d−1)τ(x))) is one-to-one on A, where d is an integer strictly larger than 2dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, the mapping Ψd,τ(·) preserves any manifold structure on A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=', it is an immersion for all compact subsets C of a smooth manifold which is contained within A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This result is a helpful generalization of Takens’ theorem, as the box counting dimension for chaotic attractors can be much smaller than the dimension of Euclidean space on which their flows are defined [55, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When a time-series projection ψ(t) of an unknown system ˙x = v(x) is observed, one can try to numerically determine a suitable embedding dimension d and time delay τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' see for example [7, 40, 42, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Choosing a proper embedding dimension and time delay is important for obtaining a reliable surrogate model of the original dynamics in time-delayed coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Notably, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2, we demonstrate that models for the velocity in time-delayed coordinates can incur excess uncertainties when the embedding dimension is not sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 Prior Work on Learning Dynamics from Invariant Measures For chaotic systems, trajectories are sensitive to initial conditions and estimation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Sometimes, the approx- imated reference velocity field {ˆv(x(ti))} cannot be accurately estimated from trajectory {x(ti)} due to the lack of observational data, slow sampling, discontinuous or inconsistent time trajectories, and noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To tackle such difficulties, instead of working with the Lagrangian trajectories, [21, 65] propose an Eulerian approach by treating the occupation measure (4) as the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When enough samples are available, the occupation measure can be treated as an approximation to the invariant measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finding the optimal parameter θ is then translated into the optimization problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The reference measure ρ∗ is the occupation measure converted from the observed trajectories {ˆx(ti)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In [21], the approximated synthetic ρε(v(θ)) is generated by first simulating the synthetic trajectories {x(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ)} based on the dynamical system and then computing its histogram following (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since this approach requires lengthy trajectory simulation, each evaluation of ρε(v(θ)) for a given θ is relatively costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, it is difficult to compute the derivative of ρε(v(θ)) with respect to θ, due to the histogram approximation of nonlinear trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As an improvement to the original idea in [21], [65] proposes a surrogate model to approximate ρε(v(θ)) that is differentiable in θ and sometimes faster to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The key idea is to solve for ρε(v(θ)) as the distributional steady-state solution to the continuity equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=', (9) with D = 0) using a finite volume upwind scheme together with the teleportation regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The gradient of the objective function J in (2) with respect to the parameter θ can be efficiently computed based on the adjoint-state method [65, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The task of learning a dynamical system from an invariant measure has also been studied in the discrete-time setting under the inverse Frobenius–Perron problem [45, 51, 54, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The Frobenius–Perron operator, also known as the transfer operator, characterizes the time-evolution of an initial measure µ0 according to some prespecified dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Given a probability measure µ, the inverse Frobenius–Perron problem seeks to construct a dynamical system for which µ is a fixed point of the associated transfer operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The most widely studied case involves recovering an ergodic map T on [0, 1] for which a prescribed absolutely continuous measure is the unique fixed point of the discrete transfer operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In this particular setting, various approaches such as topological conjugation [22] and matrix methods [50] have been introduced to solve the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The multivariate inverse Frobenius–Perron problem was also studied in [18], where ergodic maps were constructed to adhere to the statistics of two-dimensional densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, due to inherent non-uniqueness in the inverse problem, recent approaches further restrict the solution space of the discrete ergodic maps to those with a prescribed power spectrum [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To the best of our knowledge, [65] and our contributions here are the first works that numerically solve the inverse Frobenius–Perron problem in the continuous-time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, we do not assume that µ is absolutely continuous, as we use a finite-volume discretization to approximate the Frobenius–Perron operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5 3 The Forward Model and Modeling Errors A central contribution of this work is to consider a different regularized forward model than the one in [65], especially for trajectory measurements containing intrinsic noise which can be interpreted as sample paths of stochastic dynamical systems (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In those cases, the Fokker–Planck equation (7) is a better candidate as the PDE surrogate model, as it contains a diffusion term which can fit noise present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Based on the relationship between (6) and (7), one can learn both the velocity field v(x) and the diffusion tensor Σ(x) in the optimization framework (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For simplicity, we only consider a fixed diffusion constant and leave the investigation of multi-parameter inversion to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We will use (9) as the forward model to fit invariant measures generated by trajectories with intrinsic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' While, the diffusion term allows the model to fit the intrinsic noise and prevent over-fitting the noise into the target velocity component, it also controls the scaling of the reconstructed velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Indeed, when D = 0 and ˜v(x) = a v(x), we have ∇ · (ρ(x)˜v(x)) = 0 as long as ∇ · (ρ(x)v(x)) = 0, for any a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, for most cases, �v and v will not solve the stationary Fokker-Planck equation (9) for D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 2: As the mesh size of the forward model discretization is refined, we visually observe the convergence of the computed steady-state solution to (9) (top row) to the approximate physical measure obtained by binning a time trajectory based on the SDE (6) (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The Van der Pol oscillator (18) with c = 1 is used in this example, and the histograms indicate mass-per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Finite Volume Discretization We assume that our system evolves on the d-dimensional rectangular state space Ω = [a1, b1] × · · · × [ad, bd] ⊂ Rd, with a spatially dependent velocity v : Ω → Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We define ni ∈ Z+, 1 ≤ i ≤ d, to be the number of equally-spaced points along the i-th spatial dimension at which we wish to approximate the solution of (8), as well as the mesh spacing ∆xi := bi−ai ni−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We are interested in obtaining a solution to the forward problem at points of the form xk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=',kd := � a1 + k1∆x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , ad + kd∆xd � ∈ Ω, ki ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We will index our coordinates using column-major order and write xk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=',kd = xj where j = k1 + d � i=2 (ki − 1)Si , Si := i−1 � j=1 nj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (10) We will regard xj as the center of the cell Cj where Cj = d � i=1 � ai + � ki − 1 2 � ∆xi, ai + � ki + 1 2 � ∆xi � ⊂ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Following the approach in [3], we implement a first-order upwind finite volume discretization of the continuity equation, adding a diffusion term using the central difference scheme and enforcing a zero-flux boundary condition [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 6 Forward Model vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Trajectory Histogram with Diffusion = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='oo1 Forward dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Forward dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Forward dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='05 Forward dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='00200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='00175 0.' metadata={'source': 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+page_content='00000This allows us to obtain an explicit time-evolution of the probability vector ρ = [ρ1 ρ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' ρN]⊤ ∈ RN, where N = �d i=1 ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' While ρ is a discrete probability measure over the cells Cj, it also corresponds to a piecewise-constant probability density function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' With an abuse of notation, we will refer to both the piecewise-constant density and the discrete probability measure as ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We discretize the time domain with a step size ∆t chosen to satisfy the Courant–Friedrichs–Lewy (CFL) stability condition [36, Chapter 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Based on (8), the probability vector at the l-th time step evolves as ρ(l+1) = ρ(l) + Kρ(l), K = d � i=1 ∆t ∆xKi, where each Ki is a tridiagonal matrix given by Ki := .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' −vi,− j−1 + D ∆xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' vi,− j−1 − wi,+ j−1 − 2D ∆xi −vi,− j + D ∆xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' wi,+ j−1 + D ∆xi vi,− j − wi,+ j − 2D ∆xi −vi,− j+1 + D ∆xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' wi,+ j + D ∆xi vi,− j+1 − wi,+ j+1 − 2 D ∆xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' wi,+ j+1 + D ∆xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' � ��������������������������������� � ��������������������������������� Si ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Above, we have defined for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , N} the upwind velocities vi,− j := min � 0, vi j � , vi,+ j := max � 0, vi j � , wi,− j := min � 0, wi j � , wi,+ j := max � 0, wi j � , where vi j := v � xj − ei∆xi/2 � ei and wi j := v � xj + ei∆xi/2 � ei denote the i-th components of the velocity field at the center of cell faces, and {ei} is the standard basis in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that if xj is away from ∂Ω, then wi,± j = vi,± j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To enforce the zero-flux boundary condition, we set both the velocity v and diffusion D to be zero on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As a result, the columns of K each sum to zero, and the total probability ρ(l) · 1 = 1, 1 := [1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 1]⊤ ∈ RN, is conserved under time-evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since numerical artifacts cause the flux accumulation along the boundary, we also enforce ρ = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When the boundary ∂Ω is sufficiently far from the trajectory data, this artifact is insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Hereafter, we assume the uniform spatial discretization ∆xi = ∆x for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For a complete description of the finite volume scheme, we refer to [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that there are many higher-order structure-preserving schemes to solve (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' see [26] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' A more accurate numerical scheme can further reduce the forward modeling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Teleportation and Diffusion Regularization We use the finite volume discretization of the Fokker–Planck equation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 to approximate its steady-state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' After discretization, finding such stationary distributions to (9) is equivalent to solving the linear system ρ = Kρ + ρ ⇐⇒ (I + K)ρ = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since the columns of K sum to zero, we have that M := I + K is a column-stochastic Markov matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When D ̸= 0, M is a transition matrix for an ergodic Markov chain, which has a unique equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' When D = 0, to guarantee the uniqueness of the equilibrium, [65] applies the so-called teleportation regularization [20] and considers Mε := (1 − ε)M + ε U, U = N −111⊤ ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 7 There is now a unique solution to the linear system Mερ = ρ, ρ · 1 = 1, ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (11) From a computational aspect, it is useful to take advantage of the fact that M − I is sparse where I ∈ RN×N is the identity matrix, and to instead solve (1 − ε)(M − I)ρ = −N −1ε1, where we have simply rearranged terms in (11) and used the fact that ρ · 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since U is also a column stochastic Markov matrix with the uniform probability of visiting any point of the mesh, using Mε amounts to stopping the dynamics based on M at a random time and restarting it from a uniformly randomly chosen initial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The size of ε represents the restarting frequency–the smaller ε, the rarer we restart [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' On the other hand, adding the diffusion component D to the tridiagonal matrix K can be seen as another way of regularizing the noise-free Markov matrix by adding a scaled Brownian motion after each discrete evolution of the deterministic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For deterministic dynamics with D = 0, the solution to (9) might not be unique if there is more than one attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The use of teleportation connects all attractors through the “random restart”, and the solution ρε to the linear system (11) has support that connects all the disjoint attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Similarly, when D ̸= 0, the Brownian motion connects all disjoint attractors of the deterministic dynamics, giving a unique steady-state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In this scenario, the use of teleportation for the diffusive case is simply a numerical treatment to improve the conditioning of matrix M rather than to guarantee the uniqueness of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It is worth noting that both the teleportation regularization and an incorrect diffusion coefficient could be sources of modeling error when we perform parameter identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Although these regularizations enable faster evaluation of ρε(v(θ)) and better posedness of the forward problem, they may reduce the accuracy of the inverse problem solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 Numerical Diffusion In Figure 2, we illustrate the ρε computed as the steady-state solution to the Fokker–Planck equation in the top row and the approximation to physical invariant measures of the corresponding SDE in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' From Figure 2, we see that on a coarse mesh, the first-order finite volume scheme incurs a lot of numerical error, which gives a computed solution with an artificial diffusion effect and thus is often referred to as the numerical diffusion [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The amount of numerical diffusion is reduced as the mesh is refined since it is incurred by the first-order scheme, which is expected to decay as O(maxi ∆xi) in the L∞ norm as we refine the mesh [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Besides the teleportation and the modeling diffusion D, the presence of numerical diffusion is another modeling error incurred from solving the forward problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4 Gradient Calculation & Velocity Parameterization Another main contribution of this paper is to reconstruct the velocity field v(x) using large-scale parameterizations v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ), which turns an infinite-dimensional problem of searching for v(x) in a function space to a finite-dimensional optimization problem of finding θ ∈ Θ ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Here, we introduce parameterizations based on piecewise-constant, neural network, and global polynomial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We also investigate various data-fitting objective functions J that compare the mismatch between the observed and simulated invariant measures, ρ∗ and ρε(v(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We compute the gradient of such functions with respect to the coefficients θ in the parameterized velocity model v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) based on the adjoint-state method for the PDE-constrained part and the backpropagation technique [35] for the neural network part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thanks to these techniques, we can then efficiently evaluate the gradients of J with respect to θ and thus conveniently use gradient-based optimization algorithms to iteratively update θ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=', steepest descent, L-BFGS, conjugate gradient descent methods as well as stochastic methods such as Adam [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For notational simplicity, we will write ρ(v(θ)) = ρε(v(θ)) throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Gradient Calculation Through the Adjoint-State Method Recall the finite volume scheme in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 for solving (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The forward model yields a discrete measure ρ(v(θ)) = ρ(θ) = [ρ1(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' ρj(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' ρN(θ)]⊤ over the cells {Cj}, which converges to the solution to (9) in the weak sense as we refine the discretization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For the explicit form of ρ(v(θ)), we refer to [65, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Note that we have highlighted the dependence of our approximate steady-state distributional solution to the Fokker–Planck equation (9) on the velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Our goal is to solve the optimization problem (2): inf θ∈Θ J (ρ(v(θ)), ρ∗) by using gradient-based methods, where J is the cost function, and ρ∗ represents our inference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The adjoint-state method is an efficient technique by which we can evaluate the derivative ∂θJ , as the computation time is largely 8 independent of the size of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' One can derive the adjoint-state method for gradient computations by differentiating the discrete constraint [52], which in our case is the eigenvector problem g(ρ(θ), θ) = Mε(θ)ρ(θ) − ρ(θ) = 0, where ρ(θ) · 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, we will compute ∂θJ = λ⊤∂θg where λ solves (∂ρg)⊤ λ = − (∂ρJ )⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In our case, this linear system is the adjoint equation [65, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='8)] (M ⊤ ε − I)λ = − (∂ρJ )⊤ + (∂ρJ )⊤ ρ 1, (12) and the derivative ∂θJ = λ⊤� ∂θMε � ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (13) As a result, we only need to compute the derivatives ∂ρJ and ∂θMε to determine the gradient ∇θJ = (∂θJ )⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The former depends on the choice of the objective function, while the latter is based on a specific parameterization of the velocity field v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) determined by its hypothesis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 The Computation of ∂ρJ For the objective function J , we consider the quadratic Wasserstein distance, the squared L2 norm, the Kullback–Leibler (KL) Divergence, and the Jensen–Shannon (JS) Divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Quadratic Wasserstein Distance: For probability measures ρ and ρ∗ on Ω, with finite second-order moments, the squared quadratic Wasserstein distance is defined by W 2 2 (ρ, ρ∗) := inf Tρ,ρ∗∈T � Ω |x − Tρ,ρ∗(x)|2dρ(x), where T := {T : Ω → Ω : ρ(T −1(B)) = ρ∗(B), for all measurable B} is the set of maps that push ρ forward into ρ∗ [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' With an abuse of notation, we also use ρ(x) and ρ∗(x) to denote the densities of ρ and ρ∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For efficient computation of the W2 distance, we utilize the back-and-forth method [28], which instead uses the dual Kantorovich formulation [61] W 2 2 (ρ, ρ∗) = sup φ,ψ �� Ω φ(x)ρ∗(x)dx + � Ω ψ(x)ρ(x)dx � , where φ ∈ L1 ρ∗(Ω) and ψ ∈ L1 ρ(Ω) are required to satisfy φ(x) + ψ(y) ≤ |x − y|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In this case, the Fr´echet derivative of J = W 2 2 (ρ, ρ∗) with respect to ρ is given by ∂J ∂ρ = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Squared L2 Norm: The squared L2 distance as the objective function and its Fr´echet derivative are given by J = 1 2 � Ω |ρ(x) − ρ∗(x)|2dx, ∂J ∂ρ = ρ − ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' KL-Divergence: The KL-divergence and its Fr´echet derivative are given by J = DKL(ρ, ρ∗) := � Ω ρ∗(x) log �ρ∗(x) ρ(x) � dx, ∂DKL ∂ρ = −ρ∗(x) ρ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that our definition of the KL-divergence differs from many applications in which it is commonly computed as J = DKL(ρ∗, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' JS-Divergence: Defining ρ′ := (ρ + ρ∗)/2, the JS-divergence and its Fr´echet derivative are given by J = DJS(ρ, ρ∗) = 1 2DKL(ρ, ρ′) + 1 2DKL(ρ∗, ρ′), ∂DJS ∂ρ = 1 2 log � 2ρ ρ + ρ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Based on definitions of the KL and JS divergence, it is clear that we may encounter numerical instability issues if either ρ or ρ∗ is not supported on the entire domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thus, we remark that for the computation of both the KL and JS divergences, we restrict the domain Ω to regions where both ρ and ρ∗ are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This is equivalent to the definition of the KL and JS divergence based upon the so-called Csiszar divergence [57, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 The Computation of ∂θJ We have presented a few cases of ∂ρJ for different choices of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Next, we show how to obtain ∂θMε, which is the other necessary component in the adjoint-state method for gradient calculation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' see (12)-(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To begin with, we consider θ = {vi j} for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , d and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , N, which corresponds to one variant of piecewise-constant velocity parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since we are only interested in computing the gradient away from ∂Ω, we can utilize the property that wi,± j = vi,± j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' First, observe that ∂Mε ∂vi j = (1 − ε) d � ℓ=1 ∆t ∆x ∂Kℓ ∂vi j = (1 − ε) ∆t ∆x ∂Ki ∂vi j , as well as that ∂Ki ∂vi j = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' −H(vi,+ j ) −(1 − H(vi,− j )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' H(vi,+ j ) (1 − H(vi,− j ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' � ������������������������ � ������������������������ Si H(x) := �1, x > 0 0, x ≤ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Above, H(·) is the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that ∂vi jKi can only be nonzero in the (j, j), (j, j − Si), (j − Si, j), and (j − Si, j − Si)-th entries where Si is defined in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' After solving (12) for λ and applying (13), we deduce that ∂J ∂vi j = λ · ∂Mε ∂vi j ρ = (1 − ε) ∆t ∆x � λ · ∂Ki ∂vi j ρ � = (1 − ε) ∆t ∆x � H(vi,+ j )ρj−Siλj + (1 − H(vi,− j ))ρjλj − H(vi,+ j )ρj−Siλj−Si − (1 − H(vi,− j ))ρjλj−Si � = (1 − ε) ∆t ∆x (λj − λj−Si) � H(vi,+ j )ρj−Si + (1 − H(vi,− j ))ρj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (14) Equation (14) provides an efficient way for computing the gradient of the objective function with respect to the piecewise-constant velocity based on cells {Cj} from our finite-volume discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Alternatively, if the velocity v = v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) is smoothly parameterized by the vector θ = [θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , θk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , θm]⊤ ∈ Rm, for each θk, we can then evaluate ∂J ∂θk = N � j=1 d � i=1 ∂J ∂vi j ∂vi j ∂θk , ∂vi j ∂θk = ei · ∂v ∂θk ���� (xj−ei∆xi/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='θ) , (15) to determine the derivative ∂θJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' By using a similar indexing convention to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1, we can collect the terms ∂vi jJ and ∂θkvi j into the vectors ∂vJ and ∂θkv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Therefore, the double summation in (15) is achieved by the inner-product ∂vJ · ∂θkv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Note that for different θk, we only need to change ∂θkv as ∂vJ does not depend on θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Velocity Parameterization In this subsection, we apply Equations (14) and (15) to evaluate the gradients of several parameterized velocity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, we consider piecewise constant, global polynomial, and neural network parameterizations of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Piecewise-Constant Parameterization In the case of the piecewise-constant parameterization, we model the velocity as v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) = d � i=1 N � i=j vi j χCj(x) ei, θ = {vi j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (16) Here, we again use the column-major ordering from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 to accumulate vectors of cells Cj with centers xj, and velocity components, vi j = v(xj − ei∆xi/2) · ei, along the i-th direction of the cell face located at xj − ei∆x/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The parameter space of the model presented in (16) is given by {vi j}, which has size N · d, and the gradient of the parameters {vi j} can be directly evaluated by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that (16) is only one variant of piecewise-constant parameterization since the parameterization mesh is the same as the discretization mesh in the finite-volume method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' These two meshes do not have to be coupled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To reduce the numerical error from the first-order scheme, it is preferable to reduce the spacing {∆xi}, but we can keep the parameterization mesh fixed, so the size of the optimization problem does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In this case, we need to apply the chain rule (15) to obtain the final gradient after evaluating (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The model defined by (16) can be learned by gradient-based optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The regularity of the piecewise- constant model defined by (16) can be improved to a C0 function by interpolating between the values vi j using either piecewise linear or higher-order piecewise polynomial functions, as in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Global Polynomial Parameterization Though the regularity of the piecewise-constant model given by (16) can be improved by interpolation, the inverted velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) may still be highly oscillatory if the mesh size ∆x is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Modeling approaches, such as SINDy [6], learn the velocity fields of dynamical systems from a polynomial basis together with sparse regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Here, we show how the gradient derivation in (15) can be adapted to such polynomial basis parameterization of the velocity field v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) = [v1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , vd(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ)]⊤ = d � i=1 vi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The i-th component of the velocity field vi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) parameterized by a linear combination of the monomial basis of degree at most K can be written as vi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) = M � ℓ=1 ai ℓ(x⊤e1) 1ki ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (x⊤ed) dki ℓ, M = �d + K K � , θ = {ai ℓ}, (17) where the powers are represented by multi-indices ki ℓ = (1ki ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , dki ℓ), 1 ≤ ℓ ≤ M, and |ki ℓ| ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The size of θ in this case is d · M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To learn the model parameterized by (17), we can use (15) to compute the gradient ∂ai ℓJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Without loss of generality, we assume ∆xi = ∆x, for all 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The only term in (15) which explicitly depends upon the velocity paramaterization is ∂vi j ∂ai ℓ = � (xj − ei∆x/2)⊤e1 �1ki ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' � (xj − ei∆x/2)⊤ed �dki ℓ , 1 ≤ j ≤ N, where i, ai ℓ and the multi-index ki ℓ are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Note that ∂ai ℓvi′ j = 0 if i′ ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thus, we can again use gradient-based methods to infer proper polynomial coefficients {ai ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Although a global polynomial parameterization guarantees ideal C∞ regularity of the parameterized velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ), the Runge phenomenon could be a potential downside of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, as we increase the maximum degree K of the polynomial basis, we may encounter substantial interpolation errors near the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 Neural Network Parameterization Motivated by the universal approximation theory of neural networks [25], we may also choose to model each component of the velocity vi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) as a feed-forward neural network, where the tunable parameters θ make up the network’s weights 11 Figure 3: In two dimensions, both velocity components v1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ1) and v2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ2) are parameterized by fully connected neural networks with smooth activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Due to the upwind finite volume method, the adjoint-state gradient calculation for v1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ1) is performed on the mesh {xj − e1∆x/2}, whereas the gradient calculation for v2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ2) is performed on the mesh {xj − e2∆x/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Both gradients are used in conjunction with backproppogation to train the networks, which ultimately yields a smooth map x �→ (v1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ1), v2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We follow [37] to combine the adjoint-state method for the PDE constraints and the backpropagation technique to update the weights and biases of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The term ∂vi jJ in the gradient calculation (15) can be computed by first evaluating the neural network on the mesh of cell face centers oriented in the direction of ei to obtain {vi j}, which is then plugged into (14) to obtain ∂vi jJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 3 illustrates where the neural network is evaluated for gradient computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The remaining term ∂θv in (15) is then computed via the backpropagation technique [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For simplicity, we restrict ourselves to single-layer feed-forward networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, by using a smooth activation function, such as the hyperbolic tangent or the sigmoid function, we can guarantee C∞ regularity of the reconstructed velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) on the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' To enforce the zero-flux boundary condition, we manually set v = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Consequently, the neural network parameterization may lack regularity near ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, if the domain is sufficiently large, the support of the physical measure will be very far from ∂Ω, in which case we will not observe any discontinuities originating from the boundary condition while simulating the trajectories based on (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As we increase the number of nodes in the hidden layer of the neural network, both the approximation power and the potential difficulty of training the neural network are expected to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5 Numerical Results In this section, we present several numerical examples to demonstrate the utility of the proposed approach for learning dynamical systems from invariant measures with intrinsic noise1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1, we study the inverse problem for the Van der Pol oscillator with a neural network parameterization of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2, we time-delay embed a signal sampled from a Hall-effect thruster and proceed to model the dynamics in delay-coordinates based upon the time-delayed invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We then illustrate that a very low-dimensional embedding will increase the uncertainty of the learned model and that the choice of parameterization largely affects the regularity of the reconstructed velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3, we study rolling averages of weekly temperature data and perform uncertainty quantification using the learned Fokker–Planck PDE in time-delayed coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 by inverting a component of the Lorenz-63 system’s velocity using a neural network parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Van der Pol Oscillator We begin by considering the autonomous Van der Pol oscillator [23], given by � ˙x = y, ˙y = c(1 − x2)y − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' (18) Our results for learning a dynamical system with prescribed statistical properties given by the stochastically-forced Van der Pol oscillator are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In the top row, the first panel features the velocity of (18) for the choice of c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='5, the second panel shows the approximate invariant measure from a diffuse trajectory simulated via the 1For those interested, we also include an example Google Colaboratory file demonstrating the velocity inversion for a polynomial parameterization: https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='com/PDEinv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 12 j-ei△/2] v1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 01) (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 02) [αj -e2△α/2] > (*(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 01), v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 02)) < (Euler–Maruyama method, the third panel shows the diffuse dynamics which were used to approximate the invariant measure, and the fourth panel shows the dynamics of the oscillator without stochastic forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Throughout, we color trajectories by their histogrammed density to illustrate the connection between the Lagrangian and Eulerian perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 4: Learning velocity fields to reproduce the statistics of the stochastically-forced Van der Pol oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The first row features the ground truth velocity, mass-per cell, and dynamics of the Van der Pol oscillator (18) with c = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='5 and diffusion D = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The subsequent rows show the resulting optimization based on the observed density using each of the four objective functions studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 along with a neural network paramaterization utilizing a single hidden layer of 100 nodes and a hyperbolic tangent activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' During the optimization, each objective function was reduced to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='05% of its initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We emphasize that the density in the first row is approximated via the procedure in Figure 1, whereas the densities in the remaining rows are solutions of the PDE forward model from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In the following rows of Figure 4, we use a neural network parameterization to solve the inverse problem using the optimization framework from Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In the first column of Figure 4, we see that each objective function in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 is reduced below .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='05% of its initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, the reconstructed velocity for each objective function is shown to vary significantly from the true velocity shown in the first row of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This is largely due to the lack of data away from the main attracting limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In regions of the state space with no available data, we can only expect that the modeled velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) will direct trajectories towards the attracting limit cycle on which the invariant measure is supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Indeed, this is what we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, both the PDE forward model (9) and the statistics of the SDE simulation using the learned velocity match the ground truth invariant measure from the first row of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' These observations indicate that the learned velocity need not agree with the ground truth velocity to yield a dynamical system with the same invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, in the last column of Figure 4, we approximate the noise-free limit cycle by removing the diffusion term in (6), which corresponds to the simulation of an ODE rather than an SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 13 Mass per Cell Velocity Diffuse DynamicsOptimization1L3KLIterationsISIW2To reduce the computational requirements of the inversion in the final row of Figure 4, we compute J = W 2 2 on a coarsened mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Among the four objective functions in Figure 4, it is worth noting that the W2 metric does not compare the two densities pointwisely and is well-defined for comparing singular measures, which is different from the other three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The distance reflects both the local intensity differences and the global geometry mismatches [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It has also been shown that the Wasserstein metric is robust to noise [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thanks to the geometric nature of the optimal transportation problem, the Wasserstein metric is primarily sensitive to global changes such as translation and dilation and is robust to small local perturbations such as noisy measurements of ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The better stability also brings a downside as the optimization landscape can be relatively flat around the ground truth, which may lead to compromised accuracy in the velocity reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Hall-Effect Thruster We now turn to the more realistic setting of experimentally sampled time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, we study the Cathode–Pearson signal sampled from a Hall-effect thruster (HET) in its breathing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Hall-effect thrusters are in-space propulsion devices that exhibit dynamics resembling stable limit cycles while in breathing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For details about the experimental setup used to collect the data, the reader is encouraged to consult [13, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1, we utilize Takens’ theorem [59] to reformulate the large-scale optimization framework presented in Sections 3 and 4 to be compatible with scalar time-series observations, and in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 we demonstrate numerical results based upon this reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 Methods Intrinsic physical fluctuations present in the Cathode–Pearson signal indicate that the HET’s dynamics may be modeled well by a Fokker–Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Motivated by this insight, we first time-delay embed the Cathode–Pearson signal C(t) in d-dimensions to form the trajectory Cd,τ(t) := (C(t), C(t − τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' , C(t − (d − 1)τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We then use the procedure outlined in Figure 1 to compute the occupation measure ρ∗ of Cd,τ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' By viewing each dimension of the coordinate system on which the measure ρ∗ is supported as the independent variables C−kτ(t) := C(t − kτ) where 0 ≤ k ≤ d − 1, we then seek a solution to the optimization problem (2) for a velocity v = v(Cd,τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Such a velocity can then provide us with a model of the asymptotic statistics of the embedded trajectory Cd,τ(t), provided that a suitable diffusion coefficient can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We note that forming the time-delay coordinates Cd,τ(t) does require a knowledge of measurements at uniform increments in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, the available data may still be sampled slowly enough such that it is impractical to seek a direct approximation of the Lagrangian velocity through the standard approaches described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This perspective motivates our use of the approach developed in Sections 3 and 4 to learn dynamical systems from invariant measures in time-delay coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, there are a few additional considerations which arise when adapting the modeling framework presented in Sections 3 and 4 to real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Namely, we do not know the proper diffusion coefficient a priori (as was the case in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, the invariant measure that the model is based on does not contain any information about the time-scale at which the system evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Towards this, we utilize the following three-step procedure as a computationally efficient means to mitigating these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Bin the trajectory Cd,τ(t) onto a d-dimensional mesh with spacing ∆x along each axis to form the occupation measure ρ∗, assume a constant diffusion coefficient D > 0, and learn the velocity v = v(Cd,τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ), using the framework from Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Bin the trajectory Cd,τ(t) onto another d-dimensional mesh with spacing ∆ˆx ≤ ∆x to create a new occupation measure ˆρ∗ and adjust the diffusion coefficient by solving the optimization problem ˜D = arg min ˆ D∈R J (ρε(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' ˆD), ˆρ∗), (19) where the term ρε(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' ˆD) in (19) denotes the forward model evaluation with the diffusion coefficient ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Rescale both the velocity and diffusion by solving the optimization problem ˜a = arg min a∈R N � i=1 ��� ˆC(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' a) − Cd,τ(ti) ��� 2 2 , (20) where ˆC(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' a) denotes the time-ti solution of the ODE initial value problem with velocity av(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) and initial condition Cd,τ(t0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The final velocity and diffusion are then given by ˜av(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) and ˜a ˜D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 14 The three-step approach makes repeated use of the fact that ρε(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' D) = ρε(av;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' aD), for any scalar multiple a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Indeed, if the true diffusion coefficient D∗ > 0 is unknown a priori, but we instead seek a solution v(·, θ) with a different diffusion D > 0, it is guaranteed that the velocity v = (D/D∗)v∗ will still provide a solution to the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This observation motivates step one, in which an arbitrary diffusion coefficient is used to find a solution v(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) to the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As the dimensionality d is increased, solving the large scale optimization problem in step 1 on a fine mesh becomes infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As such, step one is typically performed on a coarse mesh where additional Gaussian filtering is applied to the inference measure ρ∗ to make the large-scale optimization more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The diffusion coefficient is then adjusted in step two on a finer mesh via (19) to mitigate the errors due to the Gaussian filtering, numerical diffusion, and histogram errors incurred during step one (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, in step three the scale of both the velocity and diffusion are adjusted via (20) such that the time evolution of simulated trajectories is consistent with the inference trajectory Cd,τ(t) in delay coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Since diffusion plays a relatively small role over short time-scales for the quasi-periodic HET data, we use the trajectory to calibrate a reasonable time-scaling between our model and the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' However, as the magnitude of the diffusion increases, the least squares fit in (20) will become less reliable and it may be preferable to instead minimize a transport cost between a collection of model samples and a collection of data samples at each time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Results The results of the three-step procedure in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 for learning the HET dynamics are shown in Figure 5 for an embedding dimension of d = 3 and time-delay of τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 · 10−5 seconds, or rather τ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='23 when normalizing the time-scale to the HET breathing mode frequency (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='6kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The modeled trajectory accurately reconstructs the shape of the embedded Cathode–Pearson signal but cannot capture the variable diffusion present throughout the time-delayed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We do not expect to capture such details, as we assume a constant diffusion coefficient in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Nevertheless, we regard the reconstruction of the 3D globally attracting limit cycle as a success and leave extending the model to account for the case of a non-constant diffusion tensor to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 5: Learning the velocity from embedded Cathode-Pearson signal’s invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We present the time- delay embedded signal C3,τ(t) (left), the reconstructed velocity field from the embedded signal’s occupation measure (middle), where blue indicates slow speed and red indicates fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, we show a trajectory simulated with the Euler–Maruyama method from the learned velocity and diffusion (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The velocity was parameterized by a neural network with 500 nodes in a single hidden layer and learned using the KL divergence loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The three-step procedure in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 is used to learn the model, and in step one, additional Gaussian filtering is applied to the occupation measure ρ∗ to simplify the resulting optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The dimensionality of the original HET dynamics is unknown, and as such, a sufficient embedding dimension for the Cathode–Pearson signal is unclear, though likely very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Interestingly, we can compare the model learned in Figure 5 with a 2D analog to demonstrate that when the number of time-delays is not sufficiently large, there is more uncertainty in modeling the time-delayed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This phenomenon is most evident when inspecting regions of the delayed Cathode–Pearson signal for which the 2D embedding lacks structure readily observed in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, consider a collection of nearby samples {C3,τ(ti)}n i=1 in the 3D time-delay coordinate system (C0, C−τ, C−2τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The corresponding 2D samples {C2,τ(ti)}n i=1 will also be nearby one another in the 2D time- delay coordinate system (C0, C−τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Figure 6, we initiate uniform distributions centered about these samples in both 2D and 3D time-delay coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We then evolve both the samples and initial uniform distributions forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The evolution of the ground truth samples is simply determined by the time-delayed Cathode Pearson signal Cd,τ(t), and the evolution of the uniform distributions is given by Fokker–Planck models constructed from the time-delayed Cathode Pearson signal’s invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='6time = 0 time = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='17 time = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='33 time = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='5 time (1/HET Breathing Mode Frequency) Figure 6: Comparing the model accuracy and uncertainty for the embedded Cathode–Pearson signal with 2D and 3D time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The time-evolution of the models is compared to a collection {Cd,τ(ti)}n i=1 of samples (plotted in black) from the time-delayed Cathode–Pearson signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The first two rows feature four snapshots from the 2D and 3D models from the time interval [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The bottom row compares the uncertainty of the models to the ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The time units are normalized to the inverse of a HET breathing mode frequency (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='6kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Both models utilized a neural network velocity parameterization with 500 nodes in a single hidden layer and reduced the KL divergence objective function to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1% of its initial value during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As in Figure 5, the three-step procedure in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 is used to learn the models, and in step one, additional Gaussian filtering is applied to the occupation measure ρ∗ to simplify the resulting optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The 3D visualization was plotted using [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' samples evolve in time, we observe in Figure 6 that the mean of the 3D model matches the true sample mean more closely than the 2D model, and that it has less uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Figure 7, we study the three parameterizations from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 for learning the time-delayed Cathode–Pearson signal’s velocity, now with an embedding dimension of two to allow for clearer visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It can be seen that the density associated with each velocity parameterization indeed matches the ground truth density in Figure 7, but that the velocity fields differ significantly from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The piecewise-constant velocity in Figure 7 suffers from poor regularity with discontinuities on the attracting limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As a result, we lose the connection between the Eulerian 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='6 2D model mean 3D model mean True sample mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 2D model one-o Interval 3D model one- Interval 王 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 True one-o Interval0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='000Figure 7: Comparison between the three parameterizations detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 for learning a velocity field from the time-delayed Cathode–Pearson signal’s invariant measure, using a diffusion coefficient D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The learned velocities and densities for the piecewise constant (PC), global polynomial (GP), and neural network (NN) discretizations are shown in the three columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We show the velocity field on the full state space (top row), a close-up of the velocity field’s direction near the attracting limit cycle (middle row), and the forward model output ρε(v(θ)) for each parameterization (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The resulting parameter spaces of these discretizations have dimensionality of 9800 (PC), 56 (GP), and 400 (NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The L2 loss is reduced below .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1% of its initial cost for the PC and NN discretizations and reduced below .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='7% of its initial value for the GP case when we stopped the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' and Lagrangian dynamics and cannot reconstruct zero-diffusion trajectories which form a stable limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' On the other hand, the velocities parameterized by the global polynomial and the neural network are both C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The differences among these three can clearly be seen via the zoomed-in velocity plots in the second row of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The global polynomial and neural network discretizations are both global parameterizations of the velocity, and as such, their values near the domain’s boundary are dictated by the available data in the center of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This causes the polynomial velocity to rapidly increase near the boundary, and a similar effect can also be seen for the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It is worth noting that the initial condition for the optimization in Figure 7 can play a large role in the reconstructed velocity, which is related to the optimization landscape of the nonconvex optimization problem (2) we tackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In the case of the piecewise-constant discretization, we initialize all velocities to be significantly less than the diffusion coefficient D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thus, diffusion initially dominates in the finite volume solver, and all non-boundary cells will contain nonzero mass, which allows for accurate gradient updates everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This phenomenon can also help neural network training, though it is not always necessary due to the global nature of parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, we initialize our polynomial basis to form the velocity ( ˙x, ˙y) = (−y + x(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 − x2 − y2), x + y(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 − x2 − y2)), which describes a globally attracting limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thus, to converge to the ground truth limit cycle of the time-delayed Cathode–Pearson signal, this initial velocity only needs to be translated and deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 17 Piecewise Constant Velocity Global Polynomial Velocity NeuralNetworkVelocity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 Co Co Co5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='3 Temperature Uncertainty Quantification We now study 2D time-delay embedded data of weekly rolling averages of the temperature in Ithaca, NY, between 2006 and 2020 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We view temperature fluctuations over short time scales as an intrinsic diffusion process and the approximately periodic oscillation of seasonal temperatures driven by some nonzero velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Thus, we model the 2D data in delay coordinates as a diffuse limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We again follow the procedure in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1 to learn a velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) and diffusion coefficient D, which closely matches the occupation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 8: Performing prediction and uncertainty quantification for Ithaca, NY’s temperature in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The top left plot shows the ground truth occupation measure accumulated from 13 years of weekly rolling averaged temperature observations, normalized by an affine transformation to [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The top middle and right plots show the learned model and velocity vector field, obtained using the three-step procedure in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In the bottom plot, the PDE model with a uniform initialization in the box from the top middle plot is evolved in time and used to quantify the uncertainty in the measurements of C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Observed trajectories of the temperature in delay coordinates with initial conditions displayed in the top left plot are also shown to demonstrate the effectiveness of the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' A time delay of τ = 280 days is used, and the model is trained using a neural network parameterization and the KL-divergence objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 we can use the trained model v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) to quantify measurement uncertainties through the Fokker– Planck equation (9), whose solution is a probability density in the time-delay coordinates (C0, C−τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, if we know some initial probability distribution that captures the current state of the temperature system well, we can consider the time-evolution of the distribution using our trained model to quantify the uncertainty of future temperature measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The process of evolving both the Fokker–Planck PDE from a uniform distribution and the ground truth sample paths from past temperature measurements is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The uncertainty bounds from the model accurately capture fluctuations in the training data used to form the occupation measure (plotted in black), as well as a testing sample path previously unseen by the model (plotted in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' It is also worth noting that the confidence intervals we construct may be larger than the actual range due to several factors, including additional extrinsic noise from filtering the data, modeling errors accumulated from the hypothesis space, numerical diffusion in the forward model, and a sub-optimal embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Reducing such errors may result in tighter confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Considering time-delays in higher dimensions could yield better predictions of the temperature’s transient behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 18 Occupation Measure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 CoTemperature Forecast 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 Two-o interval One-o interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='2 Mean Training Testing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 0 50 100 150 200 250 300 350 Days5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='4 Lorenz-63 System We conclude this section by studying the Lorenz-63 system [39], defined by � � � ˙x = c1(y − x) ˙y = x(c2 − z) − y ˙z = xy − c3z , (21) where we consider (c1, c2, c3) = (10, 28, 8/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For these choices of parameters, the Lorenz-63 system exhibits chaotic behavior and admits a unique physical measure [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Figure 9, we assume that the quantities ˙y and ˙z are known, and we learn a model for the velocity in the x-direction, using the stochastically-forced Lorenz-63 system’s occupation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We emphasize that the samples used to form the occupation measure are not correlated in time and were, in fact, randomly sampled from a trajectory as in [65, Figure 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This demonstrates the applicability of the proposed approach for learning the dynamics of physical processes for which measurement times are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' From the occupation measure of these samples, we invert the first component ˙x of the Lorenz-63 system’s velocity via a neural network parameterization and successfully simulate dynamical trajectories which recover the original Lorenz attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We remark that when ˙x, ˙y, and ˙z are all simultaneously inverted, the optimization is unsuccessful at reconstructing the true velocity (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' While we may be able to learn a velocity that approximately recovers the stationary state of the Lorenz-63 system in the sense of (9), the corresponding dynamical trajectories do not form a chaotic attractor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=', the physical property (3) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Whether the difficulties of inverting all velocity components of the Lorenz-63 system are due to inherent non-uniqueness in the inverse problem or simply inconvenient local minima during training is worth further investigation in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Figure 9: Neural network parameterization of ˙x using the Lorenz system’s stochastically perturbed invariant measure with D = 10 and ∆x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The top row displays the learned velocity and a single trajectory plotted both with and without diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The bottom row displays the true velocity and true trajectories both with and without diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' For visualization of the occupation measure used to learn the model displayed in the top row, we refer to [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 6 Conclusion In this paper, we performed large-scale parameter identification to model the velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) of intrinsically noisy autonomous dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We first adapted the invariant measure surrogate model ρε(v(θ)) in [65] based upon the continuity equation to the Fokker–Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This increased our modeling capacity and prevented overfitting the reconstructed velocity v(θ) while modeling intrinsically noisy trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We next extended the three-coefficient learning performed in [65] to thousands of coefficients by modeling the velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) via global polynomials, piecewise polynomials, and fully connected neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The efficient gradient computation presented in Section 4 made these large-scale parameterizations of the velocity computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We finally studied velocity inversion for invariant measures of time-delay embedded observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The method of time-delay embedding is useful for analyzing real-world data, where in many cases, only limited observations of complex systems are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' As such, we proceeded to learn the velocity in time-delay coordinates for a Hall-effect thruster system and weekly temperature measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Both systems exhibit periodic behavior with intrinsic noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' which led us to model the time-delayed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='Zero-Diffusion Trajectory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='30 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='30Lorenz-63 Samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='50 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='Zdynamics as globally attracting limit cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Using these models, we predicted future states of the systems and quantified uncertainty in forecasts by evolving the Fokker–Planck equation forward in time, similar to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In future work, there are several computational directions to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The first-order upwind finite volume discretization incurs numerical diffusion for large values of ∆x and is costly to implement as a forward model for small values of ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Higher-order methods would be both more accurate and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Moreover, most globally attracting systems admit invariant measures which are singular with respect to the Lebesgue measure, and consequently, most of the cells in the finite-volume discretization are unused, even if intrinsic noise is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Therefore, it would be more efficient to solve the Fokker–Planck equation on an unstructured mesh that adapts to the density of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Given the complex interactions of a large number of variables in real-world systems, the low-dimensional embedding presented in Section 5 may not be sufficient to resolve the full dynamics of the HET and temperature systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In many cases, time-delay embeddings of dimensionality greater than three are required, and as such, it would be useful to adapt the proposed approach to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The current framework suffers from the curse of dimensionality, so further investigation into mesh-free approaches for approximating invariant measures and solving the resulting PDE-constrained optimization (2) would be fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, the forward model ρε(v(θ)) should be adapted so that it can describe the invariant measures of multiple attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' On the theoretical side, numerous questions about the well-posedness of the inverse problem remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Perhaps the most important ones are the following questions, which are directly related to the uniqueness of the reconstructed velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Given a physical measure µ of the autonomous dynamics ˙x = v(x), when does there exist another dynamical process ˙y = w(y) which admits the same physical measure µ but satisfies v(x) ̸= w(x) on supp(µ)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Does the presence of intrinsic noise affect the answer to this question?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' If the velocity v(x) is not unique on supp(µ), is there a restriction to a reasonable function space where uniqueness can be obtained?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' A deeper investigation into the stability of the inverse problem is also needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Specifically, it is desirable to understand how perturbations to a velocity v(θ) affect the associated physical measure and our surrogate approximation via the upwind finite volume method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Finally, if the uniqueness of the velocity can be guaranteed on supp(µ), then the stability of the reconstructed velocity v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' θ) with respect to perturbations of the inference data µ would be the next interesting question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Acknowledgements J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Botvinick-Greenhouse was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Martin was partially supported by AFOSR Grants FA9550- 20RQCOR098 (PO: Leve) and FA9550-20RQCOR100 (PO: Fahroo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This work was done in part while Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Yang was visiting the Simons Institute for the Theory of Computing in Fall 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Yang acknowledges support from Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Max R¨ossler, the Walter Haefner Foundation and the ETH Z¨urich Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' This material is based upon work supported by the National Science Foundation under Award Number DMS-1913129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Chen Li for his helpful suggestions and generosity in sharing code for the approach of Section 4.' 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+page_content=' Efficient grid-based bayesian estimation of nonlinear low-dimensional systems with sparse non-gaussian pdfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Automatica, 48(7):1286–1290, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [4] Alex Blumenthal and Lai-Sang Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Equivalence of physical and SRB measures in random dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Nonlinearity, 32(4):1494–1524, mar 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [5] Steven L.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Springer, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [60] Warwick Tucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' The Lorenz attractor exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Comptes Rendus de l’Acad´emie des Sciences - Series I - Mathematics, 328(12):1197–1202, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [61] C´edric Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Topics in optimal transportation, volume 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [62] Sebastian Wallot and Dan Mønster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Calculation of average mutual information (AMI) and false-nearest neighbors (FNN) for the estimation of embedding parameters of multidimensional time series in Matlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Frontiers in psychology, 9:1679, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [63] Hua Wei, Chacha Chen, Chang Liu, Guanjie Zheng, and Zhenhui Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Learning to simulate on sparse trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 530–545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [64] Nijun Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Solutions of the inverse Frobenius–Perron problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Master’s thesis, Concordia University, July 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Unpublished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [65] Yunan Yang, Levon Nurbekyan, Elisa Negrini, Robert Martin, and Mirjeta Pasha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Optimal transport for parameter identification of chaotic dynamics via invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content='15138, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' [66] Lai-Sang Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' What are SRB measures, and which dynamical systems have them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' Journal of statistical physics, 108(5):733–754, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf'} diff --git a/fNE3T4oBgHgl3EQffQos/content/tmp_files/2301.04550v1.pdf.txt b/fNE3T4oBgHgl3EQffQos/content/tmp_files/2301.04550v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96ed482acbdc9e2b2c10287aba0a599a3df09b65 --- /dev/null +++ b/fNE3T4oBgHgl3EQffQos/content/tmp_files/2301.04550v1.pdf.txt @@ -0,0 +1,2410 @@ +Chiral Perturbation Theory +Reflections on Effective Theories of the Standard Model +B. Ananthanarayan,1, ∗ M. S. A. Alam Khan,1, † and Daniel Wyler2, ‡ +1Centre For High Energy Physics, Indian Institute of Science +Bangalore 560 012, India. +2Institute for Theoretical Physics University of Z¨urich +Winterthurerstr. 190, CH 8057 Z¨urich, Switzerland. +1 +arXiv:2301.04550v1 [hep-ph] 11 Jan 2023 + +Abstract +The pseudoscalar particles pions, kaons and the η-particle are considerably lighter than the +other hadrons such as protons or neutrons. Their lightness was understood as a consequence of +approximate chiral symmetry breaking. This led to current algebra, a way to express the relations +imposed by the symmetry breaking. It was realized by Weinberg that because of their low mass, it is +possible to formulate a purely pionic (effective) field theory at experimental energies, which carries +all information on the (non-perturbative) dynamics, symmetries, and their spontaneous breaking +of quantum chromodynamics (QCD) and allows for systematic calculations of observables. +In +this review, we trace these developments and present recent activities in this field. We make the +connection to other effective theories, more generally introduced by Wilson, as approximate field +theories at low energies. Indeed, principles and paradigms introduced first for pions have become +ubiquitous in particle physics and the standard model. Lastly, we turn to the latest development +where the present (fundamental) standard model itself is considered as an effective field theory of +a - yet to be formulated - even more fundamental theory. We also discuss important techniques +that were developed in order to turn chiral perturbation theory into a predictive framework and +briefly review some connections between lattice QCD and chiral perturbation theory (ChPT). +CONTENTS +Preamble +4 +A personal note +5 +I. Introduction +5 +II. The Chiral Lagrangian +8 +III. Extensions of chiral perturbation theory +13 +A. The η′ +14 +B. Vector mesons +14 +∗ anant@iisc.ac.in +† mohdakbar@iisc.ac.in +‡ wyler@physik.uzh.ch +2 + +C. Baryons +15 +IV. Two and Three Body Rescattering +15 +V. Generalized renormalization group and large chiral logarithms +20 +VI. Weak interactions of pseudoscalar mesons +22 +VII. Selected applications of Chiral Perturbative Theory. +25 +VIII. Other Effective Theories for the strong interactions +27 +A. Extended Effective weak Theory +28 +B. Heavy Quark Effective Theory +30 +C. NRQCD and pNRQCD +31 +D. Heavy-light mesons +33 +E. Soft Collinear Effective Theory, SCET +35 +IX. Effective theories beyond the standard model +36 +A. The Standard Model Effective Theory +36 +B. Quantum Gravity +37 +X. Miscellaneous items +38 +Feynman Integral Methods for Effective Field Theories +38 +Chiral Lagrangians and Ricci Flows +40 +Lattice QCD +41 +Outlook +42 +Acknowledgment +42 +References +43 +3 + +“And so the question naturally arose, is there a way of avoiding the machinery of +current algebra by just writing down a field theory that would automatically produce the +same results with much greater ease and perhaps physical clarity? Because after all in +using current algebra one had to always wave one’s hands and make assumptions about +the smoothness of matrix elements, whereas if you could get these results from Feynman +diagrams, you could see what the singularity structure of the matrix elements was and +make only those smoothness assumptions that were consistent with that.” +Steven Weinberg, 2020[1] +PREAMBLE +We dedicate this article to Steven Weinberg, a great physicist who influenced in many +ways the field of elementary particle physics for decades and is one of the authors of the +Standard Model. Apart from his great achievements in research, his textbooks which became +classics also testify to his outstanding teaching capacity. He was awarded the 1979 Nobel +Prize in physics along with Sheldon Glashow and Abdus Salam for the electro-weak model. +In addition, Weinberg pioneered the study of quantum field theories, behavior of Green +functions at asymptotic energies, symmetries in field theories, Goldstone mechanism, current +algebra, pion physics and effective field theories. Weinberg has explained his philosophy in +terms of phenomenological Lagrangians [2], which proved to be the cornerstone for successful +developments in precision pion physics until today. This concept of an effective Lagrangian, +that is, the low energy (large distance) manifestation of a fundamental theory, was also +developed by Ken Wilson [3] (Nobel prize 1982) quite generally. +In this picture, short- +distance degrees of freedom are systematically ‘integrated out’ and appear only as coefficients +of a theory with long-distance degrees of freedom. Weinberg’s and Wilson’s concepts are at +the base of today’s understanding of physics systems with very different energy scales, such +as in particle physics where the range goes over more than 20 orders of magnitudes [4]. +In this review, we aim to showcase the developments and richness of effective theories, +report on current progress, and encourage further work by pointing out where it is needed. +We mention important early work quite comprehensively but are more anecdotal in refer- +encing newer work; the interested reader should be able to navigate it from the references +4 + +given. +We have striven to bring under one umbrella several topics that have been separately +reviewed for various sub-communities. Our hope is that the present review will find a read- +ership that will encompass all of the working particle physics community, experimentalists +and theorists likewise, who wish to get a flavor of what has been going on under the rubric of +chiral perturbation theory and a glimpse of other effective field theories. The subject of chi- +ral perturbation has grown immensely based on rather technical and detailed computations, +in this review, we wish not to burden the text with too many equations but try to explain the +physical concepts and point the interested reader to more comprehensive reviews. In partic- +ular, we illustrate how the general principles of analyticity, unitarity, and crossing (through +dispersion relations and the analysis of experimental data based on them), can be combined +with the scattering amplitudes arising in chiral perturbation theory. This marriage has in +fact led to sufficient accuracy, thereby providing for testing the standard model at requisite +levels of precision. In addition, we have also given references to the various packages used +in the literature which readers can find interesting. +A PERSONAL NOTE +Both DW and BA have been working for over three decades on the subjects discussed here, +in particular in pion physics and Chiral Perturbation Theory (ChPT). We are pleased to +share the important lessons learned during the many years of development in the field. DW +was happy to contribute and to assist BA who has been a longtime friend and a gate opener +to India and its culture. +I. +INTRODUCTION +Many physical systems look very different when probed at different length scales or with +different energies. While ordinary matter appears to the eye in an incredibly rich diversity +of forms and textures, at the atomic scale, made for instance visible by scanning microscope +techniques, all one sees are atoms that are quite similar in different materials. Thermody- +namics, the phenomenological description of many systems, can be viewed as an effective +‘leftover’ of the microscopic theory of statistical mechanics. Such large differences appear +5 + +in many physical systems. In elementary particle physics, it is in the realm of strong inter- +actions where this can be studied particularly well. At experimental energies beyond, say, +several GeV, the relevant picture is that of the simple SU(3) gauge theory of QCD. But at +lower energies, the complicated interactions of pions and nucleons dominate and there is no +obvious trace of QCD. While we consider QCD the fundamental theory, the interactions of +pions and nucleons are described by an effective (low energy) theory, called chiral perturba- +tion theory, ChPT. So, how does one connect these two seemingly different manifestations +of the same interactions? +The key is to find properties of QCD that remain manifest also in ChPT. In this case, it +turned out that the crucial property is chiral symmetry SU(3) × SU(3) with spontaneous +breaking and a small explicit breaking term. While this symmetry is easily gleaned from +the fundamental Lagrangian of QCD, it is far less obvious at the effective level. It took +many years, from the late 1950s on, to consolidate the effects of that symmetry, and many +physicists are associated with this process. Since these developments are a fascinating part +of the history of particle physics, we will summarize some of these ideas below. +All the results obtained were cast into a bona fide field theory using external field tech- +niques by Gasser and Leutwyler in the 1980s. Apart from a rigorous formulation of the +effective Lagrangian, they gave a complete one-loop treatment of the effective Lagrangian, +also including the other pseudoscalar mesons of the eightfold way, that is the kaons and +the η. Since then, many developments have taken place. Systematic two-loop calculations +increased to quality of the predictions substantially. The role of nucleons and vector mesons +was investigated; general techniques such as dispersion relations, functional analysis meth- +ods, and rescattering theory advanced the precision of the calculations. Other methods, +such as lattice gauge theory, furnished important input. All of these represent diverse and +rich activities in physics where each would require a review article in its own right. Further- +more, the success of chiral perturbation theory encouraged the development of many other +effective theories in particle physics. By identifying high energy and low energy degrees of +freedom and using Wilson’s procedure to integrate out the high energy modes, it is possible +to arrive at an effective theory for the low energy modes in many cases. Some of the recent +theories are the heavy quark effective theory (HQET) and the soft-collinear effective the- +ory (SCET). For instance, HQET is based on the observation that in heavy quark physics +(mostly b quarks but also charmed quarks) mesons, in the limit of large (bigger than about +6 + +1 GeV) heavy quark mass mq, the relevant physics can be reliably described in a power series +in (mq)−1 where the first term is independent of (mq) 1. +While these theories are designed to understand the non-perturbative dynamics of QCD +better, more recently, the idea that even the ‘fundamental’ standard model is but the ‘low’ +energy effective manifestation of a more basic theory that would reveal itself at very high +energies. This idea is known as the standard model effective theory (SMEFT). A further +step is to view the theory of gravity, Einstein’s general relativity, as an effective theory. This +is particularly interesting for attempts to turn gravity into a quantum theory. +The contents of this review are as follows: +The next section II gives an overview of chiral perturbation theory. We recall the basic +theory of QCD and describe the construction of the effective Lagrangian, following Gasser +Leutwyler [5]. Several general observations, sometimes personal, are mixed into the text. +In section III, we show how to include particles beyond the eight light pseudoscalar +mesons, in particular the η′, the vector mesons (such as the ρ), and nucleons (baryons). +In section IV, we consider processes where ChPT does not work well and must be improved +by auxiliary methods. In particular, we look here at strong two- and three-body rescattering +where a substantial body of work exists. +In section V, we look at generalizations of the renormalization procedure for non- +renormalizable theories. The methods that have been developed might be of general interest +in going beyond ‘renormalizable’ theories. +In section VI we review the weak interactions of the pseudoscalar mesons, such as the +decays of kaons which play an important role in the understanding of fundamental effects +such as CP-violation. +In section VII, we show some applications which are of special importance. +In section VIII, we discuss effective methods to deal with the strong interactions in the +higher energy regimes where ChPT does not apply (or only in certain parts of phase space). +Then, in section IX, we show the newest development in understanding the standard +model and gravity as effective theories of an even more fundamental theory. +Finally, in section X, we collect some noteworthy recent developments that are of relevance +for ChPT. +1 Recall that the reduced mass of a system of a heavy and a light particle is largely independent of the +heavy mass +7 + +II. +THE CHIRAL LAGRANGIAN +After the initial work of Dashen, Weinstein, and Pagels [6–9], a breakthrough came from +the observations of Weinberg [2] who argued based on the principles of quantum field the- +ory and cluster decomposition and pion-pole dominance, that the lowest order effective +Lagrangian could be used to compute loops whose divergences could be absorbed into the +low-energy constants of higher order terms in the Lagrangian. This was put on a firm foot- +ing by studying the gauge invariance of the generating functional of the Green functions of +the theory by Gasser and Leutwyler [5, 10]. A scholarly exposition is given in the Scholar- +pedia article of Leutwyler [11]. In this chapter, we review their construction of the chiral +Lagrangian. +The foundation for this is QCD, the theory of the strong interactions which determine +the behavior of the observed particles in a variety of experiments where they are influenced +by ‘external fields’ or ‘sources’. These are classical objects and do not appear in loops. +Important examples of such external fields can be the masses of the quarks2, the weak +interactions, or also experimentally realized fields like a strong electromagnetic field. Since +we know that the symmetry properties are crucial, we are interested in external fields that +have well-defined transformation properties under the chiral symmetry which determines the +low energy spectrum. The objects one wants to calculate are the Green functions associated +with the external fields, from which the physical matrix amplitudes are derived in a standard +manner. +Thus, the fundamental Lagrangian involving the three light quark fields (q) has the form: +L = L0 +QCD + ¯qγµ (vµ + γ5aµ) q − ¯q (s − iγ5p) q − +θ +32π2Tr +� +Gµν ˜Gµν� +, +(1) +where +L0 +QCD = qiγµ (∂µ − iGµ) q − 1 +2g2Tr (GµνGµν) , +(2) +and Gµν is the gluon field strength tensor and ˜Gµν = +1 +2ϵµναβGαβ its dual. +The letters +vµ, aµ, s and p denote the external fields transforming as vectors, axial vectors, scalars +and pseudoscalars, respectively; the field θ transforms in a particular non-linear way. All +these quantities are x-dependent, that is vµ = vµ(x), etc. The physical Greens functions +2 In the standard model this is proportional to the vacuum expectation value of the Higgs field. +8 + +from the Lagrangian in eq. (1) are obtained by expanding the generating function around +vµ = aµ = p = 0, s = M, θ = θ0 where M is the quark mass matrix and θ0 is the +vacuum angle. We note that s = M can always be chosen to be diagonal, with real positive +elements (mu, md, ms) and an adjusted vacuum angle. For more details, see ref. [5]. The +last term in eq. (1) is odd under the CP transformation and contributes to CP violation +effects (for instance the electric dipole moment of the neutron) from the strong interaction. +These effects are found to be tiny, which requires the vacuum angle to be unnaturally small +θ0 ≲ 10−10 [12, 13]. This is the, still unresolved, ‘strong CP’ problem. It has sparked many +ideas, including the postulation of the axion, an interesting, but still hypothetical particle. +Various theoretical models of axions and the experimental bounds on the couplings with +other particles have been explored in the literature and these developments can be found in +ref. [14], for a very recent result, see ref. [15]. +Our interest is in the amplitudes at experimental particle energies below 1 GeV or so. +However, the calculations using the formulas above would be forbiddingly difficult because +at low energies, there are no free quarks (or inclusive states, such as exist at high ener- +gies), but pions or kaons (the pseudoscalar mesons), or other hadrons that are complicated +bound states of quarks and gluons. Thus, we must express the physical contents of the +QCD Lagrangian in terms of these fields. This we shall call the effective (low-energy) chiral +Lagrangian. The notion is quite general: An effective Lagrangian expresses the physics in +terms of the physical particles (or fields) at the energies relevant for the experiment consid- +ered. The form of the chiral Lagrangian is dictated by the choice of the physical (dynamical) +fields and the symmetry properties of the external fields. Because we are interested in low +energies, one considers an expansion in energy (momentum) of the particles which, because +of chiral symmetry, starts at order(p2) where p is a typical momentum of the particles. The +leading term in an energy (momentum) expansion is completely fixed due to the work of +Callan, Coleman, Wess and Zumino [16, 17] have allowed us to extract several general fea- +tures of the interactions of Goldstone bosons, quite independent of the knowledge of the +dynamics of the strong interactions, based exclusively on the (global) symmetries of the un- +derlying Lagrangian. The fact that they are (approximate) Goldstone bosons already fixes +their mutual interactions to be of the derivative type. Furthermore, the parametrization +of the degrees of freedom encoded by the physical fields requires them to be coordinates of +the coset space given by G/H, where G is the global symmetry of the Lagrangian and H is +9 + +the symmetry of the ground state. The broken generators of G not lying in H are precisely +these Goldstone boson degrees of freedom. Taking into account the symmetry properties +of the Goldstone bosons, a convenient parametrization where the chiral transformations are +linear and ensures the derivative nature of the interactions is [18]: +U ≡ ei +√ +2Φ/Fπ +(3) +where +Φ = +� +� +� +� +� +π0 +√ +2 + η8 +√ +6 +π+ +K+ +π− +− π0 +√ +2 + η8 +√ +6 +K0 +K− +K0 +− 2η8 +√ +6 +� +� +� +� +� , +(4) +which is unique up to the reparametrization of the Goldstone boson fields themselves. As +argued by Boulware and Brown [18], it is advantageous to group the pseudoscalar fields into +a 3 × 3 unitary. The point is that the external fields should couple to such operators of the +fields which transform linearly under chiral transformations in order that the interactions +can be built by a conventional loop expansion. +The results of current algebra are all captured by the effective Lagrangian: +Leff = L2 + L4 + L6 + · · · +(5) +where L2, L4 and L6 are the terms of O (p2), O (p4) and O (p6), respectively. +The leading-order term in the low energy expansion is generated by the non-linear sigma +model coupled to the external fields(v, a, s, and p, in the notation of the ref. [19]), +L2 = F 2 +π +4 ⟨DµUDµU † + χU † + χ†U⟩ , +(6) +where +DµU = ∂µU − i(vµ + aµ)U + iU(vµ − aµ), +χ = 2B (s + ip) . +(7) +The above reproduces the well-known Weinberg result for ππ scattering [20], which sets +the scale of the chiral interaction in terms of the pion decay constant. +Without dynamic external fields, s is proportional to the masses of the quarks, which are +fundamental quantities in the standard model. In fact, ChPT plays a key role in determining +these quantities in terms of the pseudoscalar masses [5, 21–23]. There is also considerable +effort to determine them from lattice QCD [24] as well as using QCD sum rules [25, 26]. +The sum rule determinations are also a powerful tool to determine the QCD parameters, +10 + +and their recent applications can be found in the book of Dominguez [27]. A summary of +the most recent determinations can be found in the PDG [28]. Apart from their importance +as fundamental parameters, a value of mu = 0 would have solved the so-called strong CP +problem (see [29] for details), but this does not seem to be the case. +In particular, in order to go beyond the leading order, Weinberg in ref. [2] argued that +the content of a field theory is dictated by the symmetries and analyticity, perturbative +unitarity, and cluster decomposition. Using this, he was able to predict the structure of the +ππ scattering amplitude and the corresponding logarithms that would have to generate the +required imaginary parts of the amplitude from the original (real-valued) tree-level ampli- +tude [2]. While this remained a thumb rule, the systematic study required the introduction +of external sources for the currents of the theory in the spirit of Julian Schwinger, who +proposed to study field theory through sources. Rather than being mere mathematical cu- +riosities, the presence of the external sources allowed a systematic computation of one-loop +generating functional through the heat-kernel technique, an established method of obtain- +ing in a compact manner the generating functional rather than an equivalent yet explicit +computation of Feynman diagrams. This functional requires regularization (dimensional) +and renormalization of the infinities that are generated by the loops. +Since the original Lagrangian in d = 4 is not renormalizable, the procedure generates +higher derivative terms not present in the original two-derivative Lagrangian. Thus new +low-energy constants are introduced into the theory with corresponding β-functions, which +are fixed from the experiment. Once fixed, at this order, any process of interest can be +computed, which makes the theory predictive. This process can be continued indefinitely to +any order in the loop expansion and/or the momentum expansion. It is the convention to +consider each loop to yield a new power of p2 and also to assign powers to the explicit mass. +The chiral Lagrangian at O(p4) [5, 10] consistent with the Lorentz invariance, C and P +symmetry is given by: +L4 =L1⟨DµU †DµU⟩2 + L2⟨DµU †DνU⟩⟨DµU †DνU⟩ ++ L3⟨DµU †DµUDνU †DνU⟩ + L4⟨DµU †DµU⟩⟨χ†U + χU †⟩ ++ L5⟨DµU †DµU(χ†U + U †χ)⟩ + L6⟨χ†U + χU †⟩2 + L7⟨χ†U − χU †⟩2 ++ L8⟨χ†Uχ†U + χU †χU †⟩ − iL9⟨F µν +R DµUDνU † + F µν +L DµU †DνU⟩ ++ L10⟨U †F µν +R UFLµν⟩ + L11⟨FRµνF µν +R + FLµνF µν +L ⟩ + L12⟨χ†χ⟩ , +(8) +11 + +where +F µν +R =∂µrν − ∂νrµ − i[rµ, rν] +(9) +F µν +L =∂µlν − ∂νlµ − i[lµ, lν] . +(10) +At this order in the momentum expansion, there are twelve new couplings Li (also called +low energy constants) that appear out of which only L3 and L7 are not divergent. +Also, the next O (p6) order Lagrangian has been worked out, see refs. [30, 31] and there +are results of O (p8) [32, 33]. Of course, the number of couplings increases, and thus the +predictive power for smaller (higher-order) effects decreases. But by choosing suitable ob- +servables, one is still able to determine some of them. +The low energy constants can be renormalized in a standard fashion and one writes: +Li = Lr +i(µ) + Γiλ(µ) +(11) +where +λ(µ) = µd−4 +4π2 +� +1 +d − 4 − 1 +2 (1 + log(4π) − γE) +� +, +(12) +where the function λ comes from performing a standard one loop calculation and the co- +efficients Γi coefficients were calculated in [5, 10]. The divergences in the bare coupling Li +cancel with the one present in the λ(µ) resulting in renormalized couplings Lr +i(µ). Their +scale dependence is given by: +Lr +i(µ2) = Lr +i(µ1) + +Γi +16π2 log +�µ1 +µ2 +� +. +(13) +The low energy constants reflect the properties of the strong interaction spectrum that has +been integrated out and contribute to their numerical values. For instance, in certain cases, +the exchange of a single vector meson accounts for the observed values; alternatively, there +might be an axial-vector meson dominance [34, 35]. Other higher angular momentum states +also make contributions but are mostly numerically less significant. Over the last years, +several papers have explored ways to determine the low energy constants, see ref. [36]. For +a recent review, see ref. [37]. +To illustrate the renormalization procedure, we consider the electromagnetic form factor +of pion which is defined as: +⟨π+(p′)|V em +µ (0)|π+(p)⟩ = (p′ + p)µF π +V (t), +t = (p′ − p)2 . +(14) +12 + +This quantity at tree-level and at one-loop is given by: +F tree +V +(t) = 1 , +(15) +F π,1−loop +V +(t) = 2φ(t, Mπ; d) + φ(t, MK; d) , +(16) +where +φ(t, M; d) = − tM d−4 +(4π)d/2 +Γ(2 − d/2) +2F 2 +π +� 1 +0 +dx x(1 − 2x) +� +1 − +t +M 2x(1 − x) +� d−4 +2 +(17) +The O(p4) contribution to F π +V comes from L9 and has the following form: +F π,L4 +V += 2L9t +F 2 +π +, +(18) +and the total contribution to F π +V is finite at this order if the L9 is tuned as: +L9 = Lr +9(µ) + λ(µ) +4 +, +(19) +(20) +this results into finite scale dependent renormalized coupling Lr +9(µ). +Now the divergent +function in eq. (17) is renormalized by writing: +φ(t, M; d) = φren(t, M, µ; d) − tλ(µ) +6F 2 +π ++ O(p4) , +(21) +and the complete expression for pion electromagnetic form factor has the form: +F π +V (t) = 1 + 2φren(t, Mπ, µ; d) + φren(t, MK, µ; d) + 2tLr +9(µ) +F 2 +π ++ O(p4) . +(22) +Other examples can also be found in the book of Donoghue, Golowich and Holstein [38]. +An important attribute of the external source technique, which was not explicitly available +in the heuristic proposal of Weinberg is the ability to also account for the electromagnetic +and weak interactions. This promotes the framework of chiral perturbation theory to an +effective theory of the Standard Model (that includes electromagnetic and weak interactions) +and not just that of the strong interaction sector with pions, kaons and the η. +III. +EXTENSIONS OF CHIRAL PERTURBATION THEORY +While the previous section treated ‘standard’ ChPT and its development, there are var- +ious extensions that are needed when specific processes are to be investigated where the +methods of the previous section are not sufficient. We present here a short overview only, +for a deeper treatment, the references given should be consulted. +13 + +A. +The η′ +Without the axial anomaly, see eq. (2), there would be 9 light mesons. Because of it, +the singlet axial current is not conserved, and the ninth meson becomes massive; indeed, +the η′ has a mass of 957 GeV, comparable to that of the nucleons. In the (hypothetical) +limit NC → ∞ the η′ is indeed massless, but NC = 3, and the η′ is heavy. However, despite +the large mass difference of the 8 pseudoscalar mesons and the η′, its influence on many +processes is substantial, and therefore, it must be included in a systematic treatment. A +successful way is to incorporate the η′ in the U-matrix, that is, taking a three-dimensional +unitary matrix [5] of form: +U(x) = e +1 +3 iφ0(x)eiφ(x) +(23) +and add a mass term for the η′. +In the absence of mixing, φ0(x) corresponds to η′. +Furthermore, as the η′ and the external field θ are both SU(3) singlet pseudoscalars, they +transform (up to a sign) in the same way under chiral transformations, and therefore the +sum φ0 + θ is invariant. This means that everywhere in the Lagrangian where there are +constants, they should be replaced by arbitrary functions of φ0 + θ. This brings of course +new uncertainties. Nevertheless, as shown in ref. [5], it is possible to draw some concrete +conclusions, in particular about the mixing of the neutral pseudoscalar mesons π0, η, and +η′. Furthermore, as can be seen from eq. (8), low energy constant L7 can be modeled by the +exchange of a chiral singlet pseudoscalar. In fact, the η′ does a good job. There are possibly +further applications of this way to include the η′ and more details can be found in Kubis et +al. [39, 40]. As mentioned, in the large NC limit, the models show new interesting aspects. +See refs. [41, 42] for a thorough investigation. +B. +Vector mesons +The next heavier hadrons after the pseudoscalar mesons are the vector mesons, such as +the ρ. Using the methods used before, in particular, that means determining the correct +transformation behavior of the vector mesons under the chiral symmetry, they can be built +into the chiral Lagrangian [34]. This does not only contribute to processes with such vector +meson, but the vectors are also resonances that contribute to the low energy constants +introduced in section II. In fact, as stated there, in many cases, they largely saturate the +14 + +constants, thus giving a very successful model for them. Some recent articles on the subject +are [43–45]. +C. +Baryons +The method can also be extended to include the baryon degrees of freedom see review +ref. [46, 47]. In the manifestly Lorentz covariant framework, a problem arises because one +cannot have a strict power-counting scheme. On the other hand, inspired by the heavy +quark effective theory, a heavy baryon version is available due to Jenkins and Manohar [48]. +A reformulated Lorentz invariant method via infra-red regularization due to Ellis [49, 50] +and Becher and Leutwyler [51], along with other versions such as extended on-mass-shell +renormalization methods that have several advantages were proposed in ref. [52]. The pion- +nucleon σ−term (see ref. [53] for latest review) obtained from the baryon ChPT has also +been useful for beyond the standard model physics considerations, especially studies related +to the dark matter searches [54]. +IV. +TWO AND THREE BODY RESCATTERING +Scattering processes provide important clues to the physics behind them. Furthermore, +scattering is often part of other processes, like decays, where the total amplitude also depends +on the (re) scattering of the decay products. For instance, in a decay K into two pions, +the pions rescatter strongly, thereby (in some cases) influencing decisively the measured +decay rate. This is important in cases where the straightforward application of ChPT is +not sufficient to explain the experimental results and must be supplemented by additional +methods, such as unitarity conditions which sum up certain higher order corrections. We +also note that there is a vast literature on scattering. +The formalism presented in section II for the one-loop ChPT can be used to calculate the +scattering amplitude involving the pseudoscalar Goldstone bosons. In the limit of isospin +conservation, it is customary to introduce amplitudes of definite isospin in the s-channel +T I(s, t), which may be related to specific physical charged states and depend on the process +15 + +of ππ or πK. These isospin amplitudes can further be decomposed into partial waves as: +T I(s, t) = 32π +� +l +(2l + 1)tI +l (s)Pl(cos(θ)) . +(24) +where tI +l (s) is partial wave amplitude, θ is the scattering angle in the center of mass frame, +and the Pl are the Legendre polynomials. The tI +l (s) are complex above the threshold and +are related by unitarity. For ππ scattering it has the following form [10]: +tI +l (s) = +� +s +s − 4m2 +π +�1/2 1 +2i +� +ηI +l (s)e2iδI +l (s) − 1 +� +, +(25) +with δI +l being the phase shift and ηI +l the elasticity parameter. For πK scattering, an anal- +ogous expression can be found in ref. [55]. It may be recalled that scattering lengths are +the lowest order shape parameters appearing in the expansion of the real part of the partial +wave amplitudes, and their expression near the threshold looks like this: +Re (tI +l (s)) = (q2)l(aI +l + bI +l q2 + O(q4)), +(26) +where q2 is the square of the momentum transfer in the center of mass frame and (q2)l +denotes the centrifugal barrier. The scattering length of the lowest waves dominates the +physical cross-section at low energies. The scattering lengths are also one of the important +quantities for the pionium as the decay rate that is sensitive to |a0 +0 − a2 +0|2[56]. DIRAC [57] +and NA48 experiments ref. [58, 59] at CERN were aimed to measure the S-wave scattering +length difference in the I = 0 and I = 2 isospin channels and the observed values were in +agreement with ChPT predictions in ref. [56]. Since pions and kaons are short-lived, one +cannot do fixed target experiments, and the obtained scattering lengths are based on phase +shift analyses. Whereas e+e− → π+π− is well studied experimentally and is related to the +I = 1 P−wave via the Watson theorem, the other phase shifts are well measured only at +higher energies from πN scattering. At low energies, they are related to the form factors +of Kl4 decays and have to be extracted using dispersion relations. In pion scattering, the +suitable framework is dispersion relations with two subtractions which suffice due to the +Froissart bound, which allows one to write a system of partial wave equations that leaves +the two S-wave scattering lengths undetermined parameters. These Roy equations and the +corresponding Roy-Steiner equations for πK scattering have been studied for over 50 years. +They also provide a very useful framework for relating the dispersion relations to the chiral +amplitudes, as shown in this section. +16 + +The O(p6) behavior of these contributions can be calculated following the work of Bijnens +et al. [60] and Colangelo, Gasser and Leutwyler [56] and the ππ scattering amplitude to can +be decomposed in the following form: +tI +ℓ(s) = tI +ℓ(s)2 + tI +ℓ(s)4 + tI +ℓ(s)6 + O(p8) . +(27) +At the leading order, the non-zero contributions from the S- and P- waves are given by: +t0 +0(s)2 = 2s − M 2 +π +32πF 2 +π +, +t1 +1(s)2 = s − 4M 2 +π +96πF 2 +π +, +t2 +0(s)2 = −s − 2M 2 +π +32πF 2 +π +. +(28) +To O(p6) accuracy, the imaginary parts of the ππ (and πK) scattering amplitude receive +contributions only from the S− and P− partial waves and can be written in terms of the +three functions of only one variable as: +A(s, t, u) = C(s, t, u) + 32π +�1 +3U 0(s) + 3 +2(s − u)U 1(t) + 3 +2U 1(u) ++ 1 +2 +� +U 2(t) + U 2(u) − U 2(s) +� � +, +(29) +where the first term must obey crossing symmetric and has the form: +C(s, t, u) = c1 + sc2 + s2c3 + (t − u)2c4 + s3c5 + s(t − u)2c6 . +(30) +It may be borne in mind that the real parts obtain contributions from the O(p4) from higher +waves as well. The ci are the subtraction constants of U i(x), which are also termed “unitarity +corrections”. For s−channel with isospin I = 0, 1, 2 have dispersion relation given by: +U 0(s) = s4 +π +� ∞ +4M2π +ds′σ (s′) t0 +0 (s′)2 {t0 +0 (s′)2 + 2 Re t0 +0 (s′)4} +s′4 (s′ − s) +(31) +U 1(s) = s3 +π +� ∞ +4M2π +ds′σ (s′) t1 +1 (s′)2 {t1 +1 (s′)2 + 2 Re t1 +1 (s′)4} +s′3 (s′ − 4M 2 +π) (s′ − s) +, +(32) +U 2(s) = s4 +π +� ∞ +4M2π +ds′σ (s′) t2 +0 (s′)2 {t2 +0 (s′)2 + 2 Re t2 +0 (s′)4} +s′4 (s′ − s) +, +(33) +more details about various quantities appearing in this equation can be found in ref. [56]. +The case when there is no three-channel crossing symmetry, and with unequal mass scat- +tering is also accessible using a combination of fixed-t and hyperbolic dispersion relations, +which were known in the literature after being suitably modified to account for chiral count- +ing, in order to saturate the dispersion relations using the imaginary parts of the relevant +S- and P- waves. +17 + +In the case of πK scattering, the structure was analyzed by Ananthanarayan and +B¨uttiker [55] and by B¨uttiker, Descotes-Genon and Moussallam [61]. +The πK scatter- +ing amplitude to one loop can be decomposed into partial waves. Once one isospin channel +amplitude is known, others or a combination of them can be obtained using the crossing +symmetry relations. Like ππ scattering, these amplitudes can also be written in terms of +functions of one variable as: +T +(s, t, u) = Z+ +t (t) + Z+ +0 (s) + Z+ +0 (u) + (t − s + ∆2 +u )Z+ +1 (u) + (t − u + ∆2 +s )Z+ +1 (s) +(34) +T −(s, t, u) = Z− +t (t) + Z− +0 (s) − Z− +0 (u) + (t − s + ∆2 +u )Z− +1 (u) − (t − u + ∆2 +s )Z− +1 (s) . +(35) +The imaginary parts of the Z’s can be written in terms of the lowest partial waves as: +Im Z± +0 (s) = 16πImf ± +0 (s) , +(36) +Im Z± +1 (s) = 12π +q2 +s +Imf ± +1 (s) , +(37) +Im Z+ +t (s) = 16π +√ +3 Imf It=1 +0 +(t) , +(38) +Im Z− +t (s) = 6 +√ +2πImf It=1 +0 +(t) +ptqt +. +(39) +The details of various quantities appearing in these equations can be found in ref. [55]. +There are processes where it is necessary to account also for 3-particle rescattering, which +is considerably more complicated. This is, for instance, the case for decays where phase space +is limited. The best-known example is the decay of η → 3π with significant data available +for the cases of exclusively neutral, as well as neutral, and charged pions, in terms of the +Dalitz plot as well as in terms of rates. This rate is sensitive to the u − d mass difference +and, therefore of special importance in the determination of the quark mass ratio (Q) [62]. +The original work of Khuri-Treiman [63] is based on the dispersive approach to study the +final state interactions in K → 3π, and a set of integral equations are obtained and later +to η → 3π by Kambor, Wisendanger and Wyler [64] and Leutwyler and Anisowich [65]. +The presence of final state interactions between the pion generates the branch cut in the +amplitudes that starts from 4m2 +π in s−, t−, and u− channels. As the centrifugal barrier +suppresses the higher partial waves, the amplitude has a resemblance with the 2 body +scattering where higher waves also start contributing from O(p8). The important difference +between the two is that the three-body scattering also involves angular averages, which +18 + +are difficult to perform. This difficulty has recently been overcome by an efficient method +provided by Gasser and Rusetsky [66]. +The scattering amplitude for η → 3π can be decomposed into the contributions from +isospin channel I = 0, 1, 2 represented by M0, M1, M2, which are the functions of one +variable in Mandelstam variables. Following the detailed analysis of refs. [65, 67–69], the +discontinuity in the amplitude has the form: +discMI(s) = θ(s − 4M 2 +π) +� +MI(s) + ˆ +MI(s) +� +sin(δI(s))e−iδI(s) +(40) +The first term in the braces receives contributions from the interactions of the s channel, +and the second term accounts for those coming from the t and u channels. The δI(s) are +the phase shifts of the ππ scattering from the leading partial waves. The t and u channel +contributions are given in terms of the angular averages of the MI’s as follows: +ˆ +M0(s) = 2 +3⟨M0⟩ + 2(s − s0)⟨M1⟩ + 2 +3κ⟨zM1⟩ + 20 +9 ⟨M2⟩ +(41) +ˆ +M1(s) = κ−1� +3⟨zM0⟩ + 9 +2(s − s0)⟨zM1⟩ − 5⟨zM2⟩ + 3 +2κ⟨z2M1⟩ +� +(42) +ˆ +M2(s) = ⟨M0⟩ − 3 +2(s − s0)⟨M1⟩ − 1 +2κ⟨zM1⟩ + 1 +3⟨M2⟩ , +(43) +where +s0 = 1 +3M 2 +η + M 2 +π +(44) +κ(s) = +� +1 − 4M 2 +π +s +� +(M 2 +η − Mπ)2 − s +� +(M 2 +η + Mπ)2 − s +(45) +⟨znMI⟩(s) = 1 +2 +� 1 +−1 +dzznMI(3 +2s0 − 1 +2s + 1 +2zκ(s)) +(46) +with I = 0, 1, 2 and n = 0, 1, 2. For more details, we refer to ref. [70]. +The details of the higher order corrections to O(p6) for the three body decay of the +η → 3π can be found in refs. [71–73]. The dispersive construction of amplitude can be found +in Kampf et al. [74, 75] and small electromagnetic corrections to this process in Ditsche, +Kubis and Meißner [76]. A detailed analysis using Dalitz plot and modified non-relativistic +effective field-theory in by Schneider, Kubis and Ditsche in ref. [77]. The determination of +quark mass ratio from these decays are presented in refs. [62, 78–80]. Cusps in K → 3π, +which are relevant for the precise determination of the pion scattering lengths, are studied +in ref. [81, 82], in η → 3π, effects of mixing of ηη′ in the η → 3π Leutwyler [83], dispersive +19 + +analysis by Leutwyler and Anisovich in ref. [65] and various topics related to three-body +decays and dispersion relations are now covered in the book of Anisovich et al. [84]. For a +detailed review, we refer to refs. [40, 70]. +V. +GENERALIZED RENORMALIZATION GROUP AND LARGE CHIRAL LOG- +ARITHMS +In section II, we touched briefly upon infinities in the low energy constants, see eq. (11). +Such infinities are, of course, well-known in QED and other quantum field theories. His- +torically such divergencies in the self-energy of an electron from classical electrodynamics +led to the birth of quantum field theory. Schwinger, Tomanaga, Feynman, and Dyson gave +a covariant description of QED which led to the consistent description to any order in the +perturbation theory. The infinities are removed by redefinition in the bare parameters of +the Lagrangian, a procedure termed renormalization. At that time, it was just a math- +ematical trick to tackle the divergences. The works of Stueckelberg and Petermann [85], +Gell-Mann and Low [86] showed that this procedure automatically incorporates the running +of renormalized coupling constants, see eq. (13). The renormalization group equations dic- +tate the running and mixing of various operators with scales and have been used as a very +useful technique that allows to sum up some of the large logarithmic corrections which are +remnants of the renormalization procedure. +Whereas the early discussion was mainly restricted to perturbation theory and renormal- +izable theories, Wilson [87–89] in the early 1970s further extended it to non-perturbative +systems in order to understand critical phenomenons and gave a deeper insight to the physics +at different scales. This has found numerous applications in various areas of physics ranging +from condensed matter, statistical physics, and cosmology to particle physics. These ideas +were later studied in great detail using the path integral by Polchinski [90]. There are var- +ious approach to the renormalization group and we refer to refs. [91–98]. An overview can +be found in ref. [99]. +While the concepts of renormalization are associated with renormalizable theories, one +might ask how they work in non-renormalizable theories, such as ChPT, where the number +of parameters increases to cancel the divergences appearing in the loop calculations. In +particular, one may ask how to order the (large) leading logarithms (LL) which arise in the +20 + +calculations. +Li and Pagels [100] in the early seventies pointed out that a large logarithm of type +m2 +π log(m2 +π) appears in one-loop calculations involving pion loops. Weinberg calculated these +logarithms in his famous paper on phenomenological Lagrangians [2] using current algebra +and the renormalization group for pion scattering. Later work of Gasser and Leutwyler [10] +where systematic one-loop extension of ChPT was performed and significant ∼ 25% con- +tribution at 1 GeV from such terms were obtained, especially the corrections to the lowest +S-wave pion scattering length. The large logarithm contributions to the two-loop can be +found in ref. [101] and have the following form: +a0 +0 = 7m2 +π +32F 2 +π +� +1 − 9 +2 +m2 +π +16π2F 2 +π +log(m2 +π/µ2) + 857 +42 +� +m2 +π +16π2F 2 +π +�2 +log2(m2 +π/µ2) +� +. +(47) +The full two-loop contributions to scattering length read [60]: +a0 +0 = +tree +� �� � +0.156 + +1 loop +� +�� +� +0.039 + 0.005 + +2 loops +� +�� +� +0.013 + 0.003 + 0.001 = +total +� �� � +0.217 +L +anal. +ki +L +anal. , +where ki are the contributions from the single as well as double chiral logarithms, which +can be evaluated using the renormalization group [2, 101]. Bijnens, Colangelo and Ecker [102, +103] extended the work on chiral double logarithms to the full meson sector. Clearly, loga- +rithmic corrections can be large in ChPT, and a tool like the renormalization group would +be useful. +Indeed, Kazakov [104] and Alvarez, Freedman, and Mukhi [105] discussed the extension of +renormalization to arbitrary (non-renormalizable) theories in order to calculate leading and +subleading divergences. These ideas were applied to ChPT by Buchler and Colangelo [106], +which required a new one-loop calculation at each order. The resummation of these large +logarithms to all orders is still an open question in ChPT. However, the chiral logarithms +have been of constant interest to understand many renormalizable and non-renormalizable +theories as a toy model. +Bissegger and Fuhrer [107] worked out a method to calculate +the chiral logarithms for two flavors to any desired order in chiral limit using analyticity, +crossing symmetry, and the Roy equations. They have also given the five-loop results for +specific two-point scalar Green functions. Kivel, Polyakov, and Vladimirov [108] provided +a method where a non-linear recurrence relation is obtained that efficiently calculates the +21 + +leading logarithm (LL) to arbitrary loops for any non-renormalizable theories. This work +was later extended for form factors in ref. [109] and some results for the LLs for the massless +O(N + 1)/O(N) σ-model are also presented. +This model, for N = 3, is equivalent to +the chiral SU(2) × SU(2) model that describes the leading low-energy interaction of pions +in the chiral limit. Later Koschinski, Polyakov, and Vladimirov [110] provided a method +to calculate the leading infrared logarithms to essentially unlimited loop order using only +the tree-level results in the non-renormalizable massless effective theory and later to sigma +models on an arbitrary Riemann manifold by Polyakov and Vladimirov in ref. [111]. The +LLs in the massive case for a non-linear O(N)-sigma model are studied by Bijnens and +Carloni in ref. [112, 113] and extended to the anomalous sector by Bijnens, Kampf and +Lanzin in ref. [114]. More recently, Ananthanarayan, Ghosh, Vladimirov, and Wyler [115] +have generalized the massive case to arbitrary order in LL corrections for various O(N) and +SU(N) models and a Mathematica code is provided that reproduces the existing results +and calculates higher-order results. Further development in two-dimensional effective field +theories can be found in ref. [116, 117] and an extension to the baryon sector in ref. [118]. +VI. +WEAK INTERACTIONS OF PSEUDOSCALAR MESONS +The ChPT formalism, especially when formulated with the external field method, is +directly adaptable to weak processes involving the pseudoscalar mesons, such as the decays +K → ππ, and others. A recent example illustrating the persistent importance of ChPT +is the rare decays involving (hypothetical) new light particles such as axions [119]. The +systematic expansion in powers of momentum and quark masses allows analyzing seriously +many ‘small’ effects. An illustration of how the weak interactions fit into the external field +method with well-defined transformation properties is given in figure 1. The large size of +MW compared to the QCD scale of a few GeV makes it clear that any interaction of gluons +that affect the W bosons is tiny: It would involve the strong coupling constant at the MW +scale and further suppression factors 1/MW. +22 + +FIG. 1: Illustration of the weak interaction of the pseudoscalar mesons. The large mass of +the W bosons is the reason why the external field method is appropriate [120]. +The basis for extending ChPT to the weak interactions was laid down in ref. [121]. It gives +a systematic treatment of ChPT for weak interactions and extended the weak interactions +Lagrangian to O (p4). It is based on several previous works; here, we mention only the +pioneering paper by Cronin [122]. +To construct the weak chiral Lagrangian, we need the form of the external field that +represents the weak interactions. The (chiral) symmetry properties of the weak interaction +follow from the fact that they arise from the symmetric product of two left-handed charged +octet currents: +L∆S=1 = g +� +J2 +1µ, J1µ +3 +� ++ + g∗ � +J1 +2µ, J3µ +1 +� ++ , +(48) +with +J2 +1µ = J1µ + iJ2µ, +J3 +1µ = J4µ + iJ5µ +(49) +and where the (numeral) indices refer to the position in the 33 flavor matrix and {, }+ +denotes the symmetric product. This implies that the weak interactions transform as (8)L +and (27L, ). We note that the CP-invariant and the CP-odd parts can be conveniently +separated in ref. [121]. +Using now the expressions +Lµ = iU +∇µU +(50) +for the left-handed meson currents, we can write the octet CP invariant effective weak +23 + +K +sylendoperator as +L(8) +WI = c2⟨λ6∇µU †∇µU⟩ = c2⟨λ6LµLµ⟩ +(51) +where the octet property is manifest in the matrix λ6 3. At O (p2), also a second operator +can be written as: +L8′ +WI = c5⟨λ6 +� +χ†U + U †χ +� +⟩ +(52) +The CP-invariant effective weak chiral Lagrangian transforming as (27L) is constructed from +the octet components of Lµ: +L27 +WI = c3 +� +3⟨ +� +Q2 +3 + Q3 +2 +� +Lµ⟩⟨Q1 +1Lµ⟩ + 2⟨Q2 +1Lµ⟩⟨Q1 +3Lµ⟩ + 2⟨Q1 +2Lµ⟩⟨Q3 +1Lµ⟩ +� +(53) +where the matrices Qi +j have a 1 in the position i, j and are zero otherwise. We note that +there is only one operator in this case. An application of this CP-invariant operator to +K → πℓℓ process at one-loop can be found in ref. [123]. +As to the CP-violating Lagrangian, it is obtained from the above by replacing λ6 by λ7 +and appropriate changes in the operators transforming as 27−plet +L− +WI =c− +2 ⟨λ7LµLµ⟩ + c− +5 ⟨λ7 +� +χ†U + U †χ +� +⟩ ++ c− +3 +� +3⟨λ7Lµ⟩⟨Q1 +1Lµ⟩ + 2i +� +⟨Q2 +1Lµ⟩⟨Q1 +3Lµ⟩ − ⟨Q1 +2Lµ⟩⟨Q3 +1Lµ⟩ +�� +(54) +We note that the ∆S = 2 operator is required for the calculation of the mass difference +in the K0- ¯K0 mixing, which transforms as a 27-plet, and it is obtained by setting the tensor +components to their appropriate values (see ref. [121]). +It is well known that the second octet operator in eq. (52) does not contribute to physical +processes. The operator is, in fact, proportional to the variation under a suitably chosen +symmetry and thus to a divergence of a conserved (Noether) current. Since the operator +does not carry momentum, the matrix element vanishes. In [121], the argument is extended +to the one-loop level. We note here, however, that in processes where the scalar external +field is not just χ, but variable, this statement might not hold. +While in ref. [121], a complete basis of the weak operators at O (p4) is given, subsequent +analyses showed that the basis could be further reduced, see ref. [124] and ref. [125]. The +complete O (p4) Lagrangian containing 37 operators can be found in those papers. +We +3 Since we consider K−decays, only the transition from an s−quark to a d−quark, that is only the Gell- +Mann matrices with elements (2, 3) contribute +24 + +also note that not all of these contribute to the decay of kaons into pions which make the +calculations simpler and the predictions better. +Much like in the strong interaction case discussed before, the application of the O (p4) +Lagrangian to physical processes is used to determine the coupling strengths of the low +energy operators, the LECs. In ref. [125] the decay K → 3π is analyzed. The order O (p4) +gets contributions from the operators mentioned and loop diagrams whose vertices are those +of the lowest order interactions. For an improved treatment, see ref. [126]. Also, kaon decays +are again considered as a laboratory for rare processes and recent progress can be found in +ref. [127]. +VII. +SELECTED APPLICATIONS OF CHIRAL PERTURBATIVE THEORY. +As already mentioned, ChPT has numerous applications in describing low-energy pro- +cesses. In some cases, the precision reached is very high and allows for testing fundamental +physics. Here we review but a few such cases. +As mentioned before in section II, the masses of the quarks can be determined quite +precisely using the chiral formalism from adequate phenomenological studies, such as of the +η → 3π decay (see section IV). Input from the lattice and QCD sum rules increases the +accuracy. These studies have confirmed that the up quark mass mu is non-zero [24]. +The predominant decay of π0 into the two photons proceeds via the chiral anomaly; +the prediction for the rate is Γ (π0 → γγ) = 7.760 eV, in remarkable agreement with +Γ (π0 → γγ) = 7.82 ± 0.14(stat.) ± 17(syst.) eV obtained from the high precision exper- +imental finding of PrimEx-II [128] experiment. +Other processes such as ππ and πK scattering require detailed analysis using SU(2) and +SU(3) versions of the ChPT. The scattering amplitude of these processes, when expanded +in terms of the partial amplitudes, results in the notion of the scattering lengths, and +their experimental inputs can be used to fix some of the low-energy constants. An explicit +expressions ππ, πK, and KK scattering lengths to O (p4) can be found in ref. [24] and +references therein. Interestingly, for the ππ interaction, the scattering lengths for the I = 0 +isospin channel have a positive sign and are larger than 3.5 times in magnitude compared +to the I = 2 isospin channel, which has a negative sign. These signs correspond to the +repulsive and attractive nature of the interactions in these channels. +Furthermore, the +25 + +phase shift analysis of the ππ scattering has been found to be a very useful ingredient in +quantifying the hadronic contributions to the anomalous magnetic moment of the muon (see +below). Readers can find further details on the form factors in refs. [129–133] and references +therein. +Of particular interest is the anomalous magnetic moment of the muon. +It is one of +the testing grounds for the standard model and has been the topic of constant interest +in the particle physics community [134, 135]. The results from the Brookhaven National +Laboratory (BNL) found tension with the predictions of the standard model a little over +3σ in ref. [136]. Further development in both the theory and experiment side has taken +place and is summarized in ref. [137]. The most recent experiment in Fermilab aimed to +study this issue with improved purity of the beam and detector components and found +agreement with the results of BNL with a smaller central value. Their combined results +have has now established the discrepancy at 4.2σ. These results can be found in a set of +publications in refs. [138–141]. The main source of the discrepancy comes from the hadronic +vacuum polarization contributions and another somewhat less numerically important but +relatively larger uncertainty known as the hadronic light by light scattering contributions. +An excellent summary of all these discussions was recently presented, see slides of [142], and +for details, we refer to ref. [129, 143–145] and references therein. Some of these hadronic +light-by-light contributions, as well as those contributions to (g−2)µ involve related processes +where transition form factors play an important role. These form factors are the complex +functions obeying the unitarity and analyticity conditions, which dictate their behavior in +the complex plane. However, their values for a given kinematical region can be fixed by +the available information from the experiments or lattice simulations. In some cases, the +Watson theorem relates the phase shift of the scattering amplitude to the phase of the form +factor. +One of them that is worth mentioning is the transition form factor for the ωπ0 +for which discrepancies between experimental data and results from dispersion theory were +reported for low energy region; see ref. [146] and references therein for details. However, +these discrepancies can be studied in a model-independent way using the method of unitarity +bounds[147, 148] combined with the functional analysis method [149] to find the bounds +on the ωπ0 form factor. These functional methods have found numerous applications in +hadron physics and are now available in the form of a textbook in ref. [150]. Recently, some +agreement between experimental data with new analysis based on subtracted Khuri-Treiman +26 + +equations has been reported for ωπ transition form factor in ref. [151]. +Of course, several other examples can be studied using chiral perturbation theory, and +many of them can also be found in the supplementary Mathematica [152] notebooks of +ref. [153] and references therein. The following publicly available codes are recommended to +study some of the processes: +• Ampcalculator by Unterdorfer and Ecker [154]. +• Phi by Orellana which calculates O (p4) corrections to one loop and already included +in FeynCalc 9.0 [155] and later versions. +• The Mathematica-based code to study the ππ scattering, and Scalar and Pseudoscalar +Form Factor and new additions to meson-meson scattering using U(3)−ChPT can be +found in the link [156]. +• Mathematica notebooks with many solved examples by Ananthanarayan, Das, and +Imsong in ref. [153] +VIII. +OTHER EFFECTIVE THEORIES FOR THE STRONG INTERACTIONS +While ChPT is designed for phenomena where momentum exchange is below 1 GeV, +one must also deal with QCD at higher energy scales. There are several effective methods +proposed and used in particle physics to account for the strong interactions, in particular +for their leading effects. +They allow for adapted calculations in processes where strong +interactions are important. With the huge harvest of ever-improving experimental data over +the last decades, such methods are, in fact, necessary to explain and exploit these results as +fully as possible. In particular, they are used to uncover a possible still more fundamental +theory than the standard model. +Characteristic for these situations is the presence of (two) very different scales, m1 ≪ m2, +that are relevant for the processes considered. Then, typically, either an expansion in the +small quantity m1/m2 is possible, or there are large logarithms of the form log(m1/m2) +originating in loops, see eq. (13) for details. +At present, the study the weak interactions and possibly other fundamental physics in- +volves three important energy scales: (1) The weak scale, MW is of the order of 100 GeV, +27 + +(2) the mass scale of the heavy quarks b and c (several GeV), and (3) the QCD scale ΛQCD +of about ∼ 1/3 GeV where the confinement effects set in. +At MW, the strong coupling constant αs is about ∼ 0.118, and the strong interactions +are perturbative (asymptotic freedom). For the heavy quark mass scale, αs is about 0.25. +This still allows for perturbative calculations, but their precision is limited. While for the b +quark mass, this treatment seems appropriate, the scale of the charm quark offers substantial +difficulties. Even more involved is the situation for the strange quarks; that is the physics +of kaons. We note, however, that because of the ‘Cabibbo suppression’, the decays of the b +and s are easier to study than those of the c quarks. For recent and updated overviews, see +refs. [157, 158] and references therein, or refs. [159–161] for charm. +We note that the methods to be described are primarily used to analyze and calculate +the effects that the strong interactions have on investigations of fundamental parameters +and theories, such as the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Of +course, there are still properties of the strong interactions themselves and it is interesting to +understand them, for instance, the spectrum and decay width of the charm quark systems. +Recent progress in this sector can be found in ref. [162, 163]. +A. +Extended Effective weak Theory +This methodology was put forward after the discovery of asymptotic freedom and the +realization that QCD, in fact, allows for perturbative calculations. It is used mainly to +investigate weak interaction processes of the heavy quarks b and c, but also the (weak) +decays of the s where it was first applied. It is an extension of the original 4-Fermi theory +and allows to include loops of the electroweak and strong interactions in a systematic way. +In particular, the strong interaction effects can be calculated reliably in the interval between +the weak scale MW and the mass of the heavy quarks, thereby taking into account the large +logarithms. Work on this began in the mid-seventies. Shifman, Altarelli, Cabibbo, Maiani, +Petronzio, Ellis, Gaillard, Lee, Gilman, Wise, and Buras are but a few that have made +important contributions and perfected the theory. For some of the original literature, see +the refs. [164–166]. We will give only a rudimentary introduction for many details of this +advanced, by now standard subject; see the book by Buras [167], which offers an in-depth +and updated treatment; and for an even more recent update, we refer to ref. [157, 168]. +28 + +The basic idea is that at energies below MW, the dynamical fields are the quarks (except +the top quark), gluons, and photons (or other light, undiscovered particles). +Thus the +weak Hamiltonian operator ˆO can be written as a series of operators consisting of the +quark fields of interest, gluons, and photons with increasing powers of 1/MW; in reality, +the important power is 1/M 2 +W. These operators must satisfy the symmetries required for +the process at hand and are usually ordered according to increasing orders of 1/M 2 +W +4. For +consistency, all operators that can contribute to the process at the desired order in the +strong and electromagnetic coupling constant must be considered. This implies that not +only the original left-left Four-Fermi operator (W-exchange) is present, but several others +are generated through loop corrections. A famous example is the so-called penguin operator +(See Fig 2). +FIG. 2: Penguin diagram contributing to B → Xsγ. +For instance, the operator for the decay B → Xsγ (Xs denotes an inclusive hadronic +state with the total strangeness of one) takes the form: +Heff(b → sγ) = −4GF +√ +2 V ∗ +tsVtb +� +6 +� +i=1 +Ci (µb) Qi + C7γ (µb) Q7γ + C8G (µb) Q8G +� +, +(55) +where the ‘magnetic’ penguin operators in the above are given by: +Q7γ = +e +16π2mb¯sασµvPRbαFµν, +Q8G = +gs +16π2mb¯sασµvPRta +αβbβGa +µv . +(56) +4 In cases where the top quark is important, there are also inverse powers of top quark mass +29 + +6 +S +M +u,c +688888 +,9,Z +8888 +u,cMore details about these equations can be found in the book of Buras [167]. Here, the +operators Q1...Q6 are four-Fermi operators. There are six instead of only one because gluon +exchanges rearrange the color order5. The coupling constants (as well as the quark field +operators) depend on the scale µ (see eq. (13)). The relevant scale for the decay at hand +is of the order of mb. On the other hand, the constants of the effective Hamiltonian can +be perturbatively calculated at the high scale, MW. Because of (weak and electromagnetic) +loops, there can be more contributing operators beyond the simple 4−Fermi interaction at +the scale mb. To connect the two scales, the renormalization group is employed. This leads +to a systematic expansion in the strong and electromagnetic coupling constants and the +summing up of the large logarithms log(mb/mW). This procedure has led to a (almost) +complete understanding of the weak parameters (such as the parameters of the Cabibbo- +Kobayashi-Maskawa matrix) and, in particular, an understanding of CP violation. The +status of deviations from the standard model in the heavy flavor sector can be found in +ref. [168]. For a detailed description of the method and the results obtained, see ref. [167]. +Note that this method best applies to inclusive hadronic decay products (that is why in the +above case, the final state is Xs, rather than an exclusive state, such as Kπ). +B. +Heavy Quark Effective Theory +While the effective weak theory described above pertains to the energy interval mb−MW, +the heavy quark effective theory, HQET, deals with scales below mb in processes involving +b quarks, such as the B-meson. Since the typical momenta inside a QCD bound state are +of the order of the strong scale ΛQCD, which is much smaller than mb, the b quark is only +lightly ‘shaken’ and can therefore be considered at rest in a first approximation. Therefore, +for an arbitrary heavy quark Q, we write the momentum of the quark as: +pµ = mQvµ + kµ +(57) +where v is the four-velocity of the hadron containing the heavy quark, and k is of the order +of ΛQCD, and thus much smaller than mQ. This decomposition allows, similar to the well- +known treatment in atomic physics, to divide the spinor into a dominant ‘upper’ component +5 this is indeed the crucial point of using an effective theory in that all operators consistent with the +symmetries must be included. +30 + +and a ‘lower’ one which is suppressed by 1/mQ. Thus, the idea is to construct an effective +theory in which the upper component(hv(x)) is dynamical, and the lower one(Hv(x)) is +integrated out. This can be achieved by suitable projections of the Q quark spinor [169]: +Ψ(x) = e−imQv·x [hv(x) + Hv(x)] . +(58) +The upper and lower component is obtained by the relation: +hv(x) = eimQv·x1 + /v +2 +Ψ(x) , +(59) +Hv(x) = eimQv·x1 − /v +2 +Ψ(x) , +(60) +and in the case of heavy antiquark, the substitution of v → −v is made. +Indeed, for +1/mQ → 0, the small the component can be integrated out [170–172] and the theory has an +extra spin-symmetry. The leading order (in 1/mQ) Lagrangian has the form: +Leff = ¯hviv · Dhv + Llight +(61) +and other terms involving the heavy quark field are rearranged as an expansion in 1/MQ, +and the Lagrangian for light degrees of freedom (quarks and gluons) is given by: +Llight = −1 +4Tr (GµνGµν) + +� +q +Ψq +� +i /D − mq +� +Ψq . +(62) +This formalism has been extensively used in the literature to extract the CKM elements +(|Vcb|, |Vub|, heavy flavor sum rules, and the description of heavy hadron decays. +More +details can be found in refs. [173–178]. +C. +NRQCD and pNRQCD +The heavy quark expansion used above is not suitable to describe a meson with two heavy +quarks (like charmonium or the Υ). In HQET, the kinetic energy is a 1/mQ effect and is +taken as a perturbation. But for a bound state, it plays an important role in balancing the +potential energy and, therefore, should be present at leading order. The necessary formalism +was provided by Bodwin, Braaten and Lepage in ref. [179] and is known as NRQCD. Such +systems also have additional scales, such as relative momenta p ≃ mv(soft) and a kinetic +energy, Ek ≃ mv2(ultrasoft scale), constructed out of the mass of heavy quark (M) and +31 + +its velocity (v ∼ αs << 1). For the bottomonium system, v2 ∼ 0.1 and for charmonium +systems, v2 ∼ 0.3. The hierarchy scales in the system are as follows: +mq(hard) ≫ mQv ≫ mQv2 . +(63) +The Lagrangian is expressed as an expansion in mQv/mQ and (mQv2/mQ), and at leading +order in 1/mQ, it has the following form: +LNRQCD = ψ† +� +iD0 + +⃗D2 +2M +� +ψ + χ† +� +iD0 − +⃗D2 +2M +� +χ + Llight +(64) +where iD0 = i∂0 − gA0 and ψ(χ) is the Pauli spinor field of fermion (antifermion). It should +be noted that the presence of the two dynamical soft and ultrasoft scales can complicate +the calculations and interfere with the power-counting and the non-perturbative effects. +The NRQCD is numerous applications in the threshold production of top-quark pairs in +electron-positron annihilation, spectroscopy of heavy charmonium and bottomonium bound +states [180], determination of heavy quark masses, strong coupling constant, and in the +understanding of the vacuum structure etc. A modified version of the NRQCD has been +recently proposed in ref. [181, 182] for the production of the J/Ψ, Ψ′, and χc. For more +details, we refer to ref. [183, 184]. +Another interesting system that can be constructed out of NRQCD is the potential +NRQCD (pNRQCD) [185–187]. It is obtained by integrating out the soft degrees of freedom. +The leading order in 1/mQ and multipole expansion in r, the Lagrangian has the following +form: +L0 +pNRQCD = Tr +� +S† � +i∂0 − V (0) +s +(r) +� +S + O† (iD0 − Vo(r)) O +� +− 1 +4F a +µνF µνa +where S and O are the singlet and octet fields. The resulting EFT has a resemblance to the +Sch¨odinger equation as the matching coefficients Vi(r) play the role of the potential between +the heavy quark. The equation of motion for the singlet case is : +i∂0S = +�p2 +m − V (0) +s +(r) +� +S +(65) +and depending on which scale is closer to ΛQCD, different versions of pNRQCD (strongly or +weakly coupled) are used for quarkonium. When there is no other scale between the soft +and ultrasoft scales known as weakly coupled pNRQCD, the leading order static potentials +32 + +have the form: +V (0) +s += −CF +αVs(r) +r +, +V (0) +o += +�CA +2 − CF +� αVo(r) +r +(66) +and Vs/o(r) has a perturbative expansion in the strong coupling constant. These potentials +have now been computed numerically to there-loop in refs. [188, 189] and analytically in +ref. [190]. Some ultrasoft contributions to static energy in the weak coupling limit are already +known to O(α4 +s) [191] and two of us have given Pad´e prediction for O (α4 +s) term to Vs(r) in +ref. [192]. The QCD static potential has been a very useful quantity in the determination +of the strong coupling constant αs as it can be calculated to very good precision on the +lattice [193]. Recent updates of αs from static energy can be found in ref. [194–196] and +references therein. There are several packages available in the literature that can be used to +study non-relativistic systems. Recently, Brambilla et al. [197] have published the publicly +available Mathematica-based package FeynOnium that can be used to study the NREFTs +to one loop. Another useful package relevant to studying the threshold quarkonium system +is QQbar threshold by Beneke et al. [198]. A detailed review on NRQCD, pNRQCD, and +a description of quarkonia from these EFTs can be found in refs. [183, 199]. +D. +Heavy-light mesons +There exist some mesonic states with heavy and light quarks, and one may ask how to +combine HQET and ChPT to study their production and decay. +This issue has indeed +been taken up by Burdmann and Donoghue [200], Wise [201] and Yan et al. [202], and is +now known as heavy meson ChPT(HMChPT). It is formulated on the fact that the mass +difference between the heavy meson and its excited state scales as ∼ 1/MQ, which can be of +the order of a few MeVs for heavy mesons such as B meson. Heavy quark symmetry relates +to the couplings of the B and B∗, and it also relates to other mesons such as D as long as +the charm quark can be treated as heavy. A meson with one heavy quark can be labeled +by the light quark spin jl and states with spin jl ± 1 +2 are degenerate due to heavy quark +spin symmetry. Due to this fact, a consistent description of a heavy light system requires +an excited state such as B∗ for B systems, as their production will require much less energy +than the pion mass. Since the energy involved are less than the pion mass, an extension to +the chiral framework can be merged with the HQET. +33 + +Degenerate triplets of spin-zero mesons Pa (a = u, d, s) and spin-one meson P ∗ +a triplets +are obtained by combining the spins of heavy and light quark spins using the heavy quark +spin symmetry. These fields can be used to define the 4 × 4 matrix Ha, given by: +Ha = (1 + /v) +2 +� +P ∗ +aµγµ − Paγ5 +� +(67) +where P ∗ +aµ is an operator that destroys a P∗a meson with velocity v and satisfies: +vµP ∗ +aµ = 0 . +(68) +Defining Ha as: +Ha ≡ γ0H† +aγ0 = +� +P ∗† +aµγµ + P † +aγ5 +� (1 + /v) +2 +, +(69) +then most general leading order Lagrangian to describe the strong interaction between +pseudo-Goldstone boson with heavy meson is given by: +L = − iTr +� +Hv · ∂H +� ++ F 2 +π +8 Tr +� +∂µU∂µU †� ++ i +2Tr +� +Hvµ � +U †∂µ + U∂µU †� +H +� ++ ig +2 Tr +� +Hγνγ5 +� +U †∂ν − U∂νU †� +H +� +− ∆ +8 Tr +� +HσµνHσµν +� ++ . . . , +(70) +where ∆ = mP ∗ − mP, g is the axial coupling constant and field U is defined in eq. (3) and +ellipses denote the higher order terms, and complete Lagrangian to one loop can be found +in ref. [203]. The Lagrangian in eq. (70) is consistent with the SU(3)L × SU(3)R, Lorentz +transformations, and the heavy quark symmetry SU(2)v. +The leading order Lagrangian in eq. (70) can be used to predict the P ∗ → Pπ transitions. +Such transitions for B meson are kinematically forbidden however, for the D system, it has +the form: +Γ +� +D∗+ → D0π+� += +g2 +6πF 2 +π +|⃗pπ|3 , +(71) +Γ +� +D∗+ → D+π−� += Γ +� +D∗0 → D0π0� += +g2 +12πF 2 +π +|⃗pπ|3 . +(72) +Using experimental input for these decays, the axial coupling g can be fixed. There are many +charmed states which have gained attention over the years as they can not be described by +the traditional methods, which require their own review. For details on the applications and +status of heavy light systems, we refer to refs. [46, 163, 203–206] and references therein. +34 + +E. +Soft Collinear Effective Theory, SCET +The (light) decay products of heavy (B) mesons typically have a large momentum of +order mb, in comparison to ΛQCD, for instance, in the decay B → Kπ. The quarks in +those fast-moving light mesons are typically on the light cone, collinear with the meson that +contains them. Deviations from collinearity are caused by QCD interactions and small, of +the order ΛQCD/mb in case of the decay of a B meson. In this situation, the effective theory is +constructed ‘around’ those collinear quarks. In an early attempt, Dugan and Grinstein [207] +constructed a ‘large energy effective theory (LEET)’, to describe the interaction of the high- +energy quarks (E around mb) with the soft gluons (energy about ΛQCD in an expansion in +q/E. Since the hadrons also contain collinear gluons, a complete theory must include them +too. In refs. [208, 209], Bauer, Fleming, Luke, Pirjol, and Stewart presented a soft collinear +effective theory (SCET). A comprehensive description of SCET is found in the book [210], +for more recent results and developments, see for instance, refs. [211, 212]. We note that +SCET, while originally applied to heavy meson decays, perfectly fits the needs of high energy +(jet) physics that is a main part of LHC-physics, see for instance in ref. [213]. +To account for the dominance of the collinear particles, light cone coordinates p = +(p+, p⊥, p−) are used. The coordinate basis for motion in the z direction is chosen to be +nµ = (1, 0, 0, 1), nµ = (1, 0, 0, 1), with n · n = 2 (the coordinates are (t,x,y,z)). The small +parameter which characterizes the perpendicular components is λ = p⊥/n · p. The momenta +are decomposed according to +pµ = +˜p +� +�� +� +n · pnµ +2 + (p⊥)µ +n · pnµ +2 = O(λ0) + O(λ1) + O(λ2) . +(73) +This decomposition into large and small components to construct an effective field the theory +looks similar to the method of regions, where the different momentum regions are first +separated and then treated differently. However, the effective field theory approach allows +for systematically including the running of operators or power corrections. The construction +of the effective theory then is similar to the theories discussed. SCET also involves three +scales like NRQCD. The quantity ˜p now acts as the label to the fields, and the large momenta +˜p are removed by defining: +ψ(x) = +� +˜p +ψn,˜p +(74) +35 + +and the derivative ∂µ on fields ψn,p gives dynamical contributions of O(λ2) like in NRQCD. +Particle moving along nµ have two large components and small components denoted by ξn,p +and ξn,p respectively. These are related to ψn,p by the following relations: +ξn,p = /n/n +4 ψn,p, +ξn,p = /n/n +4 ψn,p +(75) +satisfying the relations: +/n/n +4 ξn,p = ξn,p, +/nξn,p = 0 , +(76) +/n/n +4 ξn,p = ξn,p, +/nξn,p = 0 . +(77) +The Lagrangian constructed with the above discussion has the form: +LSCET = +� +p,p′ +� +ξn,p′ /n +2 (in · D) ξn,p + ξn,p′ /n +2 (n · p + in · D) ξn,p ++ ξn,p′ +� +/p⊥ + i /D⊥ +� +ξn,p + ξn,p′ +� +/p⊥ + i /D⊥ +� +ξn,p +� +, +(78) +where Dµ = ∂µ−igT aAa +µ is covariant derivative. More details can be found in refs. [209, 210]. +SCET is applied to a large variety of processes with collinear high-energy particles, not only +in decays of heavy mesons but increasingly in very high-energy processes such as at the +LHC. For the newest developments, see the latest SCET conference [214]. +IX. +EFFECTIVE THEORIES BEYOND THE STANDARD MODEL +A. +The Standard Model Effective Theory +So far, the standard model has proven to be essentially faultless; apart from a few cos- +mological phenomena (dark matter, matter-antimatter ratio,...) and alleged anomalies in +B meson decay [215], it reproduces all experimental results very precisely. However, it is +widely believed that there are more fundamental interactions with a typical energy scale +Λ which seems considerably higher than MW, as indicated by the absence of discoveries of +very heavy particles beyond the top quark, the W and the Z bosons and the Higgs particle +at LHC. This is reminiscent of the early days of the weak interactions when the 4-Fermi +theory HW ∼ GF(qLγµqL)(qLγµqL) was put forward and the W-boson entered indirectly +only through the Fermi constant GF ∼ 1/M 2 +W and its symmetry properties. +36 + +Similarly, in order to parameterize physics beyond the standard model, originating from +physics at a scale Λ, one considers effective operators made up of the standard model particles +(including the Higgs boson and the W, Z bosons with a coupling proportional to powers of +1/Λ, +L = +� +n +1 +ΛnOn , +(79) +where the operators On have dimension 4 + n (each On consists of many distinct operators, +each with an unknown coupling) and are composed out of standard model fields such that the +total operator is invariant under SU(3)×SU(2)×U(1). At the lowest order, (1/Λ)0, we have +just the standard model. At next order, (1/Λ)1 there is one operator [216, 217] which violates +lepton number. At the next order, (1/Λ)2, there are nearly 100 operators, see ref. [218, 219]. +In principle, the task is to determine the unknown couplings by comparing them to suitable +experimental results. Given a large number of such couplings, this is a difficult task. This +is a very active field, with several strategies to overcome the difficulties. See ref. [220] for +a comprehensive overview. For the newest developments, see the proceedings of the 2019 +conference on SMEFT-tools [221]. This conference will again be held in 2022 [222]. +B. +Quantum Gravity +One of the biggest - if not the biggest - unsolved problems in theoretical physics is how +to quantize gravity. A modest but important step can be achieved if general relativity is +viewed as a field theory. The metric gµν is promoted as the field, and the effective field +theory has the general coordinate invariance of general relativity (GR). Using the fact that +the connection, defined as: +Γαβ +λ = gλσ +2 +� +∂αgβσ + ∂βgασ − ∂σgαβ +� +, +(80) +has one derivative, and the curvature, defined in terms of the Riemann tensor (Rµναβ), given +by: +Rµνα +β = ∂µΓνα +β − ∂νΓµα +β − Γµλ +βΓνα +λ − Γνλ +βΓµα +λ , +(81) +has two derivatives. The two derivatives present in the Riemann tensor correspond to the +powers of energy when evaluated in terms of the matrix elements. It is important to note +37 + +that the various contractions of the Riemann tensor are coordinate invariant, which is also +the symmetry of the low energy theory. Hence, the Lagrangian can be constructed out of +various possible contractions of the Riemann tensor, and energy expansion can be naturally +constructed including more and more contractions of the Riemann tensor. In particular, +Donoghue [223–225] has shown how a possible extension of general relativity to a theory +with quantum degrees of freedom results naturally in an expansion in the theory of gravity, +which includes as the +S = +� +d4x√g +� +Λ + 2 +κ2R + c1R2 + c2RµνRµν + · · · + Lmatter +� +, +(82) +where Λ is cosmological constant, R = gµνRµν and Rµν = Rµναα are known as the Ricci scalar +and the Ricci tensor, respectively. So far, this theory has found only limited applications, +but it may be a guide to correct quantum gravity. A SCET inspired treatment of quantum +gravity can be found in refs. [226–228]. For more details, we refer to refs. [223–225, 229]. +X. +MISCELLANEOUS ITEMS +In this section, we cover a range of mostly technical topics which both feed into effective +theories and in whose developments effective theories have played a role. +Feynman Integral Methods for Effective Field Theories +When calculating Feynman diagrams, one integrates overall kinematically allowed values +of the internal momenta of quarks and gluons. There has been considerable effort in evaluat- +ing them to very high orders for precision physics. Many computational as well as theoretical +tools have been developed over the years. Many of these developments can be found in the +recent book of Weinzierl [230]. In a theory like QCD where the interaction of a gluon with +quarks or gluon at 1 GeV is very different for an energy scale of several GeVs. The diagram- +matic evaluation of any process gets more complicated in these multiple-parameter theories +when one goes to higher orders due to the presence of the various scales(masses and mo- +menta) in the loops. It is, therefore, reasonable to divide the integrand into regions and use +different rules for the various region. The method of regions [231] is one of the very useful +strategies for evaluating Feynman integrals in specific kinematic limits of the mass and mo- +menta. In this technique, the integrand of Feynman diagrams is expanded by identifying the +38 + +scaling behavior of the ratios of masses and momenta. Although it is not rigorously proven +to be correct, it appears to work in all known instances [232]. Interestingly, the expansion +of Feynman diagrams in various regions corresponds to an EFT in the asymptotic limits +of the parameters. In some cases, these regions may overlap and need to be systematically +subtracted (zero-bin subtraction) following the procedure of Jantzen [233]. Application of +this method in the ChPT was first made by Kaiser and Kaiser and Schweizer [234]. Now, +there exist well-dedicated codes asy.m [235], asy2.m [236] and ASPIRE algorithm [237] that +can be used to study multi-scale Feynman integrals. For more details, we refer to [232, 233] +and references therein. +The Mellin-Barnes (MB) technique is also one of the most commonly used techniques +in the literature for the analytic evaluation of the Feynman integrals and has been recently +used in the context of ChPT in refs. [238–240]. The two-loop sunset diagrams play a key role +in the analytic representation of the masses and decay constants of the pion, kaon, and η- +mesons. These diagrams are calculated using the MB technique in ref. [241–245] and further +used in evaluating some three-loop Feynman diagrams relevant for the QED corrections +to g − 2 of charged leptons in ref. [246]. The MB technique yields the final expression in +terms of generalized hypergeometric functions (pFq) and Kamp´e de F´eriet (KdF) series. +Recently, a geometric method using conic hulls is developed in ref. [239] and implemented in +the Mathematica package MBConicHulls.wl which allows systematic computation of certain +N-fold MB integral, and in the case of convergent series case, one can also find the master +series which is useful for numerical studies. This technique is used to solve certain non-trivial +conformal Feynman integrals in refs. [238, 240]. +These ChPT-inspired studies have immensely contributed to finding the new analytic +continuations of the Appell Function F4 in terms of the 2F1 in ref. [247]. These multivariate +hypergeometric functions and their properties, domain of convergences, and linear transfor- +mations are studied in mathematics literature [248–251]. One of the strategies to find the +analytic continuation of a multivariate hypergeometric function is to use the known analytic +continuations of hypergeometric functions with a lower number of variables. +The linear +transformation formulae of the one variable Gauss 2F1 function are used to find the analytic +continuations of the double variable Appell F1 in [252]. This process of finding analytic +continuations of hypergeometric series of more than one variable is automated in the Math- +ematica package Olsson.wl [253]. The package can also find the domain of convergence of +39 + +only the double-variable hypergeometric functions. The analytic continuations of the Appell +F2 functions are found using the same technique and are used to construct the numerical +package AppellF2.wl [254]. It can find the numerical value of the Appell F2 function for +real values of its arguments (i.e. +x, y) and general complex values of the Pochhammer +parameters. Some new analytic continuations of Appell F4 are obtained using the known +quadratic transformation of the Gauss 2F1 function [247]. The linear transformations of the +three variable Srivastava HC function are also found [255]. +Chiral Lagrangians and Ricci Flows +Right from the early days, the non-linear sigma model provided the fundamental building +block for the realization of chiral symmetry. Whereas for the simplest purposes, these were +based on SU(2) × SU(2) or alternatively on SO(4) general theorems for the realization of +these symmetries and the Goldstone phenomenon were established for a general group G +breaking down to H by Coleman, Wess and Zumino [16], and Coleman, Callan, Wess, and +Zumino [17]. Friedan [256, 257] studied the non-linear sigma model in 2+ϵ dimension where +fields ϕ are defined on a manifold M and the coupling are is determined by a Riemannian +metric on M. The action has the form: +S(ϕ) = Λϵ +� +dx1 +2T −1gij(ϕ(x))∂µϕi(x)∂µϕj(x) +(83) +where λ is short distance cutoff, T −1gij(ϕ(x)) is dimensionless coupling is Riemannian metric +on M. The renormalization group running of this metric at two-loop was found to be: +Λ−1 +∂ +∂Λ−1gij = βij +� +T −1g +� += −ϵ T −1gij + Rij + 1 +2T (RiklnRjkln) + O +� +T 2� +. +(84) +and the one-loop β-function was already calculated by Ecker and Honerkamp [258]. The +running of coupling in eq. (84) is known as Ricci flow introduced by Hamilton [259]. The +ideas developed by Hamilton were an attempt to solve the long-standing problem of Poincar´e +conjecture. Perelman published the proof of this conjecture in the three articles [260–262] +in 2002-3 where Ricci flow played a key role. A detailed explanation of Perelman’s proof +was published by Morgan and Tian [263] and Huai-Dong Cao, Xi-Ping Zhu [264]. +40 + +Lattice QCD +The non-renormalizable nature of the ChPT results in the increasing numbers of LECs +as one goes to higher orders and has to be fixed by inputs from other sources. Most of the +LECs can be determined from the experiments or estimated using a large Nc limit of QCD +or low energy description of strong interactions. Lattice QCD is one of the candidates at +very low energy and has provided numerous inputs and cross-checks over the years. Lattice +calculations are performed on finite lattice spacing, finite volume, and unphysical quark +masses, and ChPT provides a way to crosscheck, analyze and quantify these effects in the +continuum limit. For the brief introduction of the interplay of lattice QCD and the ChPT, +we refer to Shanahan [265] and references therein. +Among these topics, proton charge radius and the muon g − 2 anomaly has been of +constant interest over the year for their potential to provide hints to new physics beyond +the standard model at low energies. Issue of the small charge radius of proton came into +the picture in 2010 when the existing value of charge radius rp = 0.8775(51) fm from +CODATA [266] world average using the spectroscopic method and electron-proton scattering +was found to be larger than the one obtained from muonic hydrogen rp = 0.84184(67) fm +by Pohl et.al. [267]. Pohl’s result later confirmed by CREMA collaboration [268] with rp = +0.84087(39)fm. There are various theoretical models for new physics were also studied, and +some future experiments are also proposed to get more precise results, but the issue is now +believed to be settled and we refer to a very recent review by Gao and Vanderhaeghen [269] +and Hammer, Meißner [270], Bernauer [271], Peset et. al. [272] and references therein for +further details. Lattice determinations of form factors are also extensively performed and +the results are compatible with existing literature. For details of lattice determination of +proton charge radius, we refer to Ishikawa et.al. [273] and references therein for details. +Lattice methods themselves require their own review to explain various methods developed +over the years to extract the parameters of strong interaction. +For details, we refer to +Golterman [274], FLAG reviews [24, 275, 276]. +41 + +OUTLOOK +Chiral perturbation theory, ChPT, has proven very fruitful over the last 50 years. It +has provided ample predictions for understanding a great number of experimental results +involving the pseudoscalar mesons. It is still being refined to adapt to new theoretical and +experimental results, and there are still many results waiting to be improved. ChPT has +helped to understand field theory more generally; in particular, it has shed some light on +the limited role of renormalizable theories. +This direction of research is far from being +at its end, and for instance, work devoted to non-renormalizable theories will very likely +yield many interesting results [115] Thirdly ChPT has also become a valuable tool to be +used in circumstances not thought to be in its realm. +For instance, the calculation of +the anomalous magnetic moment of the muon - one of the crucial calculations in particle +physics - has benefited from results obtained by ChPT. 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C 80 (2020) no.2, 113 +[arXiv:1902.08191]. +56 + diff --git a/fNE3T4oBgHgl3EQffQos/content/tmp_files/load_file.txt b/fNE3T4oBgHgl3EQffQos/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5cfee56d12cf8d9eb1716f3520de54102d71c62b --- /dev/null +++ b/fNE3T4oBgHgl3EQffQos/content/tmp_files/load_file.txt @@ -0,0 +1,2646 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf,len=2645 +page_content='Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Ananthanarayan,1, ∗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Alam Khan,1, † and Daniel Wyler2, ‡ 1Centre For High Energy Physics, Indian Institute of Science Bangalore 560 012, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 2Institute for Theoretical Physics University of Z¨urich Winterthurerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 190, CH 8057 Z¨urich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='04550v1 [hep-ph] 11 Jan 2023 Abstract The pseudoscalar particles pions, kaons and the η-particle are considerably lighter than the other hadrons such as protons or neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Their lightness was understood as a consequence of approximate chiral symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This led to current algebra, a way to express the relations imposed by the symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It was realized by Weinberg that because of their low mass, it is possible to formulate a purely pionic (effective) field theory at experimental energies, which carries all information on the (non-perturbative) dynamics, symmetries, and their spontaneous breaking of quantum chromodynamics (QCD) and allows for systematic calculations of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this review, we trace these developments and present recent activities in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We make the connection to other effective theories, more generally introduced by Wilson, as approximate field theories at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Indeed, principles and paradigms introduced first for pions have become ubiquitous in particle physics and the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Lastly, we turn to the latest development where the present (fundamental) standard model itself is considered as an effective field theory of a - yet to be formulated - even more fundamental theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We also discuss important techniques that were developed in order to turn chiral perturbation theory into a predictive framework and briefly review some connections between lattice QCD and chiral perturbation theory (ChPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' CONTENTS Preamble 4 A personal note 5 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Introduction 5 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The Chiral Lagrangian 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Extensions of chiral perturbation theory 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The η′ 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Vector mesons 14 ∗ anant@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='in † mohdakbar@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='in ‡ wyler@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='ch 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Baryons 15 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Two and Three Body Rescattering 15 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Generalized renormalization group and large chiral logarithms 20 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Weak interactions of pseudoscalar mesons 22 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Selected applications of Chiral Perturbative Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 25 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Other Effective Theories for the strong interactions 27 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Extended Effective weak Theory 28 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Heavy Quark Effective Theory 30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' NRQCD and pNRQCD 31 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Heavy-light mesons 33 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Soft Collinear Effective Theory, SCET 35 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Effective theories beyond the standard model 36 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The Standard Model Effective Theory 36 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Quantum Gravity 37 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Miscellaneous items 38 Feynman Integral Methods for Effective Field Theories 38 Chiral Lagrangians and Ricci Flows 40 Lattice QCD 41 Outlook 42 Acknowledgment 42 References 43 3 “And so the question naturally arose, is there a way of avoiding the machinery of current algebra by just writing down a field theory that would automatically produce the same results with much greater ease and perhaps physical clarity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Because after all in using current algebra one had to always wave one’s hands and make assumptions about the smoothness of matrix elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' whereas if you could get these results from Feynman diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' you could see what the singularity structure of the matrix elements was and make only those smoothness assumptions that were consistent with that.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Steven Weinberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 2020[1] PREAMBLE We dedicate this article to Steven Weinberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' a great physicist who influenced in many ways the field of elementary particle physics for decades and is one of the authors of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Apart from his great achievements in research, his textbooks which became classics also testify to his outstanding teaching capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' He was awarded the 1979 Nobel Prize in physics along with Sheldon Glashow and Abdus Salam for the electro-weak model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In addition, Weinberg pioneered the study of quantum field theories, behavior of Green functions at asymptotic energies, symmetries in field theories, Goldstone mechanism, current algebra, pion physics and effective field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Weinberg has explained his philosophy in terms of phenomenological Lagrangians [2], which proved to be the cornerstone for successful developments in precision pion physics until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This concept of an effective Lagrangian, that is, the low energy (large distance) manifestation of a fundamental theory, was also developed by Ken Wilson [3] (Nobel prize 1982) quite generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this picture, short- distance degrees of freedom are systematically ‘integrated out’ and appear only as coefficients of a theory with long-distance degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Weinberg’s and Wilson’s concepts are at the base of today’s understanding of physics systems with very different energy scales, such as in particle physics where the range goes over more than 20 orders of magnitudes [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this review, we aim to showcase the developments and richness of effective theories, report on current progress, and encourage further work by pointing out where it is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We mention important early work quite comprehensively but are more anecdotal in refer- encing newer work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' the interested reader should be able to navigate it from the references 4 given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We have striven to bring under one umbrella several topics that have been separately reviewed for various sub-communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Our hope is that the present review will find a read- ership that will encompass all of the working particle physics community, experimentalists and theorists likewise, who wish to get a flavor of what has been going on under the rubric of chiral perturbation theory and a glimpse of other effective field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The subject of chi- ral perturbation has grown immensely based on rather technical and detailed computations, in this review, we wish not to burden the text with too many equations but try to explain the physical concepts and point the interested reader to more comprehensive reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In partic- ular, we illustrate how the general principles of analyticity, unitarity, and crossing (through dispersion relations and the analysis of experimental data based on them), can be combined with the scattering amplitudes arising in chiral perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This marriage has in fact led to sufficient accuracy, thereby providing for testing the standard model at requisite levels of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In addition, we have also given references to the various packages used in the literature which readers can find interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A PERSONAL NOTE Both DW and BA have been working for over three decades on the subjects discussed here, in particular in pion physics and Chiral Perturbation Theory (ChPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We are pleased to share the important lessons learned during the many years of development in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' DW was happy to contribute and to assist BA who has been a longtime friend and a gate opener to India and its culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' INTRODUCTION Many physical systems look very different when probed at different length scales or with different energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While ordinary matter appears to the eye in an incredibly rich diversity of forms and textures, at the atomic scale, made for instance visible by scanning microscope techniques, all one sees are atoms that are quite similar in different materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Thermody- namics, the phenomenological description of many systems, can be viewed as an effective ‘leftover’ of the microscopic theory of statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Such large differences appear 5 in many physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In elementary particle physics, it is in the realm of strong inter- actions where this can be studied particularly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At experimental energies beyond, say, several GeV, the relevant picture is that of the simple SU(3) gauge theory of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' But at lower energies, the complicated interactions of pions and nucleons dominate and there is no obvious trace of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While we consider QCD the fundamental theory, the interactions of pions and nucleons are described by an effective (low energy) theory, called chiral perturba- tion theory, ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' So, how does one connect these two seemingly different manifestations of the same interactions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The key is to find properties of QCD that remain manifest also in ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this case, it turned out that the crucial property is chiral symmetry SU(3) × SU(3) with spontaneous breaking and a small explicit breaking term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While this symmetry is easily gleaned from the fundamental Lagrangian of QCD, it is far less obvious at the effective level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It took many years, from the late 1950s on, to consolidate the effects of that symmetry, and many physicists are associated with this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since these developments are a fascinating part of the history of particle physics, we will summarize some of these ideas below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' All the results obtained were cast into a bona fide field theory using external field tech- niques by Gasser and Leutwyler in the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Apart from a rigorous formulation of the effective Lagrangian, they gave a complete one-loop treatment of the effective Lagrangian, also including the other pseudoscalar mesons of the eightfold way, that is the kaons and the η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since then, many developments have taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Systematic two-loop calculations increased to quality of the predictions substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The role of nucleons and vector mesons was investigated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' general techniques such as dispersion relations, functional analysis meth- ods, and rescattering theory advanced the precision of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Other methods, such as lattice gauge theory, furnished important input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' All of these represent diverse and rich activities in physics where each would require a review article in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Further- more, the success of chiral perturbation theory encouraged the development of many other effective theories in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' By identifying high energy and low energy degrees of freedom and using Wilson’s procedure to integrate out the high energy modes, it is possible to arrive at an effective theory for the low energy modes in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Some of the recent theories are the heavy quark effective theory (HQET) and the soft-collinear effective the- ory (SCET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For instance, HQET is based on the observation that in heavy quark physics (mostly b quarks but also charmed quarks) mesons, in the limit of large (bigger than about 6 1 GeV) heavy quark mass mq, the relevant physics can be reliably described in a power series in (mq)−1 where the first term is independent of (mq) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While these theories are designed to understand the non-perturbative dynamics of QCD better, more recently, the idea that even the ‘fundamental’ standard model is but the ‘low’ energy effective manifestation of a more basic theory that would reveal itself at very high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This idea is known as the standard model effective theory (SMEFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A further step is to view the theory of gravity, Einstein’s general relativity, as an effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This is particularly interesting for attempts to turn gravity into a quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The contents of this review are as follows: The next section II gives an overview of chiral perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We recall the basic theory of QCD and describe the construction of the effective Lagrangian, following Gasser Leutwyler [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Several general observations, sometimes personal, are mixed into the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In section III, we show how to include particles beyond the eight light pseudoscalar mesons, in particular the η′, the vector mesons (such as the ρ), and nucleons (baryons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In section IV, we consider processes where ChPT does not work well and must be improved by auxiliary methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In particular, we look here at strong two- and three-body rescattering where a substantial body of work exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In section V, we look at generalizations of the renormalization procedure for non- renormalizable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The methods that have been developed might be of general interest in going beyond ‘renormalizable’ theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In section VI we review the weak interactions of the pseudoscalar mesons, such as the decays of kaons which play an important role in the understanding of fundamental effects such as CP-violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In section VII, we show some applications which are of special importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In section VIII, we discuss effective methods to deal with the strong interactions in the higher energy regimes where ChPT does not apply (or only in certain parts of phase space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Then, in section IX, we show the newest development in understanding the standard model and gravity as effective theories of an even more fundamental theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Finally, in section X, we collect some noteworthy recent developments that are of relevance for ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 1 Recall that the reduced mass of a system of a heavy and a light particle is largely independent of the heavy mass 7 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' THE CHIRAL LAGRANGIAN After the initial work of Dashen, Weinstein, and Pagels [6–9], a breakthrough came from the observations of Weinberg [2] who argued based on the principles of quantum field the- ory and cluster decomposition and pion-pole dominance, that the lowest order effective Lagrangian could be used to compute loops whose divergences could be absorbed into the low-energy constants of higher order terms in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This was put on a firm foot- ing by studying the gauge invariance of the generating functional of the Green functions of the theory by Gasser and Leutwyler [5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A scholarly exposition is given in the Scholar- pedia article of Leutwyler [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this chapter, we review their construction of the chiral Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The foundation for this is QCD, the theory of the strong interactions which determine the behavior of the observed particles in a variety of experiments where they are influenced by ‘external fields’ or ‘sources’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These are classical objects and do not appear in loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Important examples of such external fields can be the masses of the quarks2, the weak interactions, or also experimentally realized fields like a strong electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since we know that the symmetry properties are crucial, we are interested in external fields that have well-defined transformation properties under the chiral symmetry which determines the low energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The objects one wants to calculate are the Green functions associated with the external fields, from which the physical matrix amplitudes are derived in a standard manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Thus, the fundamental Lagrangian involving the three light quark fields (q) has the form: L = L0 QCD + ¯qγµ (vµ + γ5aµ) q − ¯q (s − iγ5p) q − θ 32π2Tr � Gµν ˜Gµν� , (1) where L0 QCD = qiγµ (∂µ − iGµ) q − 1 2g2Tr (GµνGµν) , (2) and Gµν is the gluon field strength tensor and ˜Gµν = 1 2ϵµναβGαβ its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The letters vµ, aµ, s and p denote the external fields transforming as vectors, axial vectors, scalars and pseudoscalars, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' the field θ transforms in a particular non-linear way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' All these quantities are x-dependent, that is vµ = vµ(x), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The physical Greens functions 2 In the standard model this is proportional to the vacuum expectation value of the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 8 from the Lagrangian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (1) are obtained by expanding the generating function around vµ = aµ = p = 0, s = M, θ = θ0 where M is the quark mass matrix and θ0 is the vacuum angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note that s = M can always be chosen to be diagonal, with real positive elements (mu, md, ms) and an adjusted vacuum angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For more details, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The last term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (1) is odd under the CP transformation and contributes to CP violation effects (for instance the electric dipole moment of the neutron) from the strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These effects are found to be tiny, which requires the vacuum angle to be unnaturally small θ0 ≲ 10−10 [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This is the, still unresolved, ‘strong CP’ problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It has sparked many ideas, including the postulation of the axion, an interesting, but still hypothetical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Various theoretical models of axions and the experimental bounds on the couplings with other particles have been explored in the literature and these developments can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [14], for a very recent result, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Our interest is in the amplitudes at experimental particle energies below 1 GeV or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, the calculations using the formulas above would be forbiddingly difficult because at low energies, there are no free quarks (or inclusive states, such as exist at high ener- gies), but pions or kaons (the pseudoscalar mesons), or other hadrons that are complicated bound states of quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Thus, we must express the physical contents of the QCD Lagrangian in terms of these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This we shall call the effective (low-energy) chiral Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The notion is quite general: An effective Lagrangian expresses the physics in terms of the physical particles (or fields) at the energies relevant for the experiment consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The form of the chiral Lagrangian is dictated by the choice of the physical (dynamical) fields and the symmetry properties of the external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Because we are interested in low energies, one considers an expansion in energy (momentum) of the particles which, because of chiral symmetry, starts at order(p2) where p is a typical momentum of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The leading term in an energy (momentum) expansion is completely fixed due to the work of Callan, Coleman, Wess and Zumino [16, 17] have allowed us to extract several general fea- tures of the interactions of Goldstone bosons, quite independent of the knowledge of the dynamics of the strong interactions, based exclusively on the (global) symmetries of the un- derlying Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The fact that they are (approximate) Goldstone bosons already fixes their mutual interactions to be of the derivative type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Furthermore, the parametrization of the degrees of freedom encoded by the physical fields requires them to be coordinates of the coset space given by G/H, where G is the global symmetry of the Lagrangian and H is 9 the symmetry of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The broken generators of G not lying in H are precisely these Goldstone boson degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Taking into account the symmetry properties of the Goldstone bosons, a convenient parametrization where the chiral transformations are linear and ensures the derivative nature of the interactions is [18]: U ≡ ei √ 2Φ/Fπ (3) where Φ = � � � � � π0 √ 2 + η8 √ 6 π+ K+ π− − π0 √ 2 + η8 √ 6 K0 K− K0 − 2η8 √ 6 � � � � � , (4) which is unique up to the reparametrization of the Goldstone boson fields themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' As argued by Boulware and Brown [18], it is advantageous to group the pseudoscalar fields into a 3 × 3 unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The point is that the external fields should couple to such operators of the fields which transform linearly under chiral transformations in order that the interactions can be built by a conventional loop expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The results of current algebra are all captured by the effective Lagrangian: Leff = L2 + L4 + L6 + · · · (5) where L2, L4 and L6 are the terms of O (p2), O (p4) and O (p6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The leading-order term in the low energy expansion is generated by the non-linear sigma model coupled to the external fields(v, a, s, and p, in the notation of the ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [19]), L2 = F 2 π 4 ⟨DµUDµU † + χU † + χ†U⟩ , (6) where DµU = ∂µU − i(vµ + aµ)U + iU(vµ − aµ), χ = 2B (s + ip) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (7) The above reproduces the well-known Weinberg result for ππ scattering [20], which sets the scale of the chiral interaction in terms of the pion decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Without dynamic external fields, s is proportional to the masses of the quarks, which are fundamental quantities in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In fact, ChPT plays a key role in determining these quantities in terms of the pseudoscalar masses [5, 21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There is also considerable effort to determine them from lattice QCD [24] as well as using QCD sum rules [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The sum rule determinations are also a powerful tool to determine the QCD parameters, 10 and their recent applications can be found in the book of Dominguez [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A summary of the most recent determinations can be found in the PDG [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Apart from their importance as fundamental parameters, a value of mu = 0 would have solved the so-called strong CP problem (see [29] for details), but this does not seem to be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In particular, in order to go beyond the leading order, Weinberg in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [2] argued that the content of a field theory is dictated by the symmetries and analyticity, perturbative unitarity, and cluster decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Using this, he was able to predict the structure of the ππ scattering amplitude and the corresponding logarithms that would have to generate the required imaginary parts of the amplitude from the original (real-valued) tree-level ampli- tude [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While this remained a thumb rule, the systematic study required the introduction of external sources for the currents of the theory in the spirit of Julian Schwinger, who proposed to study field theory through sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Rather than being mere mathematical cu- riosities, the presence of the external sources allowed a systematic computation of one-loop generating functional through the heat-kernel technique, an established method of obtain- ing in a compact manner the generating functional rather than an equivalent yet explicit computation of Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This functional requires regularization (dimensional) and renormalization of the infinities that are generated by the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since the original Lagrangian in d = 4 is not renormalizable, the procedure generates higher derivative terms not present in the original two-derivative Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Thus new low-energy constants are introduced into the theory with corresponding β-functions, which are fixed from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Once fixed, at this order, any process of interest can be computed, which makes the theory predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This process can be continued indefinitely to any order in the loop expansion and/or the momentum expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is the convention to consider each loop to yield a new power of p2 and also to assign powers to the explicit mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The chiral Lagrangian at O(p4) [5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 10] consistent with the Lorentz invariance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' C and P symmetry is given by: L4 =L1⟨DµU †DµU⟩2 + L2⟨DµU †DνU⟩⟨DµU †DνU⟩ + L3⟨DµU †DµUDνU †DνU⟩ + L4⟨DµU †DµU⟩⟨χ†U + χU †⟩ + L5⟨DµU †DµU(χ†U + U †χ)⟩ + L6⟨χ†U + χU †⟩2 + L7⟨χ†U − χU †⟩2 + L8⟨χ†Uχ†U + χU †χU †⟩ − iL9⟨F µν R DµUDνU † + F µν L DµU †DνU⟩ + L10⟨U †F µν R UFLµν⟩ + L11⟨FRµνF µν R + FLµνF µν L ⟩ + L12⟨χ†χ⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (8) 11 where F µν R =∂µrν − ∂νrµ − i[rµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' rν] (9) F µν L =∂µlν − ∂νlµ − i[lµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' lν] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (10) At this order in the momentum expansion, there are twelve new couplings Li (also called low energy constants) that appear out of which only L3 and L7 are not divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Also, the next O (p6) order Lagrangian has been worked out, see refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [30, 31] and there are results of O (p8) [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Of course, the number of couplings increases, and thus the predictive power for smaller (higher-order) effects decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' But by choosing suitable ob- servables, one is still able to determine some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The low energy constants can be renormalized in a standard fashion and one writes: Li = Lr i(µ) + Γiλ(µ) (11) where λ(µ) = µd−4 4π2 � 1 d − 4 − 1 2 (1 + log(4π) − γE) � , (12) where the function λ comes from performing a standard one loop calculation and the co- efficients Γi coefficients were calculated in [5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The divergences in the bare coupling Li cancel with the one present in the λ(µ) resulting in renormalized couplings Lr i(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Their scale dependence is given by: Lr i(µ2) = Lr i(µ1) + Γi 16π2 log �µ1 µ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (13) The low energy constants reflect the properties of the strong interaction spectrum that has been integrated out and contribute to their numerical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For instance, in certain cases, the exchange of a single vector meson accounts for the observed values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' alternatively, there might be an axial-vector meson dominance [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Other higher angular momentum states also make contributions but are mostly numerically less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Over the last years, several papers have explored ways to determine the low energy constants, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For a recent review, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' To illustrate the renormalization procedure, we consider the electromagnetic form factor of pion which is defined as: ⟨π+(p′)|V em µ (0)|π+(p)⟩ = (p′ + p)µF π V (t), t = (p′ − p)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (14) 12 This quantity at tree-level and at one-loop is given by: F tree V (t) = 1 , (15) F π,1−loop V (t) = 2φ(t, Mπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) + φ(t, MK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) , (16) where φ(t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) = − tM d−4 (4π)d/2 Γ(2 − d/2) 2F 2 π � 1 0 dx x(1 − 2x) � 1 − t M 2x(1 − x) � d−4 2 (17) The O(p4) contribution to F π V comes from L9 and has the following form: F π,L4 V = 2L9t F 2 π , (18) and the total contribution to F π V is finite at this order if the L9 is tuned as: L9 = Lr 9(µ) + λ(µ) 4 , (19) (20) this results into finite scale dependent renormalized coupling Lr 9(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Now the divergent function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (17) is renormalized by writing: φ(t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) = φren(t, M, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) − tλ(µ) 6F 2 π + O(p4) , (21) and the complete expression for pion electromagnetic form factor has the form: F π V (t) = 1 + 2φren(t, Mπ, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) + φren(t, MK, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' d) + 2tLr 9(µ) F 2 π + O(p4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (22) Other examples can also be found in the book of Donoghue, Golowich and Holstein [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' An important attribute of the external source technique, which was not explicitly available in the heuristic proposal of Weinberg is the ability to also account for the electromagnetic and weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This promotes the framework of chiral perturbation theory to an effective theory of the Standard Model (that includes electromagnetic and weak interactions) and not just that of the strong interaction sector with pions, kaons and the η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' EXTENSIONS OF CHIRAL PERTURBATION THEORY While the previous section treated ‘standard’ ChPT and its development, there are var- ious extensions that are needed when specific processes are to be investigated where the methods of the previous section are not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We present here a short overview only, for a deeper treatment, the references given should be consulted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The η′ Without the axial anomaly, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (2), there would be 9 light mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Because of it, the singlet axial current is not conserved, and the ninth meson becomes massive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' indeed, the η′ has a mass of 957 GeV, comparable to that of the nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In the (hypothetical) limit NC → ∞ the η′ is indeed massless, but NC = 3, and the η′ is heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, despite the large mass difference of the 8 pseudoscalar mesons and the η′, its influence on many processes is substantial, and therefore, it must be included in a systematic treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A successful way is to incorporate the η′ in the U-matrix, that is, taking a three-dimensional unitary matrix [5] of form: U(x) = e 1 3 iφ0(x)eiφ(x) (23) and add a mass term for the η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In the absence of mixing, φ0(x) corresponds to η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Furthermore, as the η′ and the external field θ are both SU(3) singlet pseudoscalars, they transform (up to a sign) in the same way under chiral transformations, and therefore the sum φ0 + θ is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This means that everywhere in the Lagrangian where there are constants, they should be replaced by arbitrary functions of φ0 + θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This brings of course new uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Nevertheless, as shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [5], it is possible to draw some concrete conclusions, in particular about the mixing of the neutral pseudoscalar mesons π0, η, and η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Furthermore, as can be seen from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (8), low energy constant L7 can be modeled by the exchange of a chiral singlet pseudoscalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In fact, the η′ does a good job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are possibly further applications of this way to include the η′ and more details can be found in Kubis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' As mentioned, in the large NC limit, the models show new interesting aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' See refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [41, 42] for a thorough investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Vector mesons The next heavier hadrons after the pseudoscalar mesons are the vector mesons, such as the ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Using the methods used before, in particular, that means determining the correct transformation behavior of the vector mesons under the chiral symmetry, they can be built into the chiral Lagrangian [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This does not only contribute to processes with such vector meson, but the vectors are also resonances that contribute to the low energy constants introduced in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In fact, as stated there, in many cases, they largely saturate the 14 constants, thus giving a very successful model for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Some recent articles on the subject are [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Baryons The method can also be extended to include the baryon degrees of freedom see review ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In the manifestly Lorentz covariant framework, a problem arises because one cannot have a strict power-counting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' On the other hand, inspired by the heavy quark effective theory, a heavy baryon version is available due to Jenkins and Manohar [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A reformulated Lorentz invariant method via infra-red regularization due to Ellis [49, 50] and Becher and Leutwyler [51], along with other versions such as extended on-mass-shell renormalization methods that have several advantages were proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The pion- nucleon σ−term (see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [53] for latest review) obtained from the baryon ChPT has also been useful for beyond the standard model physics considerations, especially studies related to the dark matter searches [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' TWO AND THREE BODY RESCATTERING Scattering processes provide important clues to the physics behind them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Furthermore, scattering is often part of other processes, like decays, where the total amplitude also depends on the (re) scattering of the decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For instance, in a decay K into two pions, the pions rescatter strongly, thereby (in some cases) influencing decisively the measured decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This is important in cases where the straightforward application of ChPT is not sufficient to explain the experimental results and must be supplemented by additional methods, such as unitarity conditions which sum up certain higher order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We also note that there is a vast literature on scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The formalism presented in section II for the one-loop ChPT can be used to calculate the scattering amplitude involving the pseudoscalar Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In the limit of isospin conservation, it is customary to introduce amplitudes of definite isospin in the s-channel T I(s, t), which may be related to specific physical charged states and depend on the process 15 of ππ or πK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These isospin amplitudes can further be decomposed into partial waves as: T I(s, t) = 32π � l (2l + 1)tI l (s)Pl(cos(θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (24) where tI l (s) is partial wave amplitude, θ is the scattering angle in the center of mass frame, and the Pl are the Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The tI l (s) are complex above the threshold and are related by unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For ππ scattering it has the following form [10]: tI l (s) = � s s − 4m2 π �1/2 1 2i � ηI l (s)e2iδI l (s) − 1 � , (25) with δI l being the phase shift and ηI l the elasticity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For πK scattering, an anal- ogous expression can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It may be recalled that scattering lengths are the lowest order shape parameters appearing in the expansion of the real part of the partial wave amplitudes, and their expression near the threshold looks like this: Re (tI l (s)) = (q2)l(aI l + bI l q2 + O(q4)), (26) where q2 is the square of the momentum transfer in the center of mass frame and (q2)l denotes the centrifugal barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The scattering length of the lowest waves dominates the physical cross-section at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The scattering lengths are also one of the important quantities for the pionium as the decay rate that is sensitive to |a0 0 − a2 0|2[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' DIRAC [57] and NA48 experiments ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [58, 59] at CERN were aimed to measure the S-wave scattering length difference in the I = 0 and I = 2 isospin channels and the observed values were in agreement with ChPT predictions in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since pions and kaons are short-lived, one cannot do fixed target experiments, and the obtained scattering lengths are based on phase shift analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Whereas e+e− → π+π− is well studied experimentally and is related to the I = 1 P−wave via the Watson theorem, the other phase shifts are well measured only at higher energies from πN scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At low energies, they are related to the form factors of Kl4 decays and have to be extracted using dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In pion scattering, the suitable framework is dispersion relations with two subtractions which suffice due to the Froissart bound, which allows one to write a system of partial wave equations that leaves the two S-wave scattering lengths undetermined parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These Roy equations and the corresponding Roy-Steiner equations for πK scattering have been studied for over 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' They also provide a very useful framework for relating the dispersion relations to the chiral amplitudes, as shown in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 16 The O(p6) behavior of these contributions can be calculated following the work of Bijnens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [60] and Colangelo, Gasser and Leutwyler [56] and the ππ scattering amplitude to can be decomposed in the following form: tI ℓ(s) = tI ℓ(s)2 + tI ℓ(s)4 + tI ℓ(s)6 + O(p8) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (27) At the leading order, the non-zero contributions from the S- and P- waves are given by: t0 0(s)2 = 2s − M 2 π 32πF 2 π , t1 1(s)2 = s − 4M 2 π 96πF 2 π , t2 0(s)2 = −s − 2M 2 π 32πF 2 π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (28) To O(p6) accuracy, the imaginary parts of the ππ (and πK) scattering amplitude receive contributions only from the S− and P− partial waves and can be written in terms of the three functions of only one variable as: A(s, t, u) = C(s, t, u) + 32π �1 3U 0(s) + 3 2(s − u)U 1(t) + 3 2U 1(u) + 1 2 � U 2(t) + U 2(u) − U 2(s) � � , (29) where the first term must obey crossing symmetric and has the form: C(s, t, u) = c1 + sc2 + s2c3 + (t − u)2c4 + s3c5 + s(t − u)2c6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (30) It may be borne in mind that the real parts obtain contributions from the O(p4) from higher waves as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The ci are the subtraction constants of U i(x), which are also termed “unitarity corrections”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For s−channel with isospin I = 0, 1, 2 have dispersion relation given by: U 0(s) = s4 π � ∞ 4M2π ds′σ (s′) t0 0 (s′)2 {t0 0 (s′)2 + 2 Re t0 0 (s′)4} s′4 (s′ − s) (31) U 1(s) = s3 π � ∞ 4M2π ds′σ (s′) t1 1 (s′)2 {t1 1 (s′)2 + 2 Re t1 1 (s′)4} s′3 (s′ − 4M 2 π) (s′ − s) , (32) U 2(s) = s4 π � ∞ 4M2π ds′σ (s′) t2 0 (s′)2 {t2 0 (s′)2 + 2 Re t2 0 (s′)4} s′4 (s′ − s) , (33) more details about various quantities appearing in this equation can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The case when there is no three-channel crossing symmetry, and with unequal mass scat- tering is also accessible using a combination of fixed-t and hyperbolic dispersion relations, which were known in the literature after being suitably modified to account for chiral count- ing, in order to saturate the dispersion relations using the imaginary parts of the relevant S- and P- waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 17 In the case of πK scattering, the structure was analyzed by Ananthanarayan and B¨uttiker [55] and by B¨uttiker, Descotes-Genon and Moussallam [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The πK scatter- ing amplitude to one loop can be decomposed into partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Once one isospin channel amplitude is known, others or a combination of them can be obtained using the crossing symmetry relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Like ππ scattering, these amplitudes can also be written in terms of functions of one variable as: T +(s, t, u) = Z+ t (t) + Z+ 0 (s) + Z+ 0 (u) + (t − s + ∆2 u )Z+ 1 (u) + (t − u + ∆2 s )Z+ 1 (s) (34) T −(s, t, u) = Z− t (t) + Z− 0 (s) − Z− 0 (u) + (t − s + ∆2 u )Z− 1 (u) − (t − u + ∆2 s )Z− 1 (s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (35) The imaginary parts of the Z’s can be written in terms of the lowest partial waves as: Im Z± 0 (s) = 16πImf ± 0 (s) , (36) Im Z± 1 (s) = 12π q2 s Imf ± 1 (s) , (37) Im Z+ t (s) = 16π √ 3 Imf It=1 0 (t) , (38) Im Z− t (s) = 6 √ 2πImf It=1 0 (t) ptqt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (39) The details of various quantities appearing in these equations can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are processes where it is necessary to account also for 3-particle rescattering, which is considerably more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This is, for instance, the case for decays where phase space is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The best-known example is the decay of η → 3π with significant data available for the cases of exclusively neutral, as well as neutral, and charged pions, in terms of the Dalitz plot as well as in terms of rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This rate is sensitive to the u − d mass difference and, therefore of special importance in the determination of the quark mass ratio (Q) [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The original work of Khuri-Treiman [63] is based on the dispersive approach to study the final state interactions in K → 3π, and a set of integral equations are obtained and later to η → 3π by Kambor, Wisendanger and Wyler [64] and Leutwyler and Anisowich [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The presence of final state interactions between the pion generates the branch cut in the amplitudes that starts from 4m2 π in s−, t−, and u− channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' As the centrifugal barrier suppresses the higher partial waves, the amplitude has a resemblance with the 2 body scattering where higher waves also start contributing from O(p8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The important difference between the two is that the three-body scattering also involves angular averages, which 18 are difficult to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This difficulty has recently been overcome by an efficient method provided by Gasser and Rusetsky [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The scattering amplitude for η → 3π can be decomposed into the contributions from isospin channel I = 0, 1, 2 represented by M0, M1, M2, which are the functions of one variable in Mandelstam variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Following the detailed analysis of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [65, 67–69], the discontinuity in the amplitude has the form: discMI(s) = θ(s − 4M 2 π) � MI(s) + ˆ MI(s) � sin(δI(s))e−iδI(s) (40) The first term in the braces receives contributions from the interactions of the s channel, and the second term accounts for those coming from the t and u channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The δI(s) are the phase shifts of the ππ scattering from the leading partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The t and u channel contributions are given in terms of the angular averages of the MI’s as follows: ˆ M0(s) = 2 3⟨M0⟩ + 2(s − s0)⟨M1⟩ + 2 3κ⟨zM1⟩ + 20 9 ⟨M2⟩ (41) ˆ M1(s) = κ−1� 3⟨zM0⟩ + 9 2(s − s0)⟨zM1⟩ − 5⟨zM2⟩ + 3 2κ⟨z2M1⟩ � (42) ˆ M2(s) = ⟨M0⟩ − 3 2(s − s0)⟨M1⟩ − 1 2κ⟨zM1⟩ + 1 3⟨M2⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (43) where s0 = 1 3M 2 η + M 2 π (44) κ(s) = � 1 − 4M 2 π s � (M 2 η − Mπ)2 − s � (M 2 η + Mπ)2 − s (45) ⟨znMI⟩(s) = 1 2 � 1 −1 dzznMI(3 2s0 − 1 2s + 1 2zκ(s)) (46) with I = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 2 and n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For more details, we refer to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The details of the higher order corrections to O(p6) for the three body decay of the η → 3π can be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [71–73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The dispersive construction of amplitude can be found in Kampf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [74, 75] and small electromagnetic corrections to this process in Ditsche, Kubis and Meißner [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A detailed analysis using Dalitz plot and modified non-relativistic effective field-theory in by Schneider, Kubis and Ditsche in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The determination of quark mass ratio from these decays are presented in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [62, 78–80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Cusps in K → 3π, which are relevant for the precise determination of the pion scattering lengths, are studied in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [81, 82], in η → 3π, effects of mixing of ηη′ in the η → 3π Leutwyler [83], dispersive 19 analysis by Leutwyler and Anisovich in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [65] and various topics related to three-body decays and dispersion relations are now covered in the book of Anisovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For a detailed review, we refer to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [40, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' GENERALIZED RENORMALIZATION GROUP AND LARGE CHIRAL LOG- ARITHMS In section II, we touched briefly upon infinities in the low energy constants, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Such infinities are, of course, well-known in QED and other quantum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' His- torically such divergencies in the self-energy of an electron from classical electrodynamics led to the birth of quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Schwinger, Tomanaga, Feynman, and Dyson gave a covariant description of QED which led to the consistent description to any order in the perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The infinities are removed by redefinition in the bare parameters of the Lagrangian, a procedure termed renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At that time, it was just a math- ematical trick to tackle the divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The works of Stueckelberg and Petermann [85], Gell-Mann and Low [86] showed that this procedure automatically incorporates the running of renormalized coupling constants, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The renormalization group equations dic- tate the running and mixing of various operators with scales and have been used as a very useful technique that allows to sum up some of the large logarithmic corrections which are remnants of the renormalization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Whereas the early discussion was mainly restricted to perturbation theory and renormal- izable theories, Wilson [87–89] in the early 1970s further extended it to non-perturbative systems in order to understand critical phenomenons and gave a deeper insight to the physics at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This has found numerous applications in various areas of physics ranging from condensed matter, statistical physics, and cosmology to particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These ideas were later studied in great detail using the path integral by Polchinski [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are var- ious approach to the renormalization group and we refer to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [91–98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' An overview can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While the concepts of renormalization are associated with renormalizable theories, one might ask how they work in non-renormalizable theories, such as ChPT, where the number of parameters increases to cancel the divergences appearing in the loop calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In particular, one may ask how to order the (large) leading logarithms (LL) which arise in the 20 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Li and Pagels [100] in the early seventies pointed out that a large logarithm of type m2 π log(m2 π) appears in one-loop calculations involving pion loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Weinberg calculated these logarithms in his famous paper on phenomenological Lagrangians [2] using current algebra and the renormalization group for pion scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Later work of Gasser and Leutwyler [10] where systematic one-loop extension of ChPT was performed and significant ∼ 25% con- tribution at 1 GeV from such terms were obtained, especially the corrections to the lowest S-wave pion scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The large logarithm contributions to the two-loop can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [101] and have the following form: a0 0 = 7m2 π 32F 2 π � 1 − 9 2 m2 π 16π2F 2 π log(m2 π/µ2) + 857 42 � m2 π 16π2F 2 π �2 log2(m2 π/µ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (47) The full two-loop contributions to scattering length read [60]: a0 0 = tree � �� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='156 + 1 loop � �� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='039 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='005 + 2 loops � �� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='013 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='003 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='001 = total � �� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='217 L anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' ki L anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' , where ki are the contributions from the single as well as double chiral logarithms, which can be evaluated using the renormalization group [2, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Bijnens, Colangelo and Ecker [102, 103] extended the work on chiral double logarithms to the full meson sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Clearly, loga- rithmic corrections can be large in ChPT, and a tool like the renormalization group would be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Indeed, Kazakov [104] and Alvarez, Freedman, and Mukhi [105] discussed the extension of renormalization to arbitrary (non-renormalizable) theories in order to calculate leading and subleading divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These ideas were applied to ChPT by Buchler and Colangelo [106], which required a new one-loop calculation at each order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The resummation of these large logarithms to all orders is still an open question in ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, the chiral logarithms have been of constant interest to understand many renormalizable and non-renormalizable theories as a toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Bissegger and Fuhrer [107] worked out a method to calculate the chiral logarithms for two flavors to any desired order in chiral limit using analyticity, crossing symmetry, and the Roy equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' They have also given the five-loop results for specific two-point scalar Green functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Kivel, Polyakov, and Vladimirov [108] provided a method where a non-linear recurrence relation is obtained that efficiently calculates the 21 leading logarithm (LL) to arbitrary loops for any non-renormalizable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This work was later extended for form factors in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [109] and some results for the LLs for the massless O(N + 1)/O(N) σ-model are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This model, for N = 3, is equivalent to the chiral SU(2) × SU(2) model that describes the leading low-energy interaction of pions in the chiral limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Later Koschinski, Polyakov, and Vladimirov [110] provided a method to calculate the leading infrared logarithms to essentially unlimited loop order using only the tree-level results in the non-renormalizable massless effective theory and later to sigma models on an arbitrary Riemann manifold by Polyakov and Vladimirov in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The LLs in the massive case for a non-linear O(N)-sigma model are studied by Bijnens and Carloni in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [112, 113] and extended to the anomalous sector by Bijnens, Kampf and Lanzin in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' More recently, Ananthanarayan, Ghosh, Vladimirov, and Wyler [115] have generalized the massive case to arbitrary order in LL corrections for various O(N) and SU(N) models and a Mathematica code is provided that reproduces the existing results and calculates higher-order results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Further development in two-dimensional effective field theories can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [116, 117] and an extension to the baryon sector in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' WEAK INTERACTIONS OF PSEUDOSCALAR MESONS The ChPT formalism, especially when formulated with the external field method, is directly adaptable to weak processes involving the pseudoscalar mesons, such as the decays K → ππ, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A recent example illustrating the persistent importance of ChPT is the rare decays involving (hypothetical) new light particles such as axions [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The systematic expansion in powers of momentum and quark masses allows analyzing seriously many ‘small’ effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' An illustration of how the weak interactions fit into the external field method with well-defined transformation properties is given in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The large size of MW compared to the QCD scale of a few GeV makes it clear that any interaction of gluons that affect the W bosons is tiny: It would involve the strong coupling constant at the MW scale and further suppression factors 1/MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 22 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 1: Illustration of the weak interaction of the pseudoscalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The large mass of the W bosons is the reason why the external field method is appropriate [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The basis for extending ChPT to the weak interactions was laid down in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It gives a systematic treatment of ChPT for weak interactions and extended the weak interactions Lagrangian to O (p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is based on several previous works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' here, we mention only the pioneering paper by Cronin [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' To construct the weak chiral Lagrangian, we need the form of the external field that represents the weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The (chiral) symmetry properties of the weak interaction follow from the fact that they arise from the symmetric product of two left-handed charged octet currents: L∆S=1 = g � J2 1µ, J1µ 3 � + + g∗ � J1 2µ, J3µ 1 � + , (48) with J2 1µ = J1µ + iJ2µ, J3 1µ = J4µ + iJ5µ (49) and where the (numeral) indices refer to the position in the 33 flavor matrix and {, }+ denotes the symmetric product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This implies that the weak interactions transform as (8)L and (27L, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note that the CP-invariant and the CP-odd parts can be conveniently separated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Using now the expressions Lµ = iU +∇µU (50) for the left-handed meson currents, we can write the octet CP invariant effective weak 23 K sylendoperator as L(8) WI = c2⟨λ6∇µU †∇µU⟩ = c2⟨λ6LµLµ⟩ (51) where the octet property is manifest in the matrix λ6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At O (p2), also a second operator can be written as: L8′ WI = c5⟨λ6 � χ†U + U †χ � ⟩ (52) The CP-invariant effective weak chiral Lagrangian transforming as (27L) is constructed from the octet components of Lµ: L27 WI = c3 � 3⟨ � Q2 3 + Q3 2 � Lµ⟩⟨Q1 1Lµ⟩ + 2⟨Q2 1Lµ⟩⟨Q1 3Lµ⟩ + 2⟨Q1 2Lµ⟩⟨Q3 1Lµ⟩ � (53) where the matrices Qi j have a 1 in the position i, j and are zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note that there is only one operator in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' An application of this CP-invariant operator to K → πℓℓ process at one-loop can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' As to the CP-violating Lagrangian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' it is obtained from the above by replacing λ6 by λ7 and appropriate changes in the operators transforming as 27−plet L− WI =c− 2 ⟨λ7LµLµ⟩ + c− 5 ⟨λ7 � χ†U + U †χ � ⟩ + c− 3 � 3⟨λ7Lµ⟩⟨Q1 1Lµ⟩ + 2i � ⟨Q2 1Lµ⟩⟨Q1 3Lµ⟩ − ⟨Q1 2Lµ⟩⟨Q3 1Lµ⟩ �� (54) We note that the ∆S = 2 operator is required for the calculation of the mass difference in the K0- ¯K0 mixing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' which transforms as a 27-plet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' and it is obtained by setting the tensor components to their appropriate values (see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [121]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is well known that the second octet operator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (52) does not contribute to physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The operator is, in fact, proportional to the variation under a suitably chosen symmetry and thus to a divergence of a conserved (Noether) current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since the operator does not carry momentum, the matrix element vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In [121], the argument is extended to the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note here, however, that in processes where the scalar external field is not just χ, but variable, this statement might not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [121], a complete basis of the weak operators at O (p4) is given, subsequent analyses showed that the basis could be further reduced, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [124] and ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The complete O (p4) Lagrangian containing 37 operators can be found in those papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We 3 Since we consider K−decays, only the transition from an s−quark to a d−quark, that is only the Gell- Mann matrices with elements (2, 3) contribute 24 also note that not all of these contribute to the decay of kaons into pions which make the calculations simpler and the predictions better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Much like in the strong interaction case discussed before, the application of the O (p4) Lagrangian to physical processes is used to determine the coupling strengths of the low energy operators, the LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [125] the decay K → 3π is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The order O (p4) gets contributions from the operators mentioned and loop diagrams whose vertices are those of the lowest order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For an improved treatment, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Also, kaon decays are again considered as a laboratory for rare processes and recent progress can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' SELECTED APPLICATIONS OF CHIRAL PERTURBATIVE THEORY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' As already mentioned, ChPT has numerous applications in describing low-energy pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In some cases, the precision reached is very high and allows for testing fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Here we review but a few such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' As mentioned before in section II, the masses of the quarks can be determined quite precisely using the chiral formalism from adequate phenomenological studies, such as of the η → 3π decay (see section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Input from the lattice and QCD sum rules increases the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These studies have confirmed that the up quark mass mu is non-zero [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The predominant decay of π0 into the two photons proceeds via the chiral anomaly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' the prediction for the rate is Γ (π0 → γγ) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='760 eV, in remarkable agreement with Γ (π0 → γγ) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='14(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=') ± 17(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=') eV obtained from the high precision exper- imental finding of PrimEx-II [128] experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Other processes such as ππ and πK scattering require detailed analysis using SU(2) and SU(3) versions of the ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The scattering amplitude of these processes, when expanded in terms of the partial amplitudes, results in the notion of the scattering lengths, and their experimental inputs can be used to fix some of the low-energy constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' An explicit expressions ππ, πK, and KK scattering lengths to O (p4) can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [24] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Interestingly, for the ππ interaction, the scattering lengths for the I = 0 isospin channel have a positive sign and are larger than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='5 times in magnitude compared to the I = 2 isospin channel, which has a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These signs correspond to the repulsive and attractive nature of the interactions in these channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Furthermore, the 25 phase shift analysis of the ππ scattering has been found to be a very useful ingredient in quantifying the hadronic contributions to the anomalous magnetic moment of the muon (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Readers can find further details on the form factors in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [129–133] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Of particular interest is the anomalous magnetic moment of the muon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is one of the testing grounds for the standard model and has been the topic of constant interest in the particle physics community [134, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The results from the Brookhaven National Laboratory (BNL) found tension with the predictions of the standard model a little over 3σ in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Further development in both the theory and experiment side has taken place and is summarized in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The most recent experiment in Fermilab aimed to study this issue with improved purity of the beam and detector components and found agreement with the results of BNL with a smaller central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Their combined results have has now established the discrepancy at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These results can be found in a set of publications in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [138–141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The main source of the discrepancy comes from the hadronic vacuum polarization contributions and another somewhat less numerically important but relatively larger uncertainty known as the hadronic light by light scattering contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' An excellent summary of all these discussions was recently presented, see slides of [142], and for details, we refer to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [129, 143–145] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Some of these hadronic light-by-light contributions, as well as those contributions to (g−2)µ involve related processes where transition form factors play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These form factors are the complex functions obeying the unitarity and analyticity conditions, which dictate their behavior in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, their values for a given kinematical region can be fixed by the available information from the experiments or lattice simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In some cases, the Watson theorem relates the phase shift of the scattering amplitude to the phase of the form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' One of them that is worth mentioning is the transition form factor for the ωπ0 for which discrepancies between experimental data and results from dispersion theory were reported for low energy region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [146] and references therein for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, these discrepancies can be studied in a model-independent way using the method of unitarity bounds[147, 148] combined with the functional analysis method [149] to find the bounds on the ωπ0 form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These functional methods have found numerous applications in hadron physics and are now available in the form of a textbook in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Recently, some agreement between experimental data with new analysis based on subtracted Khuri-Treiman 26 equations has been reported for ωπ transition form factor in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Of course, several other examples can be studied using chiral perturbation theory, and many of them can also be found in the supplementary Mathematica [152] notebooks of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [153] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The following publicly available codes are recommended to study some of the processes: Ampcalculator by Unterdorfer and Ecker [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Phi by Orellana which calculates O (p4) corrections to one loop and already included in FeynCalc 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='0 [155] and later versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The Mathematica-based code to study the ππ scattering, and Scalar and Pseudoscalar Form Factor and new additions to meson-meson scattering using U(3)−ChPT can be found in the link [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Mathematica notebooks with many solved examples by Ananthanarayan, Das, and Imsong in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [153] VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' OTHER EFFECTIVE THEORIES FOR THE STRONG INTERACTIONS While ChPT is designed for phenomena where momentum exchange is below 1 GeV, one must also deal with QCD at higher energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are several effective methods proposed and used in particle physics to account for the strong interactions, in particular for their leading effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' They allow for adapted calculations in processes where strong interactions are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' With the huge harvest of ever-improving experimental data over the last decades, such methods are, in fact, necessary to explain and exploit these results as fully as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In particular, they are used to uncover a possible still more fundamental theory than the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Characteristic for these situations is the presence of (two) very different scales, m1 ≪ m2, that are relevant for the processes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Then, typically, either an expansion in the small quantity m1/m2 is possible, or there are large logarithms of the form log(m1/m2) originating in loops, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (13) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At present, the study the weak interactions and possibly other fundamental physics in- volves three important energy scales: (1) The weak scale, MW is of the order of 100 GeV, 27 (2) the mass scale of the heavy quarks b and c (several GeV), and (3) the QCD scale ΛQCD of about ∼ 1/3 GeV where the confinement effects set in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At MW, the strong coupling constant αs is about ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='118, and the strong interactions are perturbative (asymptotic freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For the heavy quark mass scale, αs is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This still allows for perturbative calculations, but their precision is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' While for the b quark mass, this treatment seems appropriate, the scale of the charm quark offers substantial difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Even more involved is the situation for the strange quarks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' that is the physics of kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note, however, that because of the ‘Cabibbo suppression’, the decays of the b and s are easier to study than those of the c quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For recent and updated overviews, see refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [157, 158] and references therein, or refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [159–161] for charm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note that the methods to be described are primarily used to analyze and calculate the effects that the strong interactions have on investigations of fundamental parameters and theories, such as the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Of course, there are still properties of the strong interactions themselves and it is interesting to understand them, for instance, the spectrum and decay width of the charm quark systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Recent progress in this sector can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [162, 163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Extended Effective weak Theory This methodology was put forward after the discovery of asymptotic freedom and the realization that QCD, in fact, allows for perturbative calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is used mainly to investigate weak interaction processes of the heavy quarks b and c, but also the (weak) decays of the s where it was first applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is an extension of the original 4-Fermi theory and allows to include loops of the electroweak and strong interactions in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In particular, the strong interaction effects can be calculated reliably in the interval between the weak scale MW and the mass of the heavy quarks, thereby taking into account the large logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Work on this began in the mid-seventies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Shifman, Altarelli, Cabibbo, Maiani, Petronzio, Ellis, Gaillard, Lee, Gilman, Wise, and Buras are but a few that have made important contributions and perfected the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For some of the original literature, see the refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [164–166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We will give only a rudimentary introduction for many details of this advanced, by now standard subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' see the book by Buras [167], which offers an in-depth and updated treatment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' and for an even more recent update, we refer to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [157, 168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 28 The basic idea is that at energies below MW, the dynamical fields are the quarks (except the top quark), gluons, and photons (or other light, undiscovered particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Thus the weak Hamiltonian operator ˆO can be written as a series of operators consisting of the quark fields of interest, gluons, and photons with increasing powers of 1/MW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' in reality, the important power is 1/M 2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These operators must satisfy the symmetries required for the process at hand and are usually ordered according to increasing orders of 1/M 2 W 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For consistency, all operators that can contribute to the process at the desired order in the strong and electromagnetic coupling constant must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This implies that not only the original left-left Four-Fermi operator (W-exchange) is present, but several others are generated through loop corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A famous example is the so-called penguin operator (See Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 2: Penguin diagram contributing to B → Xsγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For instance, the operator for the decay B → Xsγ (Xs denotes an inclusive hadronic state with the total strangeness of one) takes the form: Heff(b → sγ) = −4GF √ 2 V ∗ tsVtb � 6 � i=1 Ci (µb) Qi + C7γ (µb) Q7γ + C8G (µb) Q8G � , (55) where the ‘magnetic’ penguin operators in the above are given by: Q7γ = e 16π2mb¯sασµvPRbαFµν, Q8G = gs 16π2mb¯sασµvPRta αβbβGa µv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (56) 4 In cases where the top quark is important, there are also inverse powers of top quark mass 29 6 S M u,c 688888 ,9,Z 8888 u,cMore details about these equations can be found in the book of Buras [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Here, the operators Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='Q6 are four-Fermi operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are six instead of only one because gluon exchanges rearrange the color order5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The coupling constants (as well as the quark field operators) depend on the scale µ (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The relevant scale for the decay at hand is of the order of mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' On the other hand, the constants of the effective Hamiltonian can be perturbatively calculated at the high scale, MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Because of (weak and electromagnetic) loops, there can be more contributing operators beyond the simple 4−Fermi interaction at the scale mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' To connect the two scales, the renormalization group is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This leads to a systematic expansion in the strong and electromagnetic coupling constants and the summing up of the large logarithms log(mb/mW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This procedure has led to a (almost) complete understanding of the weak parameters (such as the parameters of the Cabibbo- Kobayashi-Maskawa matrix) and, in particular, an understanding of CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The status of deviations from the standard model in the heavy flavor sector can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For a detailed description of the method and the results obtained, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Note that this method best applies to inclusive hadronic decay products (that is why in the above case, the final state is Xs, rather than an exclusive state, such as Kπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Heavy Quark Effective Theory While the effective weak theory described above pertains to the energy interval mb−MW, the heavy quark effective theory, HQET, deals with scales below mb in processes involving b quarks, such as the B-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since the typical momenta inside a QCD bound state are of the order of the strong scale ΛQCD, which is much smaller than mb, the b quark is only lightly ‘shaken’ and can therefore be considered at rest in a first approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Therefore, for an arbitrary heavy quark Q, we write the momentum of the quark as: pµ = mQvµ + kµ (57) where v is the four-velocity of the hadron containing the heavy quark, and k is of the order of ΛQCD, and thus much smaller than mQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This decomposition allows, similar to the well- known treatment in atomic physics, to divide the spinor into a dominant ‘upper’ component 5 this is indeed the crucial point of using an effective theory in that all operators consistent with the symmetries must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 30 and a ‘lower’ one which is suppressed by 1/mQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Thus, the idea is to construct an effective theory in which the upper component(hv(x)) is dynamical, and the lower one(Hv(x)) is integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This can be achieved by suitable projections of the Q quark spinor [169]: Ψ(x) = e−imQv·x [hv(x) + Hv(x)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (58) The upper and lower component is obtained by the relation: hv(x) = eimQv·x1 + /v 2 Ψ(x) , (59) Hv(x) = eimQv·x1 − /v 2 Ψ(x) , (60) and in the case of heavy antiquark, the substitution of v → −v is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Indeed, for 1/mQ → 0, the small the component can be integrated out [170–172] and the theory has an extra spin-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The leading order (in 1/mQ) Lagrangian has the form: Leff = ¯hviv · Dhv + Llight (61) and other terms involving the heavy quark field are rearranged as an expansion in 1/MQ, and the Lagrangian for light degrees of freedom (quarks and gluons) is given by: Llight = −1 4Tr (GµνGµν) + � q Ψq � i /D − mq � Ψq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (62) This formalism has been extensively used in the literature to extract the CKM elements (|Vcb|, |Vub|, heavy flavor sum rules, and the description of heavy hadron decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' More details can be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [173–178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' NRQCD and pNRQCD The heavy quark expansion used above is not suitable to describe a meson with two heavy quarks (like charmonium or the Υ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In HQET, the kinetic energy is a 1/mQ effect and is taken as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' But for a bound state, it plays an important role in balancing the potential energy and, therefore, should be present at leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The necessary formalism was provided by Bodwin, Braaten and Lepage in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [179] and is known as NRQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Such systems also have additional scales, such as relative momenta p ≃ mv(soft) and a kinetic energy, Ek ≃ mv2(ultrasoft scale), constructed out of the mass of heavy quark (M) and 31 its velocity (v ∼ αs << 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For the bottomonium system, v2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='1 and for charmonium systems, v2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The hierarchy scales in the system are as follows: mq(hard) ≫ mQv ≫ mQv2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (63) The Lagrangian is expressed as an expansion in mQv/mQ and (mQv2/mQ), and at leading order in 1/mQ, it has the following form: LNRQCD = ψ† � iD0 + ⃗D2 2M � ψ + χ† � iD0 − ⃗D2 2M � χ + Llight (64) where iD0 = i∂0 − gA0 and ψ(χ) is the Pauli spinor field of fermion (antifermion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It should be noted that the presence of the two dynamical soft and ultrasoft scales can complicate the calculations and interfere with the power-counting and the non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The NRQCD is numerous applications in the threshold production of top-quark pairs in electron-positron annihilation, spectroscopy of heavy charmonium and bottomonium bound states [180], determination of heavy quark masses, strong coupling constant, and in the understanding of the vacuum structure etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A modified version of the NRQCD has been recently proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [181, 182] for the production of the J/Ψ, Ψ′, and χc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For more details, we refer to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [183, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Another interesting system that can be constructed out of NRQCD is the potential NRQCD (pNRQCD) [185–187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is obtained by integrating out the soft degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The leading order in 1/mQ and multipole expansion in r, the Lagrangian has the following form: L0 pNRQCD = Tr � S† � i∂0 − V (0) s (r) � S + O† (iD0 − Vo(r)) O � − 1 4F a µνF µνa where S and O are the singlet and octet fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The resulting EFT has a resemblance to the Sch¨odinger equation as the matching coefficients Vi(r) play the role of the potential between the heavy quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The equation of motion for the singlet case is : i∂0S = �p2 m − V (0) s (r) � S (65) and depending on which scale is closer to ΛQCD, different versions of pNRQCD (strongly or weakly coupled) are used for quarkonium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' When there is no other scale between the soft and ultrasoft scales known as weakly coupled pNRQCD, the leading order static potentials 32 have the form: V (0) s = −CF αVs(r) r , V (0) o = �CA 2 − CF � αVo(r) r (66) and Vs/o(r) has a perturbative expansion in the strong coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These potentials have now been computed numerically to there-loop in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [188, 189] and analytically in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Some ultrasoft contributions to static energy in the weak coupling limit are already known to O(α4 s) [191] and two of us have given Pad´e prediction for O (α4 s) term to Vs(r) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The QCD static potential has been a very useful quantity in the determination of the strong coupling constant αs as it can be calculated to very good precision on the lattice [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Recent updates of αs from static energy can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [194–196] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are several packages available in the literature that can be used to study non-relativistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Recently, Brambilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [197] have published the publicly available Mathematica-based package FeynOnium that can be used to study the NREFTs to one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Another useful package relevant to studying the threshold quarkonium system is QQbar threshold by Beneke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A detailed review on NRQCD, pNRQCD, and a description of quarkonia from these EFTs can be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [183, 199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Heavy-light mesons There exist some mesonic states with heavy and light quarks, and one may ask how to combine HQET and ChPT to study their production and decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This issue has indeed been taken up by Burdmann and Donoghue [200], Wise [201] and Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [202], and is now known as heavy meson ChPT(HMChPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is formulated on the fact that the mass difference between the heavy meson and its excited state scales as ∼ 1/MQ, which can be of the order of a few MeVs for heavy mesons such as B meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Heavy quark symmetry relates to the couplings of the B and B∗, and it also relates to other mesons such as D as long as the charm quark can be treated as heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A meson with one heavy quark can be labeled by the light quark spin jl and states with spin jl ± 1 2 are degenerate due to heavy quark spin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Due to this fact, a consistent description of a heavy light system requires an excited state such as B∗ for B systems, as their production will require much less energy than the pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since the energy involved are less than the pion mass, an extension to the chiral framework can be merged with the HQET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 33 Degenerate triplets of spin-zero mesons Pa (a = u, d, s) and spin-one meson P ∗ a triplets are obtained by combining the spins of heavy and light quark spins using the heavy quark spin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These fields can be used to define the 4 × 4 matrix Ha, given by: Ha = (1 + /v) 2 � P ∗ aµγµ − Paγ5 � (67) where P ∗ aµ is an operator that destroys a P∗a meson with velocity v and satisfies: vµP ∗ aµ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (68) Defining Ha as: Ha ≡ γ0H† aγ0 = � P ∗† aµγµ + P † aγ5 � (1 + /v) 2 , (69) then most general leading order Lagrangian to describe the strong interaction between pseudo-Goldstone boson with heavy meson is given by: L = − iTr � Hv · ∂H � + F 2 π 8 Tr � ∂µU∂µU †� + i 2Tr � Hvµ � U †∂µ + U∂µU †� H � + ig 2 Tr � Hγνγ5 � U †∂ν − U∂νU †� H � − ∆ 8 Tr � HσµνHσµν � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' , (70) where ∆ = mP ∗ − mP, g is the axial coupling constant and field U is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (3) and ellipses denote the higher order terms, and complete Lagrangian to one loop can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The Lagrangian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (70) is consistent with the SU(3)L × SU(3)R, Lorentz transformations, and the heavy quark symmetry SU(2)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The leading order Lagrangian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (70) can be used to predict the P ∗ → Pπ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Such transitions for B meson are kinematically forbidden however, for the D system, it has the form: Γ � D∗+ → D0π+� = g2 6πF 2 π |⃗pπ|3 , (71) Γ � D∗+ → D+π−� = Γ � D∗0 → D0π0� = g2 12πF 2 π |⃗pπ|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (72) Using experimental input for these decays, the axial coupling g can be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are many charmed states which have gained attention over the years as they can not be described by the traditional methods, which require their own review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For details on the applications and status of heavy light systems, we refer to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [46, 163, 203–206] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 34 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Soft Collinear Effective Theory, SCET The (light) decay products of heavy (B) mesons typically have a large momentum of order mb, in comparison to ΛQCD, for instance, in the decay B → Kπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The quarks in those fast-moving light mesons are typically on the light cone, collinear with the meson that contains them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Deviations from collinearity are caused by QCD interactions and small, of the order ΛQCD/mb in case of the decay of a B meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this situation, the effective theory is constructed ‘around’ those collinear quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In an early attempt, Dugan and Grinstein [207] constructed a ‘large energy effective theory (LEET)’, to describe the interaction of the high- energy quarks (E around mb) with the soft gluons (energy about ΛQCD in an expansion in q/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Since the hadrons also contain collinear gluons, a complete theory must include them too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [208, 209], Bauer, Fleming, Luke, Pirjol, and Stewart presented a soft collinear effective theory (SCET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A comprehensive description of SCET is found in the book [210], for more recent results and developments, see for instance, refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [211, 212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We note that SCET, while originally applied to heavy meson decays, perfectly fits the needs of high energy (jet) physics that is a main part of LHC-physics, see for instance in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' To account for the dominance of the collinear particles, light cone coordinates p = (p+, p⊥, p−) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The coordinate basis for motion in the z direction is chosen to be nµ = (1, 0, 0, 1), nµ = (1, 0, 0, 1), with n · n = 2 (the coordinates are (t,x,y,z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The small parameter which characterizes the perpendicular components is λ = p⊥/n · p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The momenta are decomposed according to pµ = ˜p � �� � n · pnµ 2 + (p⊥)µ +n · pnµ 2 = O(λ0) + O(λ1) + O(λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (73) This decomposition into large and small components to construct an effective field the theory looks similar to the method of regions, where the different momentum regions are first separated and then treated differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, the effective field theory approach allows for systematically including the running of operators or power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The construction of the effective theory then is similar to the theories discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' SCET also involves three scales like NRQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The quantity ˜p now acts as the label to the fields, and the large momenta ˜p are removed by defining: ψ(x) = � ˜p ψn,˜p (74) 35 and the derivative ∂µ on fields ψn,p gives dynamical contributions of O(λ2) like in NRQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Particle moving along nµ have two large components and small components denoted by ξn,p and ξn,p respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These are related to ψn,p by the following relations: ξn,p = /n/n 4 ψn,p, ξn,p = /n/n 4 ψn,p (75) satisfying the relations: /n/n 4 ξn,p = ξn,p, /nξn,p = 0 , (76) /n/n 4 ξn,p = ξn,p, /nξn,p = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (77) The Lagrangian constructed with the above discussion has the form: LSCET = � p,p′ � ξn,p′ /n 2 (in · D) ξn,p + ξn,p′ /n 2 (n · p + in · D) ξn,p + ξn,p′ � /p⊥ + i /D⊥ � ξn,p + ξn,p′ � /p⊥ + i /D⊥ � ξn,p � , (78) where Dµ = ∂µ−igT aAa µ is covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' More details can be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [209, 210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' SCET is applied to a large variety of processes with collinear high-energy particles, not only in decays of heavy mesons but increasingly in very high-energy processes such as at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For the newest developments, see the latest SCET conference [214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' EFFECTIVE THEORIES BEYOND THE STANDARD MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The Standard Model Effective Theory So far, the standard model has proven to be essentially faultless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' apart from a few cos- mological phenomena (dark matter, matter-antimatter ratio,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=') and alleged anomalies in B meson decay [215], it reproduces all experimental results very precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' However, it is widely believed that there are more fundamental interactions with a typical energy scale Λ which seems considerably higher than MW, as indicated by the absence of discoveries of very heavy particles beyond the top quark, the W and the Z bosons and the Higgs particle at LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This is reminiscent of the early days of the weak interactions when the 4-Fermi theory HW ∼ GF(qLγµqL)(qLγµqL) was put forward and the W-boson entered indirectly only through the Fermi constant GF ∼ 1/M 2 W and its symmetry properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 36 Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' in order to parameterize physics beyond the standard model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' originating from physics at a scale Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' one considers effective operators made up of the standard model particles (including the Higgs boson and the W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Z bosons with a coupling proportional to powers of 1/Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' L = � n 1 ΛnOn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (79) where the operators On have dimension 4 + n (each On consists of many distinct operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' each with an unknown coupling) and are composed out of standard model fields such that the total operator is invariant under SU(3)×SU(2)×U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At the lowest order, (1/Λ)0, we have just the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At next order, (1/Λ)1 there is one operator [216, 217] which violates lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' At the next order, (1/Λ)2, there are nearly 100 operators, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [218, 219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In principle, the task is to determine the unknown couplings by comparing them to suitable experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Given a large number of such couplings, this is a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This is a very active field, with several strategies to overcome the difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' See ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [220] for a comprehensive overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For the newest developments, see the proceedings of the 2019 conference on SMEFT-tools [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This conference will again be held in 2022 [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Quantum Gravity One of the biggest - if not the biggest - unsolved problems in theoretical physics is how to quantize gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A modest but important step can be achieved if general relativity is viewed as a field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The metric gµν is promoted as the field, and the effective field theory has the general coordinate invariance of general relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Using the fact that the connection, defined as: Γαβ λ = gλσ 2 � ∂αgβσ + ∂βgασ − ∂σgαβ � , (80) has one derivative, and the curvature, defined in terms of the Riemann tensor (Rµναβ), given by: Rµνα β = ∂µΓνα β − ∂νΓµα β − Γµλ βΓνα λ − Γνλ βΓµα λ , (81) has two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The two derivatives present in the Riemann tensor correspond to the powers of energy when evaluated in terms of the matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is important to note 37 that the various contractions of the Riemann tensor are coordinate invariant, which is also the symmetry of the low energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Hence, the Lagrangian can be constructed out of various possible contractions of the Riemann tensor, and energy expansion can be naturally constructed including more and more contractions of the Riemann tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In particular, Donoghue [223–225] has shown how a possible extension of general relativity to a theory with quantum degrees of freedom results naturally in an expansion in the theory of gravity, which includes as the S = � d4x√g � Λ + 2 κ2R + c1R2 + c2RµνRµν + · · · + Lmatter � , (82) where Λ is cosmological constant, R = gµνRµν and Rµν = Rµναα are known as the Ricci scalar and the Ricci tensor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' So far, this theory has found only limited applications, but it may be a guide to correct quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A SCET inspired treatment of quantum gravity can be found in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [226–228].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For more details, we refer to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [223–225, 229].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' MISCELLANEOUS ITEMS In this section, we cover a range of mostly technical topics which both feed into effective theories and in whose developments effective theories have played a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Feynman Integral Methods for Effective Field Theories When calculating Feynman diagrams, one integrates overall kinematically allowed values of the internal momenta of quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There has been considerable effort in evaluat- ing them to very high orders for precision physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Many computational as well as theoretical tools have been developed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Many of these developments can be found in the recent book of Weinzierl [230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In a theory like QCD where the interaction of a gluon with quarks or gluon at 1 GeV is very different for an energy scale of several GeVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The diagram- matic evaluation of any process gets more complicated in these multiple-parameter theories when one goes to higher orders due to the presence of the various scales(masses and mo- menta) in the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is, therefore, reasonable to divide the integrand into regions and use different rules for the various region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The method of regions [231] is one of the very useful strategies for evaluating Feynman integrals in specific kinematic limits of the mass and mo- menta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In this technique, the integrand of Feynman diagrams is expanded by identifying the 38 scaling behavior of the ratios of masses and momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Although it is not rigorously proven to be correct, it appears to work in all known instances [232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Interestingly, the expansion of Feynman diagrams in various regions corresponds to an EFT in the asymptotic limits of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' In some cases, these regions may overlap and need to be systematically subtracted (zero-bin subtraction) following the procedure of Jantzen [233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Application of this method in the ChPT was first made by Kaiser and Kaiser and Schweizer [234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Now, there exist well-dedicated codes asy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='m [235], asy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='m [236] and ASPIRE algorithm [237] that can be used to study multi-scale Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For more details, we refer to [232, 233] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The Mellin-Barnes (MB) technique is also one of the most commonly used techniques in the literature for the analytic evaluation of the Feynman integrals and has been recently used in the context of ChPT in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [238–240].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The two-loop sunset diagrams play a key role in the analytic representation of the masses and decay constants of the pion, kaon, and η- mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These diagrams are calculated using the MB technique in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [241–245] and further used in evaluating some three-loop Feynman diagrams relevant for the QED corrections to g − 2 of charged leptons in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The MB technique yields the final expression in terms of generalized hypergeometric functions (pFq) and Kamp´e de F´eriet (KdF) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Recently, a geometric method using conic hulls is developed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [239] and implemented in the Mathematica package MBConicHulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='wl which allows systematic computation of certain N-fold MB integral, and in the case of convergent series case, one can also find the master series which is useful for numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This technique is used to solve certain non-trivial conformal Feynman integrals in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [238, 240].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These ChPT-inspired studies have immensely contributed to finding the new analytic continuations of the Appell Function F4 in terms of the 2F1 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [247].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' These multivariate hypergeometric functions and their properties, domain of convergences, and linear transfor- mations are studied in mathematics literature [248–251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' One of the strategies to find the analytic continuation of a multivariate hypergeometric function is to use the known analytic continuations of hypergeometric functions with a lower number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The linear transformation formulae of the one variable Gauss 2F1 function are used to find the analytic continuations of the double variable Appell F1 in [252].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This process of finding analytic continuations of hypergeometric series of more than one variable is automated in the Math- ematica package Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='wl [253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The package can also find the domain of convergence of 39 only the double-variable hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The analytic continuations of the Appell F2 functions are found using the same technique and are used to construct the numerical package AppellF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='wl [254].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It can find the numerical value of the Appell F2 function for real values of its arguments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' x, y) and general complex values of the Pochhammer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Some new analytic continuations of Appell F4 are obtained using the known quadratic transformation of the Gauss 2F1 function [247].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The linear transformations of the three variable Srivastava HC function are also found [255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Chiral Lagrangians and Ricci Flows Right from the early days, the non-linear sigma model provided the fundamental building block for the realization of chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Whereas for the simplest purposes, these were based on SU(2) × SU(2) or alternatively on SO(4) general theorems for the realization of these symmetries and the Goldstone phenomenon were established for a general group G breaking down to H by Coleman, Wess and Zumino [16], and Coleman, Callan, Wess, and Zumino [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Friedan [256, 257] studied the non-linear sigma model in 2+ϵ dimension where fields ϕ are defined on a manifold M and the coupling are is determined by a Riemannian metric on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The action has the form: S(ϕ) = Λϵ � dx1 2T −1gij(ϕ(x))∂µϕi(x)∂µϕj(x) (83) where λ is short distance cutoff, T −1gij(ϕ(x)) is dimensionless coupling is Riemannian metric on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The renormalization group running of this metric at two-loop was found to be: Λ−1 ∂ ∂Λ−1gij = βij � T −1g � = −ϵ T −1gij + Rij + 1 2T (RiklnRjkln) + O � T 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (84) and the one-loop β-function was already calculated by Ecker and Honerkamp [258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The running of coupling in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' (84) is known as Ricci flow introduced by Hamilton [259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' The ideas developed by Hamilton were an attempt to solve the long-standing problem of Poincar´e conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Perelman published the proof of this conjecture in the three articles [260–262] in 2002-3 where Ricci flow played a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' A detailed explanation of Perelman’s proof was published by Morgan and Tian [263] and Huai-Dong Cao, Xi-Ping Zhu [264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 40 Lattice QCD The non-renormalizable nature of the ChPT results in the increasing numbers of LECs as one goes to higher orders and has to be fixed by inputs from other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Most of the LECs can be determined from the experiments or estimated using a large Nc limit of QCD or low energy description of strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Lattice QCD is one of the candidates at very low energy and has provided numerous inputs and cross-checks over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Lattice calculations are performed on finite lattice spacing, finite volume, and unphysical quark masses, and ChPT provides a way to crosscheck, analyze and quantify these effects in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For the brief introduction of the interplay of lattice QCD and the ChPT, we refer to Shanahan [265] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Among these topics, proton charge radius and the muon g − 2 anomaly has been of constant interest over the year for their potential to provide hints to new physics beyond the standard model at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Issue of the small charge radius of proton came into the picture in 2010 when the existing value of charge radius rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='8775(51) fm from CODATA [266] world average using the spectroscopic method and electron-proton scattering was found to be larger than the one obtained from muonic hydrogen rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='84184(67) fm by Pohl et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Pohl’s result later confirmed by CREMA collaboration [268] with rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='84087(39)fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' There are various theoretical models for new physics were also studied, and some future experiments are also proposed to get more precise results, but the issue is now believed to be settled and we refer to a very recent review by Gao and Vanderhaeghen [269] and Hammer, Meißner [270], Bernauer [271], Peset et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [272] and references therein for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Lattice determinations of form factors are also extensively performed and the results are compatible with existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For details of lattice determination of proton charge radius, we refer to Ishikawa et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [273] and references therein for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Lattice methods themselves require their own review to explain various methods developed over the years to extract the parameters of strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For details, we refer to Golterman [274], FLAG reviews [24, 275, 276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 41 OUTLOOK Chiral perturbation theory, ChPT, has proven very fruitful over the last 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It has provided ample predictions for understanding a great number of experimental results involving the pseudoscalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' It is still being refined to adapt to new theoretical and experimental results, and there are still many results waiting to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' ChPT has helped to understand field theory more generally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' in particular, it has shed some light on the limited role of renormalizable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' This direction of research is far from being at its end, and for instance, work devoted to non-renormalizable theories will very likely yield many interesting results [115] Thirdly ChPT has also become a valuable tool to be used in circumstances not thought to be in its realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' For instance, the calculation of the anomalous magnetic moment of the muon - one of the crucial calculations in particle physics - has benefited from results obtained by ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We, therefore, believe that chiral perturbation theory, albeit an established and mature technology, has considerable potential to be improved and gateway to many other developments in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank Dilip K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Ghosh and Sourov Roy for inviting us to write this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' AK is supported by a fellowship from the Ministry of Human Resources Development, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We thank Souvik Bera for clarifying remarks and Sumit Banik for help with the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' We also thank the referee for the valuable comments that have improved this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 42 [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Weinberg, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Phys.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='19, 191801 (2022) [arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content='01454 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Coleman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Wess and B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' 175 (1968), 2195-2199 [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Halprin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Lee and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} +page_content=' Sorba, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQffQos/content/2301.04550v1.pdf'} 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a/h9E2T4oBgHgl3EQfHwY9/content/tmp_files/2301.03671v1.pdf.txt b/h9E2T4oBgHgl3EQfHwY9/content/tmp_files/2301.03671v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff5a9853a0d14f3eb4c848773c1ed78bc9be9324 --- /dev/null +++ b/h9E2T4oBgHgl3EQfHwY9/content/tmp_files/2301.03671v1.pdf.txt @@ -0,0 +1,1462 @@ +1 + +Dislocation-induced structural and luminescence degradation in InAs +quantum dot emitters on silicon + +Eamonn T. Hughes1, Gunnar Kusch2, Jennifer Selvidge1, Bastien Bonef1, Justin Norman1, Chen Shang1, +John E. Bowers1, Rachel A. Oliver2, Kunal Mukherjee3,a + +1Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA +2Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge +CB3 0FS, United Kingdom +3Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA + +ABSTRACT +We probe the extent to which dislocations reduce carrier lifetimes and alter luminescence and +growth morphology in InAs quantum dots (QD) grown on silicon. These heterostructures are key +ingredients to achieving a highly reliable monolithically integrated light source on silicon +necessary for photonic integrated circuits. We find up to 20–30% shorter carrier lifetimes at +spatially resolved individual dislocations from both the QD ground and excited states at room +temperature using time-resolved cathodoluminescence spectroscopy. These lifetimes are +consistent with differences in the intensity measured under steady-state excitation suggesting that +trap-assisted recombination limits the minority carrier lifetime, even away from dislocations. Our +techniques also reveal the dramatic growth of misfit dislocations in these structures under carrier +injection fueled by recombination-enhanced dislocation glide and III-V/Si residual strain. Beyond +these direct effects of increased nonradiative recombination, we find the long-range strain field of +misfit dislocations deeper in the defect filter layers employed during III-V/Si growth alter the QD +growth environment and introduce a crosshatch-like variation in the QD emission color and +intensity when the filter layer is positioned close to the QD emitter layer. Sessile threading +dislocations generate even more egregious hillock defects that also reduce emission intensities by +altering layer thicknesses, as measured by transmission electron microscopy and atom probe +tomography. Our work presents a more complete picture of the impacts of dislocations relevant +for the development of light sources for scalable silicon photonic integrated circuits. + + + + + + +_________________________ +a Corresponding author: kunalm@stanford.edu + +2 + +I. INTRODUCTION +Understanding how dislocations affect the +properties of optoelectronic devices like lasers and +photodetectors is central to efforts in direct epitaxial +heterogeneous integration of active devices for +silicon photonics.1 In practice, getting lasers to +operate efficiently and reliably in the presence of +dislocations remains a key challenge in III-V/Si +integration to serve the growing communications +market at scale.2 While advances in experimental and +computational +methods +continue +to +uncover +structure-property relationships of dislocation- +induced +electronic +states +and +nonradiative +recombination of charge carriers in semiconductors +like Ge,3 GaAs,4 and GaN,5,6 there is a growing +realization that the impact of dislocations goes +beyond this static and often idealized picture. +Dislocations may affect a heterostructure device +even before device operation by altering the local +composition or growth rates during synthesis, +exemplified by prior work on dislocation-induced +phase separation in alloys7,8 and roughening +surfaces.9,10 Dislocations continue to modify device +behavior long after fabrication by diffusing or +transporting dopants and other impurities during +device operation11,12 or, more dramatically, by +damaging devices via recombination-enhanced +dislocation motion where dislocations inject point +defects and subsequently increase in length over time +via dislocation climb.13–16 +Understanding +these +broader +impacts +of +dislocations will further the development of self- +assembled epitaxial InAs quantum dot (QD) lasers +on silicon.17–20 These are the most dislocation +tolerant datacom-band lasers directly grown on +silicon, but we need to continue to improve reliability +at high current and high temperatures as well as +improve manufacturability and uniformity. Both +tasks necessitate a detailed representation of +dislocation behavior and their local environment. +One important consideration is the direct impact of +dislocations on QD formation as the epitaxial growth +window for QDs is narrow, and hence more sensitive +to perturbations than conventional III-V quantum +well +(QW) +heterostructures. +Additionally, +nonradiative recombination of charge carriers at +dislocations in QD systems remains to be fully +characterized. Finally, InAs QD devices retain a +sizeable thermal strain due to the silicon substrate +that continues to drive recombination-enhanced +dislocation motion during operation.21 +In this work, we use a combination of +microanalysis techniques on a model shallow (near +to the growth surface) layer of InAs QDs on Si to +show that dislocations not only reduce excess carrier +lifetimes +and +emission +intensities +at +room +temperature, but they also introduce non-trivial +crosshatch- and hillock-induced compositional shifts +that locally alter the QD energy levels and intensity. +Properly accounting for these effects in laser design +and growth can yield improved laser performance +and reliability. Our work also anticipates the +complex effects of dislocations in the next generation +of III-V lasers grown directly on silicon beyond +datacom wavelengths such as recent works in the +visible22 and mid-infrared.23 + +II. +EXPERIMENTAL +METHODS +AND +SAMPLE DETAILS +The InAs QD model structure investigated here +was +previously +reported +in +a +multi-modal +characterization study.21 Briefly, we use molecular +beam epitaxy (MBE) to synthesize the structure +depicted in Fig. 1a with an active layer consisting of +a single shallow InAs QD layer embedded in a 7 nm +In0.15Ga0.85As quantum well and capped by a 100 nm +thick GaAs layer. The QD layer is not intentionally +doped. The active layer is grown on a GaAs-on-Si +template used for an earlier generation of QD lasers. +The template consists of two separate defect-filter +structures—a 200 nm thick continuous InGaAs layer +and a 10-period strained-layer superlattice of +10nm/10nm In0.1Ga0.9As/GaAs (see Fig. 5e for a +cross-sectional +scanning +transmission +electron +microscopy +(STEM) +image). +The +threading +dislocation density in the sample is 7×106 cm-2, and +the InAs QD density is approximately 5×1010 cm-2. +We also label the locations of misfit dislocation + +3 + +networks in Fig. 1a, which will be relevant for later +analysis. The growth conditions (temperature, V/III +ratio, growth rate) of the various layers have been +described previously.21 +Optical characterization on the nanoscale was +performed by cathodoluminescence spectroscopy +(CL). The CL measurements were carried out in an +Attolight Allalin 4027 Chronos dedicated CL +scanning +electron +microscope +(SEM). +CL +hyperspectral maps were recorded with an Andor +Kymera 328i spectrometer with a focal length of 328 +mm, a 150-lines-per-mm grating blazed at 1250 nm, +and an Andor 512 px InGaAs diode array camera. +Time-resolved CL measurements were performed by +triggering the electron gun with the third harmonic of +a Nd:YAG laser (355 nm) with a frequency of 80.6 +MHz and a pulse width of 7 ps. All CL time decay +curves were recorded with a time-correlated single +photon counting (TCSPC) setup resulting in a time +resolution of about 100 ps. All CL measurements +were performed at room temperature with an +acceleration voltage of 6 kV (interaction region is +~75 nm radius sphere tangential to sample surface) +and a beam current of 30 nA for continuous wave +measurements and between 15 pA and 90 pA for +pulsed measurements. +Atom probe tips were created using an FEI +Helios Dualbeam Nanolab 600 focused ion beam +(FIB) microscope using standard 30 kV annular +milling steps and a 2 kV broad-area polish to form +the final tip shape. Tips were evaporated using a +Cameca 3000X HR Local Electrode Atom Probe +(LEAP) at 40 K with laser pulsing at a 532 nm +wavelength, a 200 kHz repetition rate, and a laser +pulse energy of 0.20 nJ. TEM foils were prepared +using the FEI Helios Dualbeam FIB and imaged +using a ThermoFisher Talos in STEM mode using a +bright field detector with a collection angle of 17 +mrad. +Electron channeling contrast imaging (ECCI) +was performed on a ThermoFisher Apreo S SEM +using a three-beam g=040 and g=220 channeling +condition. Figure 1b shows a plan-view ECCI image +of the structure showing numerous long segments of +misfit dislocations along with threading dislocations, +which together form the subjects of our study. We +have previously determined that these misfit +dislocations form just below the 7 nm InGaAs +QW/GaAs.21 The origins of these misfit dislocations, +which appear in layers grown nominally below the +critical thickness for dislocation glide, is also +important to contextualize our results. Briefly, these +misfit dislocations form not during growth, but after +growth as the sample cools due to a combination of: +(1) residual tensile strain in the III-V layers due to +thermal expansion mismatch with silicon and (2) +local pinning of the threading dislocation segment by +the InAs QDs.24 The formation process is illustrated +in Fig. 1c where only unpinned threading dislocation +segments glide below the QD layer to form misfit +dislocations. We have identified these misfit +dislocations as being primarily responsible for +degradation in early generations of GaAs-based +lasers on silicon and, more recently, in InAs quantum +dot lasers on silicon where their effects can now be +largely mitigated using strained indium-containing +trapping layers.19 + + + + +Fig. 1. (a) Structure of sample characterized in this study. (b) +Electron-channeling contrast imaging (ECCI) from the sample +surface showing a moderate density of threading dislocations +and misfit dislocations located just below the QD layer. (c) +Illustration of the misfit dislocation formation process in which +thermal expansion misfit stress generated during cooldown +propels free threading dislocations below the QD layer to glide. +The TD segment is pinned through the QD layer and cannot +follow the lower thread segment, so a misfit dislocation forms +here. + +(a) +(b) +Emitter region +GaAs (100 nm) +InAs QDs in Ino.15Ga0.85As QW (7 nm) +Threading +Misfit +Misfit dislocation +dislocations +dislocations +networks GaAs (650 nm) +Dislocation filter layers +10 nm/10 nm +Ino1GaogAs/GaAs (10x) +2 μm +GaAs (300 nm) +[110] +Ino.1Gao.gAs (200 nm) +(c) +GaAs Cap +QD layer +GaAs (1600 nm) +III-V/Si +GaAs on GaP/Si4 + +III. RESULTS AND DISCUSSION +A. Recombination dynamics at dislocations +We use time-resolved CL using a pulsed primary +electron beam to probe the effect of dislocations on +carrier recombination at room temperature. Our +results show the misfit dislocations lying close to the +InAs QD layers (Fig. 1c) are potent nonradiative +recombination sites. Figures 2a-d shows CL intensity +decay traces as a function of increasing probe current +collected at a dislocation-free region and a region +with misfit dislocations. The signal is spectrally +filtered to separately track the CL intensity decay of +the GS (Fig. 2a-b) and ES (Fig. 2c-d) luminescence +at 1250 nm and 1167 nm, respectively, with a 2 nm +bandwidth, hence we directly probe only the +occupation of dots emitting in these narrow ranges +and indirectly probe most remaining dots via their +carrier exchange with the wetting layer, due to fast +carrier equilibration at room temperature. The insets +in these figures show that the recombination lifetime +in both regions, obtained by fitting to a single- +exponential decay, are in the 0.2–0.3 ns range and do +not vary much with probe current. Upon initial +inspection, we find the expected outcome that +carriers recombine faster near the misfit dislocation, +noting a 20% shorter GS recombination lifetime at +the lowest probe current. The ES luminescence +decays about 30% faster at the misfit dislocation. +Figures 2e and 2f show a steady steady-state- +excitation CL luminescence map (GS) and a +corresponding pulsed-excitation carrier lifetime map +obtained from each site. Comparing the two, we see +a clear correlation between the CL intensity and +luminescence lifetimes, typical of defect-limited +recombination. +Figure 2g follows the TRCL decay along a trace +that is orthogonal to a misfit dislocation (or group of +misfit dislocations) at the center of the distance axis. +When carriers are injected directly over the +dislocations, nonradiative recombination reduces +carrier concentration even at the shortest resolvable +time scales (~100 ps, estimated from the signal rise- +time), leading to a lowered initial peak intensity at +t≈0 s. We may assume that minimal carrier diffusion +takes place within this time, so the roughly 1 µm +lateral extent of reduced intensity is the convolution +of the defect size and the cross section probed by the +electron beam. Carriers injected further away from +the misfit dislocation should eventually diffuse +towards this defect, leading to a widening of the +reduced intensity valley with time. Yet, we find that +the lateral extent of reduced intensity remains +constant even on the longer time scale of 1–2 ns as +the luminescence decays, visualized as a trench of +apparent constant width in Fig. 2g. Using this +information, we obtain an upper bound for the +diffusivity of carriers in this system using 𝐿! = √𝐷𝜏, +estimating an ambipolar diffusivity 𝐷 of less than +40 cm2/s for the measured recombination lifetime +𝜏 = 0.25 ns in dislocation-free regions (Fig 2a). This +corresponds to a diffusion length, 𝐿!, of less than 1 +μm, which is shorter than reported values for +quantum-well systems in GaAs and reinforces a key +mechanism behind the dislocation tolerance of InAs +QDs.25,26 At this time, we are unable to resolve the +properties of isolated threading dislocations, but their +impact appears minimal compared to misfit +dislocations. +We expect these short carrier lifetimes are set by +trap-assisted recombination away from dislocations +that, naturally, become even shorter at dislocations. +Bimberg et al. use PL to measure a spontaneous +recombination lifetime, 𝜏", in the GS of InAs QDs of +1.8 ns, which is independent of injection over a pulse +excitation range of 0.1–100 kW/cm2 at 77 K and only +weakly temperature dependent.27 Fiore et al measure +an effective lifetime of 1.8 ns from a single InAs QD +layer in an In0.15Ga0.85As quantum well using PL at +room temperature at very low excitation of 9 +W/cm2.28 The much shorter recombination lifetimes +measured in our experiments possibly points to +elevated point defect concentrations even in regions +away from dislocations. Under these constraints, the +internal quantum efficiency of spontaneous emission +is 𝜂 = +#!" +#"$#!" ≈ +#!" +#" , and the recombination lifetime is +𝜏 = +#!"#" +#!"$#" ≈ 𝜏%". +Hence, +the +steady +state +luminescence of the GS is proportional to the + +5 + +recombination lifetime 𝜏. This is indeed borne out in +our experiments where the steady-state GS +luminescence peak near misfit dislocations is darker +by about 25% (see Section 3.2), comparable to the +reduction in lifetime. We see a similar trend for the +ES. +In addition to the faster decay at dislocations, +there are some features present across the system that +are worth noting. Figures 2c-d show a consistently +faster decay of the ES intensity compared to the GS +both near to and away from dislocations. Dissimilar +decay behavior of the ES and GS arise when their +occupancy is not in steady state equilibrium with +each other and is expected at low temperatures.29,30 +Nevertheless, previous work has shown that the ES +and GS start to mirror each other at temperatures +above 120 K (for a 60 meV GS-ES energy +separation) as the states come into equilibrium with +each other.29 Although our QDs have a slightly larger +ES-GS energy separation (about 70 meV), finding +dissimilar decay at room temperature is unexpected. +Osborne et al. report an anomalous situation in strong +electrically pumped InAs dots-in-a-well structure at +room temperature where they see the ESs between +dots in quasi-equilibrium and the same for the GSs, +but unexpectedly, within each dot the ES and GS are +not in equilibrium.31 That is, the ES and GS have +different quasi-Fermi energy separations under bias +even at room temperature. More work is needed to +understand if a similar situation arises in our system +that could lead to dissimilar ES and GS decay even +at room temperature. +We also note that the GS and ES intensity decay +are also slightly non-exponential both near to and +away from dislocations as the intensity reduces. +Several groups have reported biexponential decay +(i.e., a fast and a slow component) of the ES +luminescence at cryogenic temperatures.30,32 In our + + +Fig. 2. (a-d) Cathodoluminescence intensity decay traces at room temperature as a function of probe current from 15–90 pA for the +ground state (a) near to and (b) away from misfit dislocations, and the excited state (c) near to and (d) away from misfit dislocations. +The insets show the 1/e lifetimes for each decay trace. (e) Continuous wave cathodoluminescence intensity and (f) 1/e decay lifetime +of the same region obtained using a pulsed electron source. The one-to-one correspondence between these two regions demonstrates +that nonradiative recombination via dislocation-related traps limits spontaneous emission. (g) Time-position trace of +cathodoluminescence intensity across a misfit dislocation (located at 5 µm) taken from the yellow dashed rectangle marked in (e) +and (f). A constant width region of reduced intensity corresponding to the misfit dislocation indicates minimal lateral diffusion in +the InAs QD system within the experiment window. + +(a) GS: MD-Free +(b) GS: With MD +(e) Cw-lntensity (a.u.) +(f) Lifetime (ns) +0.3 +0.5 +0.3 +18 +0 +103 +103 +0 +0.4 +0.2 +0 +0.2 +0 +6 +0.15 +0.3 +102 +0.1 +102 +0.1 +0 +50 +100 +4 +0 +50 +100 +Probe current (pA) +Probe current (pA) +0.2 +2 +(a.u.) +101 +101 +0.1 +2 μm +um +Intensity ( +0 +1 +2 +3 +0 +1 +2 +3 +0 +0 +(c) ES: MD-Free +(d) ES: With MD +(g) +0.3 +0.3 +5 +d +103 +103 +4 +0 +0.2 +0.2 +0 +0 +0 +2 +102 +102 +0.1 +0.1 +50 +100 : +50 +100 +0 +0 +Probe current (pA) +Probe current (pA) +0.5 +Time +1.5 +101 +101 +10 +8 +16 +2.0 +4 +(ns) +12 +2.5° +0 +1 +2 +3 +0 +1 +2 +3 +0 +Distance (μm) +Time (ns)6 + +room-temperature case, it is likely that the origin of +non-exponential behavior lies in nonradiative +recombination in a disordered system. If dot sizes are +inhomogeneous, the dots with deeper confinement +lose carriers to traps at a slower rate than shallow +dots, once again hinting that global equilibrium is not +achieved even at room temperature in these high +excitation conditions. We cannot be more definitive +about this since our probe directly follows the carrier +concentration only over a small range of QD sizes +(set by the instrument spectral bandwidth of 2 nm) +but still probes other QD sizes indirectly through +carrier thermalization and recapture. + +B. In-situ view of recombination-enhanced +dislocation glide +The process of nonradiative recombination at +dislocations in InAs QDs so far assumes a fixed +number of dislocations. However, this is not true in +practice. Mismatch in the thermal expansion +coefficient of the III-V layers and Si leads to growing +tensile strain during cooldown after growth, causing +the multi-micron-thick III-V layers to exceed the +critical thickness for dislocation glide. While +threading dislocations in the epilayers do glide to a +certain extent and result in the misfit dislocations +characterized earlier, they effectively freeze once +temperatures drop below 300 °C, typically leaving a +residual strain of about 0.15% at room temperature. +It is now well known that nonradiative carrier +recombination at the dislocation core can revive +glide even at room temperature via aptly termed +recombination-enhanced +dislocation +glide.33–35 +Figure 3a shows a time-lapse sequence of +panchromatic cathodoluminescence (CL) images +collected +in +plan-view, +primarily +imaging +luminescence from the QDs. The sequence of images +shows the lengthening of certain misfit dislocation +segments along 〈110〉 directions after repeated scans. +The primary electron beam generates electron-hole +pairs that recombine nonradiatively at dislocations +and, under the right circumstances, lengthen misfit +dislocations by recombination-enhanced dislocation +glide. We also see significantly more extension of +misfit dislocations along the [1310] direction over the +[110]. In undoped GaAs, α-type dislocation glide is +much faster than β- and screw-type dislocations.36 +Thus, we are likely primarily seeing reverse-glide of +α-type threading dislocations.37 +We probe the impact of the newly grown misfit +dislocation in the region marked using the yellow- +dotted box (Figure 3a) on QD luminescence in situ. +Figure 3b shows luminescence spectra collected over +this boxed region before and after the single misfit +dislocation grows under it. We measure about a 25% +decrease in GS peak luminescence and a 40% +decrease in ES luminescence. This difference is +reasonable as the lower steady-state carrier +concentration near the dislocation implies relatively +fewer ES states are filled over GS states. While the +newly grown defect reduces the local emission +intensity, interestingly, there is no accompanying +shift in the luminescence spectrum due to the strong +and local strain field of the dislocation. We think this +is a consequence of the large interaction volume of +the electron beam compared to the extent of the strain +field: the dislocation strain field locally affects only + + + +Fig. 3. (a) Time-lapse images of recombination-enhanced +dislocation glide induced by the scanning electron beam and +residual strain in the III-V layer due to thermal expansion +mismatch with the silicon substrate. The growing misfit +dislocation +contrast +is +captured +using +panchromatic +cathodoluminescence (CL) mapping. The time-lapse was +generated from a 6 kV 30 nA scanning electron beam rastered +over a 256 um2 area. Each frame in the figure is separated by +30 minutes of scan time. (b) The integrated CL spectra from the +yellow dashed rectangle in (a) capture the impact of a misfit +dislocation growing. + +200 +(a) +(b) +GS +Before +150 +MD +CL intensity +100 +ES +After +50 +MD +0 +1100 +1150 +1200 +1250 +1300 +2 μm +[110] +Wavelength (nm7 + +a small number of QDs whereas carrier generation, +diffusion, and nonradiative recombination affect a +much large number of QDs. + +C. Impact of remote misfit dislocations on +quantum dot formation +In surveying a wider area of the sample, we find +large spatial inhomogeneities in QD emission +wavelength and intensity that are distinct from the +more local nonradiative effects of dislocations +described thus far. Our observation is a potentially +important consequence of growth on silicon as the +uniformity of emission is key for laser gain and +optical isolation. Figure 4a shows a map of the peak +GS emission wavelength, respectively, from this +sample, exhibiting wide, blue-shifted wavelength +bands in a crosshatch-like pattern aligned to the +〈110〉 directions. The bands are spaced much wider +than the beam interaction cross-section of 100-200 +nm diameter convolved with a 1 μm carrier diffusion +radius, which points to a long-range effect rather than +the typical inhomogeneous broadening from dot-to- +dot variation. Each pixel in the map probes +luminescence collectively from several hundred QDs +(hence already inhomogeneously broadened). A +similar sample grown on a GaAs substrate does not +exhibit these wide bands of wavelength variation +(Fig. 4b), confirming their origin in growth on +silicon. Along these blue-shifted bands, the GS +emission intensity is also moderately reduced by 10- +15% (Fig. 4c). We reiterate that these features are not +to be confused with the much more prominent dark +regions stemming from the local misfit dislocation +network, since, as is clear here and as shown +previously in Fig. 3b, these local misfit dislocations +are not associated with a wavelength shift. For the +sample on GaAs (Fig. 4d), the GS emission intensity +is +much +more +uniform, +as +expected. +The +corresponding maps for the excited state are shown +in Figure S1 and show comparable features to the +ground state but with a clearer correlation between +blue-shifted bands and reduced emission. + + +Fig. 4. (a-b) Peak emission wavelength of the ground state for InAs QDs grown (a) on silicon and (b) on GaAs collected using +steady-state cathodoluminescence hyperspectral imaging. (c-d) Total emission intensity (Gaussian fit) from the ground state (c) on +silicon and (d) on GaAs. In addition to sharply reduced intensity at misfit dislocations, a crosshatching in emission intensity and +emission wavelength occurs with a reduced intensity in blue-shifted regions. (e) Comparison of a typical pixel spectrum (determined +as spectrum with the median GS peak wavelength) (red) to the distribution of peak wavelengths for all spectra in the CL map (black). +(f) This same comparison for the sample on GaAs. Comparing (e) and (f), the GaAs sample clearly has a smaller distribution of peak +wavelengths; however, both are small compared to the FWHM of the typical spectrum, so overall broadening due to the larger +distribution on silicon is muted. + +Gs Wavelength (nm) +Gs Intensity (a.u.) +250 +(e) +Median +(a) +(c) +1260 +spectrum +200 +(by peak +2 +wavelength) +Intensity +150 +1250 +Silicon +100 +substrate +1240 +Peak +50 +wavelength +1230 +distribution +4 μm +4 μm +1 +0 +1200 +1250 +1300 +0 +200 +(b) +(d) +(f) +1290 +:n: +150 +2 +1280 +GaAs +100 +substrate +1270 +50 +d +4 μm +4 μm +1260 +0 +0 +1220 1240 1260 1280 1300 1320 +Wavelength (nm)8 + +We hypothesize that these darkened, blue-shifted +bands arise from the misfit dislocation network lying +at the threading dislocation filters layers 650 nm +below the QDs, which generates long-range strain +fields that alter the growth, and hence emission +wavelength and intensity, of the InAs QDs. This +points to the important role of dislocation strain +fields in influencing the motion of adatoms, +particularly indium, during growth and in subtly +altering QD formation. The presence of a network of +misfit dislocations is known to alter growth rates,9,38 +generate compositional variations in III-V alloy +metamorphic layers,8 and introduce fluctuating +surface step densities.39 +One might expect the significant spatial variation +in GS emission seen in CL to be detected by a more +routine, spatially unresolved photoluminescence +(PL) experiment as a broadened emission peak, but +this may often not be the case. We examine the +magnitude of this effect in Fig. 4e where we compare +the GS peak of a typical pixel to the distribution of +all spectra peak wavelengths, weighted by peak +intensity. Convolving these two approximately +Gaussian distributions gives an approximation of the +FWHM when sampling a large area, as is done for +typical PL measurements. Despite the significant +distribution of peak wavelengths, averaging the +spectra over the entire CL map only broadens the +FWHM by 1.0 meV or about 2% compared to a +typical single pixel FWHM of 44.3 meV. This can be +understood by recalling that when convolving two +Gaussians, the FWHMs combine as the root of the +sum of the squares, so the broadening effect of the +relatively tight peak wavelength distribution is +greatly suppressed. Comparing to the sample grown +on GaAs (Fig. 4f), where these spatial variations are +absent, the broadening is negligible with a FWHM of +38.5 meV for both the median pixel and the entire +image. While the broadening is certainly larger for +the sample on silicon, it is still too small to +distinguish from typical sample-to-sample variation. +Therefore, +spectral +measurements +made +by +photoluminescence (PL), a commonly relied upon +tool for assessing growth quality, will in many cases +be ineffective at detecting this non-uniform +crosshatched emission. Further, the associated +intensity reduction can also be obscured because PL +intensities are generally not comparable between +samples and particularly between different substrate +types due to differences in reflection at the interface. +However, micro-PL mapping with a sufficiently +small spot size should be capable of detecting these +local wavelength and intensity variations. +Solutions to reduce crosshatch nonuniformity +require either reducing adatom diffusivity8 (by +increasing the V/III ratio, for example) or increasing +the spacing between the misfit dislocation network +and the active layer.7 During growth of our single- +QD-layer sample, the nearest misfit dislocation +network lies 650 nm below the QDs at the defect +filter layer as shown in Fig. 5e (remember that the +other sparse misfit dislocation network adjacent to +the QDs only forms later during cooldown). +Fortunately, the nearest misfit dislocation network in +a typical QD laser is often about twice as distant due +to a thick lower AlGaAs cladding. Indeed, we see no +crosshatch-like spatial variations (Fig. S2) in a CL +map of the active layer from a laser bar, despite a +modest density of misfit dislocations formed by post- +growth thermal glide. This confirms our hypothesis +of long-range strain fields from the buffer as the +underlying cause behind crosshatched emission +wavelength. Even so, future laser designs, intended +to better couple the optical mode from the III-V gain +region into silicon and to reduce the likelihood of +film cracking call for much thinner buffers and +cladding layers.40–42 If such lasers are directly grown +on silicon, the misfit dislocation network may be +close enough to the active region to result in +undesirable luminescence broadening. + +D. +Growth +modification +near +threading +dislocations +We have seen that the distant misfit dislocation +network influences QD growth itself by altering +some combination of the composition, morphology, + +9 + +and thickness of the layers. Yet, the influence of +these remote misfit dislocations must be small +compared to threading dislocations continuously +intersecting the growth surface at a point that may +not change much over time. This allows growth +impacts to accumulate, in some cases forming +growth mounds or hillocks due to locally accelerated +growth at spiral step edges. We locate a cluster of +threading dislocations shown in Figure 5a using +ECCI and place a fiducial marker to co-locate this +site in CL and APT. Some threading dislocations +appear at the center of hillocks, demonstrating their +potential impact on surface morphology. Fig. 5b +shows significantly dimmer and blue-shifted +emission from the hillock center compared to a +region away from the hillock, with no clear GS or ES +peaks identified from the former. Fig. 5c shows that +the region near the hillock with strongly blue-shifted +QD peak emission wavelength overlaps almost +exactly with the region of reduced intensity in Fig. +5d. This correspondence likely arises as carriers +more easily thermalize out of the GS of the shallower +blue-shifted QDs to recombine nonradiatively at the +cluster of adjacent threading dislocations. Still, we +note that it is primarily the hillock and not the +threading dislocations themselves that induce these +changes: individual threading dislocations not +associated with hillocks elsewhere in the film do not +show such blue-shifted emission. Furthermore, it +appears that clusters of immobile threading +dislocations are required to form hillocks large +enough to see these effects, so reducing threading +dislocation densities should significantly reduce their +incidence. +Hillocks arise due to the spiraling nature of +surface steps at threading dislocations that have a +screw-component to their Burgers vector. The +increased density of step edges surrounding the +hillock provides additional nucleation sites for QDs, +which may result in a greater number of smaller (in +volume), and hence bluer-emitting, QDs. We see +tentative evidence for this in cross-sectional STEM +of a region containing a threading dislocation with a +hillock shown in Fig. 5e. When tilted away from the +zone axis, the growth plane containing the QDs at the +defect-free region is viewed at an angle in projection +(Fig. 5f). When viewing the hillock region in this +same tilt condition, the QD growth plane is viewed +edge on (Fig. 5g), indicating this growth plane is +inclined relative to the zone axis, since this narrow +slice of QDs are grown along the side of a hillock. It + + + +Fig. 5. (a) Electron contrast channeling image (ECCI) of a +cluster of threading dislocations forming a hillock. (b) +Emission spectra from the center and away from the hillock. (c) +Peak emission wavelength and (d) peak emission intensity +surrounding the hillock region shown in (a). (e) Cross-sectional +scanning transmission electron microscopy (STEM) of a region +containing a hillock capturing a threading dislocation and a +perceived local widening of the active region. High +magnification view at (f) a defect-free region showing a low- +angle side view of individual InAs QDs due to manual tilting +of the foil and (g) the hillock containing a threading dislocation +for the same foil tilt, but here, the QDs are viewed edge-on due +to compensating tilt of the growth plane surrounding the +hillock. + +1000 +(a) +CL Intensity (a.u.) +(b) +Defect-free +Hillock +500 +[110] +1 μm +[110] +0 +1100 +1200 +1300 +(nm) +Wavelength (nm) (a.u.) × 104 +(c) +1270 +(d) +10 +1260 +1250 +5 +1240 +1μm +1230 +1 μm +1220 +0 +(e) +(f) +Defect-free +Hillock +QD +location +layer +50nm +(g) +Hillock +Dislocation +filterlayers +500nm +50nm10 + +is also worth considering why hillocks do not feature +prominently +in +conventional +III-V +lattice- +mismatched (metamorphic) growth but do so in our +samples. Typically, threading dislocations glide +rapidly to relieve strain during growth and tend not +to stay in one place long enough to yield a hillock. +We speculate that a combination of near on-axis +(001) substrate (limiting the density of contending +steps) and sessile threading dislocations that arise at +the GaAs/Si interface or by dislocation reactions +result in hillocks. +To probe the structural and compositional +changes caused by these hillocks in more detail, we +extract tips for laser-pulsed APT from the TD- +impacted hillock region and from a nominally TD- +free region next to it. Note that the shallow 100 nm +depth of our QDs that enables CL imaging (and +ECCI), dramatically reduces the likelihood of +capturing a QD in the APT tip since the conical tip +diameter is very small near the top. Indeed, we see +from the top-down views in Fig. 6a that neither tip +has regions of high indium concentration as would +be expected from a QD, indicating that both tips +probe only the InGaAs QW that encases the QDs. +Nevertheless, +the +fluctuations +are +essentially +consistent with those of a random alloy of InGaAs, +as shown in Fig. S3 and no evidence for phase +separation or clustering is seen. On the other hand, +the cross-sectional indium profiles of each tip in Fig. +6b reveal that the QW in the defective region is +significantly thicker than in the TD-free region. +Collapsing these down to one-dimensional vertical +profiles of indium composition, averaged laterally +over the center of the tip, we see in Fig. 6c that the +QW in the hillock is about 8-9 nm thick with a +tapering indium profile, contrasted with a 7 nm thick +QW with a slightly less tapered profile seen in the tip +from the TD-free region. Some of this taper, along +with indium concentrations elevated above the +expected 15% nominal value, may be explained by +the unresolved InAs wetting layer (consistent with +other APT43,44 and STEM45 studies) that lies 2 nm +above the base of the QW. However, the additional +thickness of the QW is an effect of the hillock. +Reiterating that the hillock regions contain a +higher density of surface steps, if the availability of +steps limits the incorporation of adatoms, any +asymmetry between the diffusivity of indium and +gallium may lead to preferential incorporation of +indium in such hillocks. However, the vertical + + +Fig. 6. (a) Top view and (b) side view of the lateral indium +composition in the nominally In0.15Ga0.85As quantum well that +surrounds the InAs quantum dots at a threading dislocation +containing hillock, similar to that in Fig. 5 (left) and at a +neighboring threading dislocation (TD)-free region (right). +Data collected using site-selective laser atom probe +tomography informed by cathodoluminescence and electron +channeling contrast imaging. No quantum dots were captured +in the analysis due to the limited cross-sectional area of the APT +tip possible from the 100 nm shallow structure. (c) Vertical +composition trace through the quantum well showing a region +of tapered but similar composition profiles for the two sites, but +increased thickness for the defective region. Error bars +representing one standard deviation are indicated by the dotted +lines. + + +(a) +Hillock region +Defect-free region +0.25 +site) +0.20 +Il dnojb) +0.15 +0.10 +% +0.05 +10 nm +(b) +0.25 +[001] +0.20 +site +0.15 +(group +0.10 +In +0.05 +% +5 nm +0 +(c) +0.30 +Hillock region +Defect-free +0.25 +0.20 +[001] +0.15 +fraction ( +0.10 +0.05 +0.00 +-2 +0 +2 +4 +6 +8 +10 +12 +Position from bottom of QW (nm)11 + +profiles show near identical indium incorporation for +the first 7 nm in both sites; the hillock simply extends +this tapered profile for an additional 1–2 nm. This +suggests that the growth rate increases at the hillock +without any alteration in the composition, and both +indium and gallium are quite mobile on the growth +surface and incorporate at the hillock without +preference. Without direct access to the composition +or shape of the InAs QDs, we may only infer how the +altered QW affects the emission spectra. In addition +to easier thermalization from smaller blue-shifted +QDs, a locally thicker QW may have a ground state +closer in energy to the QDs and enhance carrier +thermalization out of the dots. Taken together, these +analyses demonstrate the serious impact both distant +misfit dislocations and local threading dislocations +can have in altering the growth of QDs and their +surrounding structures, ultimately broadening their +size distribution (and hence their emission spectrum) +and further aggravating nonradiative recombination. +Therefore, these effects must be closely considered +when tuning device design to optimize performance +and reliability. + +IV. CONCLUSIONS +With +the +large +untapped +potential +of +heterogeneous integration of dissimilar materials by +direct growth, it is important to understand the +microscale effect of dislocations on the final devices. +We have quantified how dislocations affect +spontaneous-emission luminescence in InAs QDs on +silicon by facilitating defect-assisted recombination +using +time-resolved +cathodoluminescence +spectroscopy on a model InAs QD structure on +silicon. +We +find +a +significantly +reduced +recombination lifetime for both the ground and +excited states at misfit dislocations but also find +recombination to be limited by defects in regions +away from dislocations. Yet, the impact of +dislocations goes much beyond simple nonradiative +recombination. We find, using hyperspectral CL +imaging and atom probe tomography, alterations in +QD and QW growth that form pockets of blue-shifted +emission arising from long range misfit dislocation +strain fields and short-range threading dislocation +spiral growth. Both yield reduced emission +homogeneity that increases susceptibility to carrier +losses. Our work shows how new characterization +tools may enable a more complete understanding of +the impact of dislocations on devices. InAs quantum +dots, currently yielding the most reliable devices, are +now part of a series of III-V laser devices being +synthesized on silicon spanning the visible to the +mid-infrared. As the field matures, we expect to see +multi-modal microstructural characterization of the +kind employed in this work to rise to prominence in +those devices as well. + +SUPPLEMENTARY MATERIAL +See supplementary material for (Fig. S1) +cathodoluminescence peak excited state wavelength +and intensity maps for samples on GaAs and silicon, +corresponding to the ground state maps in Fig 4a-d, +(Fig. S2) a cathodoluminescence wavelength map of +an InAs QD laser active region after milling away the +upper cladding, and (Fig. S3) an atom probe +compositional frequency distribution comparison of +the hillock and defect-free regions. + +ACKNOWLEDGEMENTS +The sample growth was supported by ARPA-E, +U.S. Department of Energy, under Award No. DE- +AR0001043. This material is based upon work +supported by the National Science Foundation (NSF) +Graduate Research Fellowship under Grant No. +1650114. APT and TEM studies were performed at +the UCSB MRL Shared Experimental Facilities, +supported by the MRSEC Program of the NSF under +Award No. DMR 1720256; a member of the NSF- +funded Materials Research Facilities Network. CL +studies were supported by the EPSRC under +EP/R025193/1. K.M. acknowledges additional +support from NSF CAREER award under grant no. +DMR-2036520. + +CONFLICTS OF INTEREST +The authors have no conflicts to disclose. + + +12 + +DATA AVAILABILITY +The data that support the findings of this study +are available from the corresponding author upon +reasonable request. + +REFERENCES +1 A.Y. Liu and J. Bowers, IEEE Journal of +Selected Topics in Quantum Electronics 24, 1 +(2018). +2 N. Margalit, C. Xiang, S.M. Bowers, A. +Bjorlin, R. Blum, and J.E. Bowers, Appl. Phys. +Lett. 118, 220501 (2021). +3 E. Simoen, B. Hsu, G. Eneman, E. Rosseel, R. +Loo, H. Arimura, N. Horiguchi, W.-C. Wen, H. +Nakashima, C. Claeys, A. Oliveira, P. Agopian, and +J. 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Heidelberger, C. Sorace-Agaskar, J.J. +Plant, D. Kharas, R.B. Swint, P. Dhingra, M.L. Lee, +and P.W. Juodawlkis, in 2021 IEEE Photonics +Conference (IPC) (2021), pp. 1–2. +42 C. Shang, K. Feng, E.T. Hughes, A. Clark, M. +Debnath, R. Koscica, G. Leake, J. Herman, D. +Harame, P. Ludewig, Y. Wan, and J.E. Bowers, +(2022). +43 M. Müller, A. Cerezo, G.D.W. Smith, L. +Chang, and S.S.A. Gerstl, Appl. Phys. Lett. 92, +233115 (2008). +44 A.D. Giddings, J.G. Keizer, M. Hara, G.J. +Hamhuis, H. Yuasa, H. Fukuzawa, and P.M. +Koenraad, Phys. Rev. B 83, 205308 (2011). +45 P. Wang, A.L. Bleloch, M. Falke, P.J. +Goodhew, J. Ng, and M. Missous, Appl. Phys. Lett. +89, 072111 (2006). + + + +14 + +Supplementary Material +Dislocation-induced structural and luminescence degradation in InAs quantum dot +emitters on silicon +Eamonn T. Hughes1, Gunnar Kusch2, Jennifer Selvidge1, Bastien Bonef1, Justin Norman1, Chen +Shang1, John E. Bowers1, Rachel A. Oliver2, Kunal Mukherjee3 +1Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA +2Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, +Cambridge CB3 0FS, United Kingdom +3Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA + + + +Figure S1. (a-b) Excited-state peak-emission wavelength cathodoluminescence map for the sample (a) on silicon +and (b) on GaAs. (c-d) Corresponding excited-state cathodoluminescence intensity maps for the sample (a) on silicon +and (b) on GaAs. + + + +Figure S2. Stitched cathodoluminescence map from a five-layer QD laser grown on silicon after milling away +upper cladding using a focused ion beam microscope. The spacing between the active region and uppermost defect +filter layer (which hosts a misfit dislocation network) is much larger here than in the single QD structure in the main +text. Consequently, the effects of extended misfit dislocation strain fields are weaker, so no distinct crosshatch pattern +is visible. Even so, there are wide variations in peak emission wavelength and several strongly blue shifted regions, +possibly due to hillocks formed by sessile threading dislocation clusters. + +Es Wavelength (nm) +Es Intensity (a.u.) +1.0 +(a) +(c) +1180 +1170 +Silicon +0.5 +substrate +1160 +4 μm +1150 +4 μm +0 +(b) +1200 +(d) +1.5 +1190 +GaAs +1.0 +substrate +1180 +0.5 +1170 +4 μm +4 μm +0Gs Wavelength (nm) +1305 +1300 +1295 +1290 +1285 +[110] +1280 +4 μm +1275 +127015 + + + + +Figure S3. Compositional frequency distribution measured from the bottom 2 nm of the QW analyzed in the two +atom probe tomography specimens, which roughly aligns with the expected location of any QDs and the wetting layer. +The dashed curve is a binomial fit representing the expected compositional distribution for a random alloy. The p- +values estimate the probability that the observed distributions represent a random alloy, therefore, both alloys appear +to be randomly distributed with no indication of a quantum dot or partial quantum dot present in either. The bin size +for composition measurements is 50 atoms. + + + +Hillock region +Defect-free region +30 +30 +p = 0.66 +p = 0.75 +20 +20 +Counts +Counts +10 +10 +0 +10 +20 +30 +0 +10 +20 +30 +Composition (%) +Composition (%) \ No newline at end of file diff --git a/h9E2T4oBgHgl3EQfHwY9/content/tmp_files/load_file.txt b/h9E2T4oBgHgl3EQfHwY9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..739a32589f990a79f0de4e74c6e435231e6d53cf --- /dev/null +++ b/h9E2T4oBgHgl3EQfHwY9/content/tmp_files/load_file.txt @@ -0,0 +1,934 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf,len=933 +page_content='1 Dislocation-induced structural and luminescence degradation in InAs quantum dot emitters on silicon Eamonn T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Hughes1, Gunnar Kusch2, Jennifer Selvidge1, Bastien Bonef1, Justin Norman1, Chen Shang1, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Bowers1, Rachel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Oliver2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Kunal Mukherjee3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='a 1Materials Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' University of California Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' California 93106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' USA 2Department of Materials Science and Metallurgy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 27 Charles Babbage Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Cambridge CB3 0FS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' United Kingdom 3Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Stanford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' California 94305,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' USA ABSTRACT We probe the extent to which dislocations reduce carrier lifetimes and alter luminescence and growth morphology in InAs quantum dots (QD) grown on silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' These heterostructures are key ingredients to achieving a highly reliable monolithically integrated light source on silicon necessary for photonic integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We find up to 20–30% shorter carrier lifetimes at spatially resolved individual dislocations from both the QD ground and excited states at room temperature using time-resolved cathodoluminescence spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' These lifetimes are consistent with differences in the intensity measured under steady-state excitation suggesting that trap-assisted recombination limits the minority carrier lifetime, even away from dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Our techniques also reveal the dramatic growth of misfit dislocations in these structures under carrier injection fueled by recombination-enhanced dislocation glide and III-V/Si residual strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Beyond these direct effects of increased nonradiative recombination, we find the long-range strain field of misfit dislocations deeper in the defect filter layers employed during III-V/Si growth alter the QD growth environment and introduce a crosshatch-like variation in the QD emission color and intensity when the filter layer is positioned close to the QD emitter layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Sessile threading dislocations generate even more egregious hillock defects that also reduce emission intensities by altering layer thicknesses, as measured by transmission electron microscopy and atom probe tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Our work presents a more complete picture of the impacts of dislocations relevant for the development of light sources for scalable silicon photonic integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' _________________________ a Corresponding author: kunalm@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='edu 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' INTRODUCTION Understanding how dislocations affect the properties of optoelectronic devices like lasers and photodetectors is central to efforts in direct epitaxial heterogeneous integration of active devices for silicon photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1 In practice, getting lasers to operate efficiently and reliably in the presence of dislocations remains a key challenge in III-V/Si integration to serve the growing communications market at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2 While advances in experimental and computational methods continue to uncover structure-property relationships of dislocation- induced electronic states and nonradiative recombination of charge carriers in semiconductors like Ge,3 GaAs,4 and GaN,5,6 there is a growing realization that the impact of dislocations goes beyond this static and often idealized picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Dislocations may affect a heterostructure device even before device operation by altering the local composition or growth rates during synthesis, exemplified by prior work on dislocation-induced phase separation in alloys7,8 and roughening surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='9,10 Dislocations continue to modify device behavior long after fabrication by diffusing or transporting dopants and other impurities during device operation11,12 or, more dramatically, by damaging devices via recombination-enhanced dislocation motion where dislocations inject point defects and subsequently increase in length over time via dislocation climb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='13–16 Understanding these broader impacts of dislocations will further the development of self- assembled epitaxial InAs quantum dot (QD) lasers on silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='17–20 These are the most dislocation tolerant datacom-band lasers directly grown on silicon, but we need to continue to improve reliability at high current and high temperatures as well as improve manufacturability and uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Both tasks necessitate a detailed representation of dislocation behavior and their local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' One important consideration is the direct impact of dislocations on QD formation as the epitaxial growth window for QDs is narrow, and hence more sensitive to perturbations than conventional III-V quantum well (QW) heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Additionally, nonradiative recombination of charge carriers at dislocations in QD systems remains to be fully characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Finally, InAs QD devices retain a sizeable thermal strain due to the silicon substrate that continues to drive recombination-enhanced dislocation motion during operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='21 In this work, we use a combination of microanalysis techniques on a model shallow (near to the growth surface) layer of InAs QDs on Si to show that dislocations not only reduce excess carrier lifetimes and emission intensities at room temperature, but they also introduce non-trivial crosshatch- and hillock-induced compositional shifts that locally alter the QD energy levels and intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Properly accounting for these effects in laser design and growth can yield improved laser performance and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Our work also anticipates the complex effects of dislocations in the next generation of III-V lasers grown directly on silicon beyond datacom wavelengths such as recent works in the visible22 and mid-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='23 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' EXPERIMENTAL METHODS AND SAMPLE DETAILS The InAs QD model structure investigated here was previously reported in a multi-modal characterization study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='21 Briefly, we use molecular beam epitaxy (MBE) to synthesize the structure depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1a with an active layer consisting of a single shallow InAs QD layer embedded in a 7 nm In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='85As quantum well and capped by a 100 nm thick GaAs layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The QD layer is not intentionally doped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The active layer is grown on a GaAs-on-Si template used for an earlier generation of QD lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The template consists of two separate defect-filter structures—a 200 nm thick continuous InGaAs layer and a 10-period strained-layer superlattice of 10nm/10nm In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='9As/GaAs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5e for a cross-sectional scanning transmission electron microscopy (STEM) image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The threading dislocation density in the sample is 7×106 cm-2, and the InAs QD density is approximately 5×1010 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We also label the locations of misfit dislocation 3 networks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1a, which will be relevant for later analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The growth conditions (temperature, V/III ratio, growth rate) of the various layers have been described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='21 Optical characterization on the nanoscale was performed by cathodoluminescence spectroscopy (CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The CL measurements were carried out in an Attolight Allalin 4027 Chronos dedicated CL scanning electron microscope (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' CL hyperspectral maps were recorded with an Andor Kymera 328i spectrometer with a focal length of 328 mm, a 150-lines-per-mm grating blazed at 1250 nm, and an Andor 512 px InGaAs diode array camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Time-resolved CL measurements were performed by triggering the electron gun with the third harmonic of a Nd:YAG laser (355 nm) with a frequency of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='6 MHz and a pulse width of 7 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' All CL time decay curves were recorded with a time-correlated single photon counting (TCSPC) setup resulting in a time resolution of about 100 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' All CL measurements were performed at room temperature with an acceleration voltage of 6 kV (interaction region is ~75 nm radius sphere tangential to sample surface) and a beam current of 30 nA for continuous wave measurements and between 15 pA and 90 pA for pulsed measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Atom probe tips were created using an FEI Helios Dualbeam Nanolab 600 focused ion beam (FIB) microscope using standard 30 kV annular milling steps and a 2 kV broad-area polish to form the final tip shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Tips were evaporated using a Cameca 3000X HR Local Electrode Atom Probe (LEAP) at 40 K with laser pulsing at a 532 nm wavelength, a 200 kHz repetition rate, and a laser pulse energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='20 nJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' TEM foils were prepared using the FEI Helios Dualbeam FIB and imaged using a ThermoFisher Talos in STEM mode using a bright field detector with a collection angle of 17 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Electron channeling contrast imaging (ECCI) was performed on a ThermoFisher Apreo S SEM using a three-beam g=040 and g=220 channeling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figure 1b shows a plan-view ECCI image of the structure showing numerous long segments of misfit dislocations along with threading dislocations, which together form the subjects of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We have previously determined that these misfit dislocations form just below the 7 nm InGaAs QW/GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='21 The origins of these misfit dislocations, which appear in layers grown nominally below the critical thickness for dislocation glide, is also important to contextualize our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Briefly, these misfit dislocations form not during growth, but after growth as the sample cools due to a combination of: (1) residual tensile strain in the III-V layers due to thermal expansion mismatch with silicon and (2) local pinning of the threading dislocation segment by the InAs QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='24 The formation process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1c where only unpinned threading dislocation segments glide below the QD layer to form misfit dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We have identified these misfit dislocations as being primarily responsible for degradation in early generations of GaAs-based lasers on silicon and, more recently, in InAs quantum dot lasers on silicon where their effects can now be largely mitigated using strained indium-containing trapping layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) Structure of sample characterized in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (b) Electron-channeling contrast imaging (ECCI) from the sample surface showing a moderate density of threading dislocations and misfit dislocations located just below the QD layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (c) Illustration of the misfit dislocation formation process in which thermal expansion misfit stress generated during cooldown propels free threading dislocations below the QD layer to glide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The TD segment is pinned through the QD layer and cannot follow the lower thread segment, so a misfit dislocation forms here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) (b) Emitter region GaAs (100 nm) InAs QDs in Ino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='85As QW (7 nm) Threading Misfit Misfit dislocation dislocations dislocations networks GaAs (650 nm) Dislocation filter layers 10 nm/10 nm Ino1GaogAs/GaAs (10x) 2 μm GaAs (300 nm) [110] Ino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='gAs (200 nm) (c) GaAs Cap QD layer GaAs (1600 nm) III-V/Si GaAs on GaP/Si4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Recombination dynamics at dislocations We use time-resolved CL using a pulsed primary electron beam to probe the effect of dislocations on carrier recombination at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Our results show the misfit dislocations lying close to the InAs QD layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1c) are potent nonradiative recombination sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figures 2a-d shows CL intensity decay traces as a function of increasing probe current collected at a dislocation-free region and a region with misfit dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The signal is spectrally filtered to separately track the CL intensity decay of the GS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 2a-b) and ES (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 2c-d) luminescence at 1250 nm and 1167 nm, respectively, with a 2 nm bandwidth, hence we directly probe only the occupation of dots emitting in these narrow ranges and indirectly probe most remaining dots via their carrier exchange with the wetting layer, due to fast carrier equilibration at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The insets in these figures show that the recombination lifetime in both regions, obtained by fitting to a single- exponential decay, are in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 ns range and do not vary much with probe current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Upon initial inspection, we find the expected outcome that carriers recombine faster near the misfit dislocation, noting a 20% shorter GS recombination lifetime at the lowest probe current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The ES luminescence decays about 30% faster at the misfit dislocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figures 2e and 2f show a steady steady-state- excitation CL luminescence map (GS) and a corresponding pulsed-excitation carrier lifetime map obtained from each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Comparing the two, we see a clear correlation between the CL intensity and luminescence lifetimes, typical of defect-limited recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figure 2g follows the TRCL decay along a trace that is orthogonal to a misfit dislocation (or group of misfit dislocations) at the center of the distance axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' When carriers are injected directly over the dislocations, nonradiative recombination reduces carrier concentration even at the shortest resolvable time scales (~100 ps, estimated from the signal rise- time), leading to a lowered initial peak intensity at t≈0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We may assume that minimal carrier diffusion takes place within this time, so the roughly 1 µm lateral extent of reduced intensity is the convolution of the defect size and the cross section probed by the electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Carriers injected further away from the misfit dislocation should eventually diffuse towards this defect, leading to a widening of the reduced intensity valley with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Yet, we find that the lateral extent of reduced intensity remains constant even on the longer time scale of 1–2 ns as the luminescence decays, visualized as a trench of apparent constant width in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Using this information, we obtain an upper bound for the diffusivity of carriers in this system using 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' = √𝐷𝜏, estimating an ambipolar diffusivity 𝐷 of less than 40 cm2/s for the measured recombination lifetime 𝜏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='25 ns in dislocation-free regions (Fig 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This corresponds to a diffusion length, 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=', of less than 1 μm, which is shorter than reported values for quantum-well systems in GaAs and reinforces a key mechanism behind the dislocation tolerance of InAs QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='25,26 At this time, we are unable to resolve the properties of isolated threading dislocations, but their impact appears minimal compared to misfit dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We expect these short carrier lifetimes are set by trap-assisted recombination away from dislocations that, naturally, become even shorter at dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Bimberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' use PL to measure a spontaneous recombination lifetime, 𝜏", in the GS of InAs QDs of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='8 ns, which is independent of injection over a pulse excitation range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1–100 kW/cm2 at 77 K and only weakly temperature dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='27 Fiore et al measure an effective lifetime of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='8 ns from a single InAs QD layer in an In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='85As quantum well using PL at room temperature at very low excitation of 9 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='28 The much shorter recombination lifetimes measured in our experiments possibly points to elevated point defect concentrations even in regions away from dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Under these constraints, the internal quantum efficiency of spontaneous emission is 𝜂 = #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='" #"$#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='" ≈ #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='" #" , and the recombination lifetime is 𝜏 = #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' "#" #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' "$#" ≈ 𝜏%".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Hence, the steady state luminescence of the GS is proportional to the 5 recombination lifetime 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This is indeed borne out in our experiments where the steady-state GS luminescence peak near misfit dislocations is darker by about 25% (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2), comparable to the reduction in lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We see a similar trend for the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' In addition to the faster decay at dislocations, there are some features present across the system that are worth noting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figures 2c-d show a consistently faster decay of the ES intensity compared to the GS both near to and away from dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Dissimilar decay behavior of the ES and GS arise when their occupancy is not in steady state equilibrium with each other and is expected at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='29,30 Nevertheless, previous work has shown that the ES and GS start to mirror each other at temperatures above 120 K (for a 60 meV GS-ES energy separation) as the states come into equilibrium with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='29 Although our QDs have a slightly larger ES-GS energy separation (about 70 meV), finding dissimilar decay at room temperature is unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Osborne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' report an anomalous situation in strong electrically pumped InAs dots-in-a-well structure at room temperature where they see the ESs between dots in quasi-equilibrium and the same for the GSs, but unexpectedly, within each dot the ES and GS are not in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='31 That is, the ES and GS have different quasi-Fermi energy separations under bias even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' More work is needed to understand if a similar situation arises in our system that could lead to dissimilar ES and GS decay even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We also note that the GS and ES intensity decay are also slightly non-exponential both near to and away from dislocations as the intensity reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Several groups have reported biexponential decay (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=', a fast and a slow component) of the ES luminescence at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='30,32 In our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a-d) Cathodoluminescence intensity decay traces at room temperature as a function of probe current from 15–90 pA for the ground state (a) near to and (b) away from misfit dislocations, and the excited state (c) near to and (d) away from misfit dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The insets show the 1/e lifetimes for each decay trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (e) Continuous wave cathodoluminescence intensity and (f) 1/e decay lifetime of the same region obtained using a pulsed electron source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The one-to-one correspondence between these two regions demonstrates that nonradiative recombination via dislocation-related traps limits spontaneous emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (g) Time-position trace of cathodoluminescence intensity across a misfit dislocation (located at 5 µm) taken from the yellow dashed rectangle marked in (e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' A constant width region of reduced intensity corresponding to the misfit dislocation indicates minimal lateral diffusion in the InAs QD system within the experiment window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) GS: MD-Free (b) GS: With MD (e) Cw-lntensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=') (f) Lifetime (ns) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 18 0 103 103 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2 0 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1 0 50 100 4 0 50 100 Probe current (pA) Probe current (pA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2 2 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=') 101 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1 2 μm um Intensity ( 0 1 2 3 0 1 2 3 0 0 (c) ES: MD-Free (d) ES: With MD (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 5 d 103 103 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='2 0 0 0 2 102 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='1 50 100 : 50 100 0 0 Probe current (pA) Probe current (pA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 101 101 10 8 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='0 4 (ns) 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5° 0 1 2 3 0 1 2 3 0 Distance (μm) Time (ns)6 room-temperature case, it is likely that the origin of non-exponential behavior lies in nonradiative recombination in a disordered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' If dot sizes are inhomogeneous, the dots with deeper confinement lose carriers to traps at a slower rate than shallow dots, once again hinting that global equilibrium is not achieved even at room temperature in these high excitation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We cannot be more definitive about this since our probe directly follows the carrier concentration only over a small range of QD sizes (set by the instrument spectral bandwidth of 2 nm) but still probes other QD sizes indirectly through carrier thermalization and recapture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' In-situ view of recombination-enhanced dislocation glide The process of nonradiative recombination at dislocations in InAs QDs so far assumes a fixed number of dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' However, this is not true in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Mismatch in the thermal expansion coefficient of the III-V layers and Si leads to growing tensile strain during cooldown after growth, causing the multi-micron-thick III-V layers to exceed the critical thickness for dislocation glide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' While threading dislocations in the epilayers do glide to a certain extent and result in the misfit dislocations characterized earlier, they effectively freeze once temperatures drop below 300 °C, typically leaving a residual strain of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15% at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' It is now well known that nonradiative carrier recombination at the dislocation core can revive glide even at room temperature via aptly termed recombination-enhanced dislocation glide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='33–35 Figure 3a shows a time-lapse sequence of panchromatic cathodoluminescence (CL) images collected in plan-view, primarily imaging luminescence from the QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The sequence of images shows the lengthening of certain misfit dislocation segments along 〈110〉 directions after repeated scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The primary electron beam generates electron-hole pairs that recombine nonradiatively at dislocations and, under the right circumstances, lengthen misfit dislocations by recombination-enhanced dislocation glide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We also see significantly more extension of misfit dislocations along the [1310] direction over the [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' In undoped GaAs, α-type dislocation glide is much faster than β- and screw-type dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='36 Thus, we are likely primarily seeing reverse-glide of α-type threading dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='37 We probe the impact of the newly grown misfit dislocation in the region marked using the yellow- dotted box (Figure 3a) on QD luminescence in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figure 3b shows luminescence spectra collected over this boxed region before and after the single misfit dislocation grows under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We measure about a 25% decrease in GS peak luminescence and a 40% decrease in ES luminescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This difference is reasonable as the lower steady-state carrier concentration near the dislocation implies relatively fewer ES states are filled over GS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' While the newly grown defect reduces the local emission intensity, interestingly, there is no accompanying shift in the luminescence spectrum due to the strong and local strain field of the dislocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We think this is a consequence of the large interaction volume of the electron beam compared to the extent of the strain field: the dislocation strain field locally affects only Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) Time-lapse images of recombination-enhanced dislocation glide induced by the scanning electron beam and residual strain in the III-V layer due to thermal expansion mismatch with the silicon substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The growing misfit dislocation contrast is captured using panchromatic cathodoluminescence (CL) mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The time-lapse was generated from a 6 kV 30 nA scanning electron beam rastered over a 256 um2 area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Each frame in the figure is separated by 30 minutes of scan time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (b) The integrated CL spectra from the yellow dashed rectangle in (a) capture the impact of a misfit dislocation growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 200 (a) (b) GS Before 150 MD CL intensity 100 ES After 50 MD 0 1100 1150 1200 1250 1300 2 μm [110] Wavelength (nm7 a small number of QDs whereas carrier generation, diffusion, and nonradiative recombination affect a much large number of QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Impact of remote misfit dislocations on quantum dot formation In surveying a wider area of the sample, we find large spatial inhomogeneities in QD emission wavelength and intensity that are distinct from the more local nonradiative effects of dislocations described thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Our observation is a potentially important consequence of growth on silicon as the uniformity of emission is key for laser gain and optical isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figure 4a shows a map of the peak GS emission wavelength, respectively, from this sample, exhibiting wide, blue-shifted wavelength bands in a crosshatch-like pattern aligned to the 〈110〉 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The bands are spaced much wider than the beam interaction cross-section of 100-200 nm diameter convolved with a 1 μm carrier diffusion radius, which points to a long-range effect rather than the typical inhomogeneous broadening from dot-to- dot variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Each pixel in the map probes luminescence collectively from several hundred QDs (hence already inhomogeneously broadened).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' A similar sample grown on a GaAs substrate does not exhibit these wide bands of wavelength variation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 4b), confirming their origin in growth on silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Along these blue-shifted bands, the GS emission intensity is also moderately reduced by 10- 15% (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We reiterate that these features are not to be confused with the much more prominent dark regions stemming from the local misfit dislocation network, since, as is clear here and as shown previously in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 3b, these local misfit dislocations are not associated with a wavelength shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' For the sample on GaAs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 4d), the GS emission intensity is much more uniform, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The corresponding maps for the excited state are shown in Figure S1 and show comparable features to the ground state but with a clearer correlation between blue-shifted bands and reduced emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a-b) Peak emission wavelength of the ground state for InAs QDs grown (a) on silicon and (b) on GaAs collected using steady-state cathodoluminescence hyperspectral imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (c-d) Total emission intensity (Gaussian fit) from the ground state (c) on silicon and (d) on GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' In addition to sharply reduced intensity at misfit dislocations, a crosshatching in emission intensity and emission wavelength occurs with a reduced intensity in blue-shifted regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (e) Comparison of a typical pixel spectrum (determined as spectrum with the median GS peak wavelength) (red) to the distribution of peak wavelengths for all spectra in the CL map (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (f) This same comparison for the sample on GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Comparing (e) and (f), the GaAs sample clearly has a smaller distribution of peak wavelengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' however, both are small compared to the FWHM of the typical spectrum, so overall broadening due to the larger distribution on silicon is muted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Gs Wavelength (nm) Gs Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=') 250 (e) Median (a) (c) 1260 spectrum 200 (by peak 2 wavelength) Intensity 150 1250 Silicon 100 substrate 1240 Peak 50 wavelength 1230 distribution 4 μm 4 μm 1 0 1200 1250 1300 0 200 (b) (d) (f) 1290 :n: 150 2 1280 GaAs 100 substrate 1270 50 d 4 μm 4 μm 1260 0 0 1220 1240 1260 1280 1300 1320 Wavelength (nm)8 We hypothesize that these darkened,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' blue-shifted bands arise from the misfit dislocation network lying at the threading dislocation filters layers 650 nm below the QDs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' which generates long-range strain fields that alter the growth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' and hence emission wavelength and intensity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' of the InAs QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This points to the important role of dislocation strain fields in influencing the motion of adatoms, particularly indium, during growth and in subtly altering QD formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The presence of a network of misfit dislocations is known to alter growth rates,9,38 generate compositional variations in III-V alloy metamorphic layers,8 and introduce fluctuating surface step densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='39 One might expect the significant spatial variation in GS emission seen in CL to be detected by a more routine, spatially unresolved photoluminescence (PL) experiment as a broadened emission peak, but this may often not be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We examine the magnitude of this effect in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 4e where we compare the GS peak of a typical pixel to the distribution of all spectra peak wavelengths, weighted by peak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Convolving these two approximately Gaussian distributions gives an approximation of the FWHM when sampling a large area, as is done for typical PL measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Despite the significant distribution of peak wavelengths, averaging the spectra over the entire CL map only broadens the FWHM by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='0 meV or about 2% compared to a typical single pixel FWHM of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='3 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This can be understood by recalling that when convolving two Gaussians, the FWHMs combine as the root of the sum of the squares, so the broadening effect of the relatively tight peak wavelength distribution is greatly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Comparing to the sample grown on GaAs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 4f), where these spatial variations are absent, the broadening is negligible with a FWHM of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 meV for both the median pixel and the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' While the broadening is certainly larger for the sample on silicon, it is still too small to distinguish from typical sample-to-sample variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Therefore, spectral measurements made by photoluminescence (PL), a commonly relied upon tool for assessing growth quality, will in many cases be ineffective at detecting this non-uniform crosshatched emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Further, the associated intensity reduction can also be obscured because PL intensities are generally not comparable between samples and particularly between different substrate types due to differences in reflection at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' However, micro-PL mapping with a sufficiently small spot size should be capable of detecting these local wavelength and intensity variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Solutions to reduce crosshatch nonuniformity require either reducing adatom diffusivity8 (by increasing the V/III ratio, for example) or increasing the spacing between the misfit dislocation network and the active layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='7 During growth of our single- QD-layer sample, the nearest misfit dislocation network lies 650 nm below the QDs at the defect filter layer as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5e (remember that the other sparse misfit dislocation network adjacent to the QDs only forms later during cooldown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Fortunately, the nearest misfit dislocation network in a typical QD laser is often about twice as distant due to a thick lower AlGaAs cladding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Indeed, we see no crosshatch-like spatial variations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' S2) in a CL map of the active layer from a laser bar, despite a modest density of misfit dislocations formed by post- growth thermal glide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This confirms our hypothesis of long-range strain fields from the buffer as the underlying cause behind crosshatched emission wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Even so, future laser designs, intended to better couple the optical mode from the III-V gain region into silicon and to reduce the likelihood of film cracking call for much thinner buffers and cladding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='40–42 If such lasers are directly grown on silicon, the misfit dislocation network may be close enough to the active region to result in undesirable luminescence broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Growth modification near threading dislocations We have seen that the distant misfit dislocation network influences QD growth itself by altering some combination of the composition, morphology, 9 and thickness of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Yet, the influence of these remote misfit dislocations must be small compared to threading dislocations continuously intersecting the growth surface at a point that may not change much over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This allows growth impacts to accumulate, in some cases forming growth mounds or hillocks due to locally accelerated growth at spiral step edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We locate a cluster of threading dislocations shown in Figure 5a using ECCI and place a fiducial marker to co-locate this site in CL and APT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Some threading dislocations appear at the center of hillocks, demonstrating their potential impact on surface morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5b shows significantly dimmer and blue-shifted emission from the hillock center compared to a region away from the hillock, with no clear GS or ES peaks identified from the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5c shows that the region near the hillock with strongly blue-shifted QD peak emission wavelength overlaps almost exactly with the region of reduced intensity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This correspondence likely arises as carriers more easily thermalize out of the GS of the shallower blue-shifted QDs to recombine nonradiatively at the cluster of adjacent threading dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Still, we note that it is primarily the hillock and not the threading dislocations themselves that induce these changes: individual threading dislocations not associated with hillocks elsewhere in the film do not show such blue-shifted emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Furthermore, it appears that clusters of immobile threading dislocations are required to form hillocks large enough to see these effects, so reducing threading dislocation densities should significantly reduce their incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Hillocks arise due to the spiraling nature of surface steps at threading dislocations that have a screw-component to their Burgers vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The increased density of step edges surrounding the hillock provides additional nucleation sites for QDs, which may result in a greater number of smaller (in volume), and hence bluer-emitting, QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We see tentative evidence for this in cross-sectional STEM of a region containing a threading dislocation with a hillock shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' When tilted away from the zone axis, the growth plane containing the QDs at the defect-free region is viewed at an angle in projection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' When viewing the hillock region in this same tilt condition, the QD growth plane is viewed edge on (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5g), indicating this growth plane is inclined relative to the zone axis, since this narrow slice of QDs are grown along the side of a hillock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' It Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) Electron contrast channeling image (ECCI) of a cluster of threading dislocations forming a hillock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (b) Emission spectra from the center and away from the hillock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (c) Peak emission wavelength and (d) peak emission intensity surrounding the hillock region shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (e) Cross-sectional scanning transmission electron microscopy (STEM) of a region containing a hillock capturing a threading dislocation and a perceived local widening of the active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' High magnification view at (f) a defect-free region showing a low- angle side view of individual InAs QDs due to manual tilting of the foil and (g) the hillock containing a threading dislocation for the same foil tilt, but here, the QDs are viewed edge-on due to compensating tilt of the growth plane surrounding the hillock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1000 (a) CL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=') (b) Defect-free Hillock 500 [110] 1 μm [110] 0 1100 1200 1300 (nm) Wavelength (nm) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=') × 104 (c) 1270 (d) 10 1260 1250 5 1240 1μm 1230 1 μm 1220 0 (e) (f) Defect-free Hillock QD location layer 50nm (g) Hillock Dislocation filterlayers 500nm 50nm10 is also worth considering why hillocks do not feature prominently in conventional III-V lattice- mismatched (metamorphic) growth but do so in our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Typically, threading dislocations glide rapidly to relieve strain during growth and tend not to stay in one place long enough to yield a hillock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We speculate that a combination of near on-axis (001) substrate (limiting the density of contending steps) and sessile threading dislocations that arise at the GaAs/Si interface or by dislocation reactions result in hillocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' To probe the structural and compositional changes caused by these hillocks in more detail, we extract tips for laser-pulsed APT from the TD- impacted hillock region and from a nominally TD- free region next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Note that the shallow 100 nm depth of our QDs that enables CL imaging (and ECCI), dramatically reduces the likelihood of capturing a QD in the APT tip since the conical tip diameter is very small near the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Indeed, we see from the top-down views in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 6a that neither tip has regions of high indium concentration as would be expected from a QD, indicating that both tips probe only the InGaAs QW that encases the QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Nevertheless, the fluctuations are essentially consistent with those of a random alloy of InGaAs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' S3 and no evidence for phase separation or clustering is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' On the other hand, the cross-sectional indium profiles of each tip in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 6b reveal that the QW in the defective region is significantly thicker than in the TD-free region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Collapsing these down to one-dimensional vertical profiles of indium composition, averaged laterally over the center of the tip, we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 6c that the QW in the hillock is about 8-9 nm thick with a tapering indium profile, contrasted with a 7 nm thick QW with a slightly less tapered profile seen in the tip from the TD-free region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Some of this taper, along with indium concentrations elevated above the expected 15% nominal value, may be explained by the unresolved InAs wetting layer (consistent with other APT43,44 and STEM45 studies) that lies 2 nm above the base of the QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' However, the additional thickness of the QW is an effect of the hillock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Reiterating that the hillock regions contain a higher density of surface steps, if the availability of steps limits the incorporation of adatoms, any asymmetry between the diffusivity of indium and gallium may lead to preferential incorporation of indium in such hillocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' However, the vertical Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) Top view and (b) side view of the lateral indium composition in the nominally In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='85As quantum well that surrounds the InAs quantum dots at a threading dislocation containing hillock, similar to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 5 (left) and at a neighboring threading dislocation (TD)-free region (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Data collected using site-selective laser atom probe tomography informed by cathodoluminescence and electron channeling contrast imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' No quantum dots were captured in the analysis due to the limited cross-sectional area of the APT tip possible from the 100 nm shallow structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (c) Vertical composition trace through the quantum well showing a region of tapered but similar composition profiles for the two sites, but increased thickness for the defective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Error bars representing one standard deviation are indicated by the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a) Hillock region Defect-free region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='25 site) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='20 Il dnojb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='10 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='05 10 nm (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='25 [001] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='20 site 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15 (group 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='10 In 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='05 % 5 nm 0 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='30 Hillock region Defect-free 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='20 [001] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='15 fraction ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='00 2 0 2 4 6 8 10 12 Position from bottom of QW (nm)11 profiles show near identical indium incorporation for the first 7 nm in both sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' the hillock simply extends this tapered profile for an additional 1–2 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This suggests that the growth rate increases at the hillock without any alteration in the composition, and both indium and gallium are quite mobile on the growth surface and incorporate at the hillock without preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Without direct access to the composition or shape of the InAs QDs, we may only infer how the altered QW affects the emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' In addition to easier thermalization from smaller blue-shifted QDs, a locally thicker QW may have a ground state closer in energy to the QDs and enhance carrier thermalization out of the dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Taken together, these analyses demonstrate the serious impact both distant misfit dislocations and local threading dislocations can have in altering the growth of QDs and their surrounding structures, ultimately broadening their size distribution (and hence their emission spectrum) and further aggravating nonradiative recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Therefore, these effects must be closely considered when tuning device design to optimize performance and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' CONCLUSIONS With the large untapped potential of heterogeneous integration of dissimilar materials by direct growth, it is important to understand the microscale effect of dislocations on the final devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We have quantified how dislocations affect spontaneous-emission luminescence in InAs QDs on silicon by facilitating defect-assisted recombination using time-resolved cathodoluminescence spectroscopy on a model InAs QD structure on silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We find a significantly reduced recombination lifetime for both the ground and excited states at misfit dislocations but also find recombination to be limited by defects in regions away from dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Yet, the impact of dislocations goes much beyond simple nonradiative recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' We find, using hyperspectral CL imaging and atom probe tomography, alterations in QD and QW growth that form pockets of blue-shifted emission arising from long range misfit dislocation strain fields and short-range threading dislocation spiral growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Both yield reduced emission homogeneity that increases susceptibility to carrier losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Our work shows how new characterization tools may enable a more complete understanding of the impact of dislocations on devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' InAs quantum dots, currently yielding the most reliable devices, are now part of a series of III-V laser devices being synthesized on silicon spanning the visible to the mid-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' As the field matures, we expect to see multi-modal microstructural characterization of the kind employed in this work to rise to prominence in those devices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL See supplementary material for (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' S1) cathodoluminescence peak excited state wavelength and intensity maps for samples on GaAs and silicon, corresponding to the ground state maps in Fig 4a-d, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' S2) a cathodoluminescence wavelength map of an InAs QD laser active region after milling away the upper cladding, and (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' S3) an atom probe compositional frequency distribution comparison of the hillock and defect-free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The sample growth was supported by ARPA-E, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Department of Energy, under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' DE- AR0001043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' This material is based upon work supported by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 1650114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' APT and TEM studies were performed at the UCSB MRL Shared Experimental Facilities, supported by the MRSEC Program of the NSF under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' DMR 1720256;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' a member of the NSF- funded Materials Research Facilities Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' CL studies were supported by the EPSRC under EP/R025193/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' acknowledges additional support from NSF CAREER award under grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' DMR-2036520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' CONFLICTS OF INTEREST The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 12 DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' REFERENCES 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Liu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Bowers, IEEE Journal of Selected Topics in Quantum Electronics 24, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Margalit, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' D: Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 55, 404006 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 21 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Selvidge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 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+page_content=' Missous, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 89, 072111 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' 14 Supplementary Material Dislocation-induced structural and luminescence degradation in InAs quantum dot emitters on silicon Eamonn T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Hughes1, Gunnar Kusch2, Jennifer Selvidge1, Bastien Bonef1, Justin Norman1, Chen Shang1, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Bowers1, Rachel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Oliver2, Kunal Mukherjee3 1Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA 2Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom 3Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (a-b) Excited-state peak-emission wavelength cathodoluminescence map for the sample (a) on silicon and (b) on GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' (c-d) Corresponding excited-state cathodoluminescence intensity maps for the sample (a) on silicon and (b) on GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Stitched cathodoluminescence map from a five-layer QD laser grown on silicon after milling away upper cladding using a focused ion beam microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The spacing between the active region and uppermost defect filter layer (which hosts a misfit dislocation network) is much larger here than in the single QD structure in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Consequently, the effects of extended misfit dislocation strain fields are weaker, so no distinct crosshatch pattern is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Even so, there are wide variations in peak emission wavelength and several strongly blue shifted regions, possibly due to hillocks formed by sessile threading dislocation clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Es Wavelength (nm) Es Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='0 (a) (c) 1180 1170 Silicon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 substrate 1160 4 μm 1150 4 μm 0 (b) 1200 (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 1190 GaAs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='0 substrate 1180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='5 1170 4 μm 4 μm 0Gs Wavelength (nm) 1305 1300 1295 1290 1285 [110] 1280 4 μm 1275 127015 Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Compositional frequency distribution measured from the bottom 2 nm of the QW analyzed in the two atom probe tomography specimens, which roughly aligns with the expected location of any QDs and the wetting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The dashed curve is a binomial fit representing the expected compositional distribution for a random alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The p- values estimate the probability that the observed distributions represent a random alloy, therefore, both alloys appear to be randomly distributed with no indication of a quantum dot or partial quantum dot present in either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' The bin size for composition measurements is 50 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content=' Hillock region Defect-free region 30 30 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='66 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} +page_content='75 20 20 Counts Counts 10 10 0 10 20 30 0 10 20 30 Composition (%) Composition (%)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E2T4oBgHgl3EQfHwY9/content/2301.03671v1.pdf'} diff --git a/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf b/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf new file mode 100644 index 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b/iNE0T4oBgHgl3EQfpgG_/content/tmp_files/2301.02541v1.pdf.txt @@ -0,0 +1,1668 @@ +The one step fixed-lag particle smoother +as a strategy to improve +the prediction step of particle filtering +S. Nyobe*1,2, F. Campillo3, S. Moto1,2, V. Rossi4,5 +1MIBA research Unity, Faculty of Science, University of Yaound´e, Cameroon +2UMI 209, UMMISCO-Cameroon, IRD, Sorbonne University, France +3Inria, MathNeuro Team, Montpellier, France +4RU Forets and Societies, CIRAD, Yaound´e, Cameroon +5National Advanced School of Engineering, University of Yaound´e, Cameroon +*E-mail : samuel.nyobe@facsciences-uy1.cm +Abstract +Sequential Monte Carlo methods have been a major advance in the field of numerical filtering +for stochastic dynamical state-space systems with partial and noisy observations. However, these +methods still have some weaknesses. One of its main weaknesses concerns the degeneracy of these +particle filters due to the impoverishment of the particles. Indeed, during the prediction step of +these filters, the particles explore the state space, and if this exploration phase is not done correctly, +a large part of the particles will end up in areas that are weakly weighted by the new measurement +and will be mostly eliminated, only a few particles will be kept, leading to a degeneracy of the +filter. In order to improve this last step within the framework of the classic bootstrap particle filter, +we propose a simple approximation of the one step fixed-lag smoother. At each time iteration, +we propose to perform additional simulations during the prediction step in order to improve the +likelihood of the selected particles. Note that we aim to propose an algorithm that is almost as fast +and of the same order of complexity as the bootstrap particle filter, and which is robust in poorly +conditioned filtering situations. We also investigate a robust version of this smoother. +Keywords +particle filter, robust particle filter, regime switching particle filter, one step fixed-lag particle +smoother, extended Kalman filter, unscented Kalman filter. +I +INTRODUCTION +Since the 1980s sequential Monte Carlo (SMC) methods, also called particle filter (PF) methods +[2, 4, 6, 7, 11], have met with vast success and broad usage in the context of nonlinear filtering +for state-space models with partial and noisy observations, also known as hidden Markov mod- +els (HMM) for state-space models. The success of these methods is due to their ability to take +into account, in a numerically realistic and efficient way, the non-linearity of the dynamics and +the non-Gaussianity of the underlying conditional distributions. +The exact (i.e., non approximated) dynamics of these HMMs takes the form of a sequential +Bayes formula represented in Eqs. (3)-(4) also called Bayesian filter. In the case of linear +1 +arXiv:2301.02541v1 [stat.AP] 6 Jan 2023 + +models with additive Gaussian noise, the Kalman filter gives an exact solution of this problem +as well as an efficient algorithm to compute it. There are a few cases where this filter can be +computed explicitly in a finite-dimensional way, but in the majority of cases it is necessary to +use numerical approximation methods. Historically, the first method proposed is the extended +Kalman filter (EKF) [1] and its variants, which consist in linearizing the model around the +current estimate and applying the Kalman filter. But in the case of strongly nonlinear models, +the EKF often diverges. Since the 1980s, Monte Carlo methods have become very effective +alternatives, they are now recognized as a powerful tool in the estimation with Bayesian filters +in nonlinear/non-Gaussian hidden Markov models. +Among Monte Carlo methods, the PF techniques rely on an online importance sampling ap- +proximation of the sequential Bayes formula (3)-(4). At each time iteration, the PF builds a set +of particles i.e. an independently and identically distributed sample from an approximation of +the theoretical solution of the Bayesian filter. +A PF time iteration classically includes two steps: a prediction step consisting in exploring the +state space by moving the particles according to the state equation, followed by a correction +step consisting in first weighting the particles according to their correspondence with the new +observation, i.e. according to their likelihood, then resampling these particles according to their +weight. These two steps can be understood respectively as the mutation and selection steps of +genetic algorithms. +The first really efficient PF algorithm, namely the bootstrap particle filter (BPF) or sampling- +importance-resampling filter proposed in 1993 [11], follows exactly these two steps. +The resampling step is essential, without it we notice after a few time iterations an impoverish- +ment of the particles: very few particles, even one, will concentrate all the likelihood; the other +particles then become useless, the filter then loses track of the current state. +This resampling step is therefore essential, but still it does not prevent another type of particle +degeneracy [2, 7, 17, 21, 27]. The latter appears when the particles explore areas of the state +space that are not in agreement with the new observation. This is due to the fact that in its +classical version, especially in the case of BPF, the prediction step propagates the particles in +the state space without taking into account the next observation. Thus, when weighting via +likelihood, a large proportion of the particles may have almost zero weight, this phenomenon is +aggravated by the machine epsilon. In the worst case, all the particles can have a zero weight, +the filter has then completely lost track of the state evolution. +To overcome this problem, it is relevant to take into account the following observation during +the prediction step. +To overcome this problem it is relevant to take into account the observations in the prediction +step consisting in exploring the state space. The auxiliary particle filter (APF) [22, 23] is one +of the first attempts to take into account the observation in the prediction step. One can consult +the review articles [9, 10] concerning the possibilities of improvement of particle filters. +Another possibility to overcome this difficulty, without using the next observation, is to use +robust versions of particle filters, such as the regime switching method [15] or the averaging- +model approach [19, 25], which allow model uncertainties to be taken into account. +The present paper aims to propose a new PF algorithm, called predictive bootstrap particle +smoother (PBPS), to further improve the prediction step but keeping the simplicity of the BPF. +2 + +The principle of the PBPS is to take into account the current and the next observations for the +prediction and correction steps. At each prediction step, we make additional explorations in +order to better determine the likelihood according to the next observation. It can be seen as a +simple approximation of the one step fixed-lag smoother. The additional explorations contribute +to the correction step in order to improve the likelihood of the selected particles with the current +and next observation. +We want to propose an algorithm that is only a little more complex than the BPF, this is why we +do not consider more complex algorithms, like the smoother with a deeper time-lag. We also +want to propose an efficient algorithm in case of unfavorable filtering situations such as when +observability is poorly conditioned. +In Section II state-space models and the (exact) formulas for nonlinear filters and smoothers +are presented. In Section III, we first present the classical bootstrap particle filter (BPF), then +the predictive bootstrap particle smoother (PBPS) and its “robustfified” version called regime +switching predictive bootstrap particle smoother (RS-PBPS). +In Section IV we present simulations in two test cases where observability is weakly condi- +tioned. The first test case is a classical example in state space dimension 1 for which we com- +pare the PBPS to the extended Kalman filter (EKF) [1], the unscented Kalman filter (UKF) [13, +26], the bootstrap particle filter (BPF) [11], and the the auxiliary particle filter (APF) [22]. The +second is more realistic, it is a classical problem of bearing-only 2D tracking with a state space +of dimension 4. In this case we first compare the PBPS to the BPF and the APF filters. In +this example there are often model uncertainties, so it is relevant to use robust filters, and we +compare the RS-PBPS to “robustified” versions of the particle filters. +II +NONLINEAR FILTERING AND SMOOTHING +2.1 +The state space model +We consider a Markovian state-space model with state process (Xk)k≥0 taking values in Rn and +observation process (Yk)k≥1 taking values in Rd. We suppose that conditionally on (Xk)k≥0, +the observations Yk are independent. +The ingredients of the state-space model are: +qk(x|x′) +def= pXk|Xk−1=x′(x) , +(state transition kernel) +ψk(x|y) +def= pYk|Xk=x(y) , +(local likelihood function) +ρ0(x) +def= pX0(x) , +(initial distribution) +for any x, x′ ∈ Rn, y ∈ Rd. +These ingredients can be made more explicit in the case of the following state space model: +Xk = fk−1(Xk−1) + gk−1(Xk−1) Wk−1 , +(1) +Yk = hk(Xk) + Vk , +(2) +for 1 ≤ k ≤ K, where Xk (resp. Yk, Wk, Vk) takes values in Rn (resp. Rd, Rm, Rd), fk +(resp. gk, hk) is a differentiable and at most linear growth, uniformly in k, from Rn to Rn (resp. +Rd, Rn×m, Rd), gk being also bounded. Random sequences Wk and Vk are independently and +3 + +identically distributed centered white Gaussian noises with respective variances Qk and Rk; +Wk, Vk, X0 being independent. In this case: +qk(x|x′) = +1 +� +(2 π)n det Qk−1 +exp +� +−1 +2 +� +x − fk−1(x′) +�∗ � +gk(x′) Qk−1 gk(x′)∗�−1 � +x − fk−1(x′) +�� +and +ψk(x|y) ∝ exp +� +−1 +2 +� +y − hk(x) +�∗ R−1 +k +� +y − hk(x) +�� +(ψk has to be known up to a multiplicative constant). +2.2 +Nonlinear filtering +Nonlinear filtering aims at determining the conditional distribution ηk of the current state Xk +given the past observations Y1, . . . , Yk, namely: +ηk(x) +def= pXk|Y1:k=y1:k(x) , x ∈ Rn +for any k ≥ 1 and y1:k ∈ (Rd)k, here we use the notation: +“Y1:k” for (Y1, . . . , Yk) +(e.g. Y1:k = y1:k means Yℓ = yℓ for all ℓ = 1, . . . , k). +The nonlinear filter allows us to determine ηk from ηk−1 using the classical two-step recursive +Bayes formula: +Prediction step. The predicted distribution ηk−(x) +def= pXk|Y1:k−1=y1:k−1(x) of Xk given +Y1:k−1 = y1:k−1 is given by: +ηk−(x) = +� +Rn qk(x|x′) ηk−1(x′) dx′ , +(3) +for x ∈ Rn. +Correction step. The new observation Yk = yk allows the predicted distribution to be +updated in order to obtain ηk according to the Bayes formula: +ηk(x) = +ψk(x|yk) ηk−(x) +� +Rn ψk(x′|yk) ηk−(x′) dx′ , +(4) +for x ∈ Rn. +Note that in the correction step (4) , ηk is proportional to the product of ηk−1 and the local +likelihood function, that is: +ηk(x) ∝ ψk(x|yk) ηk−(x) . +4 + +2.3 +Nonlinear smoothing +To get ¯ηk−1(x) the conditional distribution of Xk−1 given Y0:k = y0:k, we consider the extended +state vector Xk = (Xk, Xk−1), it’s a Markov with transition: +Qk(x′, x′′|xk−1, xk−2) = += pXk,Xk−1|Xk−1=xk−1,Xk−2=xk−2(x′, x′′) += pXk|Xk−1=x′′,Xk−1=xk−1,Xk−2=xk−2(x′) pXk−1|Xk−1=xk−1,Xk−2=xk−2(x′′) += pXk|Xk−1=xk−1,Xk−2=xk−2(x′) δxk−1(x′′) += pXk|Xk−1=xk−1(x′) δxk−1(x′′) += qk(x′|xk−1) δxk−1(x′′) . +The conditional distribution ηk(x′, x′′) of Xk = (Xk, Xk−1) given Y0:k = y0:k, we apply the +previous filter formula: +ηk−(x′, x′′) = +�� +Qk(x′, x′′|xk−1, xk−2) ηk−1(xk−1, xk−2) dxk−1 dxk−2 += +�� +qk(x′|xk−1) δxk−1(x′′) ηk−1(xk−1, xk−2) dxk−1 dxk−2 += +�� +qk(x′|xk−1) δxk−1(x′′) ηk−1(x′′, xk−2) dxk−1 dxk−2 , +and the distribution of Yk given (Xk = xk, Xk−1 = xk−1) is the distribution of Yk given Xk = +xk, hence: +ηk(x′, x′′) ∝ ψk(x′|yk) ηk−(x′, x′′) , +(5) +which the x′′-marginal distribution gives the conditional distribution of Xk−1 given Y0:k = y0:k. +2.4 +Approximations +The main difficulty encountered by the nonlinear filter (3)-(4) lies in the two integrations. These +integrations can be solved explicitly only in the linear/Gaussian case, leading to the Kalman fil- +ter, and in a very few other specific nonlinear/non-Gaussian cases. In the latter cases, the optimal +filter can be solved explicitly in the form of a finite dimensional filter; hence in the vast major- +ity of cases, it is necessary to use approximation techniques [4]. Among the approximation +techniques, we will consider the extended Kalman filter (EKF) [1], the unscented Kalman filter +(UKF) [13, 26] and particle filter techniques, also called sequential Monte Carlo techniques [6, +11], see [8] for a recent overview. +Concerning the particle filters, as we will see in the next section, it is important to notice that +on the one hand we do not need to know the analytical expressions of the state transition kernel +and of the initial distribution, we just need to be able to sample (efficiently) from them; on +the other hand, we do need the analytical expression of the local likelihood function (up to a +multiplicative constant), indeed, for a given y, we need to compute ψk(x|y) for a very large +number of x values. +5 + +III +PARTICLE APPROXIMATIONS +The particle approximation ηN +k of ηk is a Monte Carlo empirical approximation of the form: +ηN +k (x) = +N +� +i=1 +ωi +k δξi +k(x) , x ∈ Rn . +composed of N particles ξi +k in Rn and weights ωi +k, the weights are positive and sum to one [6]. +Ideally the particles are sampled from ηk and the importance weight are all equal to 1/N. +3.1 +The bootstrap particle filter +The bootstrap particle filter (BPF) filter is a classical sequential importance resampling method: +suppose we have a good approximation ηN +k−1 = 1 +N +�N +i=1 δξi +k−1 of ηN +k−1. We can apply the predic- +tion step (3) to ηN +k−1 and get: +� +Rn qk(x|x′) ηN +k−1(x′) dx′ = +N +� +i=1 +ωi +k−1 qk(x|ξi +k−1) +and then apply the correction step (4) and get: +�N +i=1 ωi +k−1 ψk(x|yk) qk(x|ξi +k−1) +� +Rn +�N +i=1 ωi +k−1 ψk(x′|yk) qk(x′|ξi +k−1) dx′ . +Both the two last expressions are mixture of the densities qk( · |ξi +k−1) and therefor not of the +particle type. The bootstrap particle filter (BPF) proposed by [11] is the simplest method to +propose a particle approximation. For the prediction step we use the sampling technique: +ξi +k− ∼ qk( · |ξi +k−1) , i = 1 : N +(independently) +which is: +ξi +k− = fk−1(ξi +k−1) + gk−1(ξi +k−1) wi +where wi are i.i.d. N(0, Rk−1) samples. Then we let +ηN +k−(x) +def= +N +� +i=1 +ωi +k−1 δξi +k−(x) . +Through the correction step (4), this approximation ηN +k− gives �N +i=1 ωi +k δξi +k− where: +ωi +k +def= +ψk(ξi +k−|yk) ωi +k− +�N +j=1 ψk(ξj +k−|yk) ωj +k− +(6) +are the updated weights taking account of the new observation yk through the likelihood func- +tion ψk,yk. This correction step must be completed by a resampling of the particles ξi +k− according +to the importance weights ωi +k: +ξi +k ∼ +N +� +j=1 +ωj +k δξj +k− , i = 1 : N +(independently) +(7) +6 + +leading the to the particle approximation: +ηN +k (x) +def= +N +� +i=1 +ωi +k δξi +k(x) , with ωi +k = 1 +N . +Algorithm III.1 gives a summary of the BPF, in this version a resampling is performed at each +time step. The multinomial resampling step (7), the basic idea proposed in [11], could be deeply +improved, there are several resampling techniques, see [5, 16] for more details. +Algorithm III.1 Bootstrap particle filter (BPF). +1: ξ1:N +0 +iid∼ ρ0 +# initialization +2: return ξ1:N +0 +3: for k = 1 : K do +4: +ξi +k− ∼ qk( · |ξi +k−1) , i = 1 : N +# particles evolution +5: +ωi +k ← ψk(ξi +k−|yk) , i = 1 : N +# likelihood +6: +ωi +k ← ωi +k/�N +j=1 ωj +k , i = 1 : N +# renormalization +7: +ξ1:N +k +iid∼ �N +j=1 ωj +k δξj +k− +# resampling +8: +return ξ1:N +k +9: end for +It is not necessary to resample the particles systematically at each time iteration like in Algo- +rithm III.1. However, this resampling step should be done regularly in terms of time iterations +to avoid degeneracy of the weights [6]. +We will consider another degeneracy problem. Suppose that in step (6), the local likelihood +values ψk(ξi +k−|yk) associated with the predicted particles ξi +k− are all very small, or even equal to +zero due to rounding in floating point arithmetic. This normalization step (6) is then impossible. +This problem occurs when the filter loses track of the true state Xk, i.e. the likelihood of the +predicted particles ξi +k− w.r.t. the observation yk are all negligible, in this case the observation yk +appears as an outlier. This occurs especially when the period of time between two successive +instants of observation is very large compared to the dynamics of the state process. Strategies +to overcome this weakness encompass the iterated extended Kalman particle filter [18] or the +iterated unscented Kalman particle filter [12]; see [10] and [17] for reviews of the subject. +3.2 +The predictive bootstrap particle smoother +The purpose of the predictive bootstrap filter (PBPS) is to improve the correction step (6) of the +BPF by using both observations Yk = yk and Yk+1 = yk+1 at each iteration k. In a way, the +PBPS can be seen as an approximation of the distribution of Xk given Y1:k+1 = y1:k+1, the one +step fixed-lag smoother presented in Section 2.3. +In the proposed algorithm, the prediction step consists in a first step to propagate at time k the +ηN +k−1 particles by simulating the transition kernel sampler, then in a second step to extend these +particles according to a one-step-ahead sampler. Then the correction step consists in updating +the weights of the N particles of ηN +k− according to their likelihood with yk and the likelihood of +their one-step ahead offspring particles with yk+1. +The iteration k − 1 → k of the filter is more precisely: +7 + +Prediction step. Like for the BPF we sample N particles and compute N normalized likelihood +weights: +˜ξi +k ∼ qk( · |ξi +k−1) , ˜ωi +k ∝ ψk(ξi +k−|yk) ωi +k− , i = 1 : N . +Again we will resample the particles ˜ξ1:N +k +, but instead of doing so according to the weights +˜ω1:N +k +, we will first modify the latter ones. For each index i, we generate a one-step-ahead +offspring particles and compute its weight: +˜ξi +k+1 ∼ ˜qk+1( · |˜ξi +k) , +˜ωi +k+1 +def= ψk+1(˜ξi +k+1|yk+1). +Note that the likelihood weights ˜ωi +k+1 depend on the next observation yk+1. See later for +the choice of one-step-ahead sampler ˜qk+1. +Correction step. We compute the weights at time k according to Eq. (5): +ωi +k +def= ˜ωi +k ˜ωi +k+1 +for i = 1 : N. +Then, N particles ˜ξ1:N +k +are resampled according to the weights ω1:N +k +. +The PBPS is depicted in Algorithm III.2. +Choice of one-step-ahead sampler ˜qk+1. +For the special case of the system (1)-(2) we can +choose a simpler “deterministic sampler”: +˜ξi +k+1 = fk(˜ξi +k) +which corresponds to the one-step-ahead mean: +˜ξi +k+1 = E +� +Xk+1 +��Xk = ˜ξi +k +� += E +� +fk(Xk) + gk(Xk) Wk +��Xk = ˜ξi +k +� += fk(˜ξi +k) + gk(˜ξi +k) E(Wk|Xk = ˜ξi +k) +� +�� +� +=0 += fk(˜ξi +k) +This choice greatly reduces the computation burden and it is sufficient, as we will see, in lin- +ear state equation. For highly nonlinear state equation, we can choose ˜qk+1 = qk+1, which +corresponds to simulate the state dynamic. +3.3 +The regime switching predictive bootstrap particle smoother +In most applications, both the state model and the observation model are only partially known. +As we will see in Section 4.2, applying a filter without care in such situations can be delicate. +There are several techniques to make these filters more robust to model mismatch such as the +dynamic model averaging (DMA) [19, 25] or the regime switching (RS) [15]. We will use +both DMA and RS techniques in the simulations of Section 4.2, but we introduce only the RS +technique in the present section by proposing a “robustified” version of PBPS, called regime +switching PBPS (RS-PBPS). +The partial knowledge on the state-space model introduced in Section 2.1 can be on the state +model, i.e. on the transition kernel qk, as well as on the observation model, i.e. on the likelihood +function Ψk. +8 + +Algorithm III.2 predictive bootstrap particle smoother (PBPS). +1: ξ1:N +0 +iid∼ ρ0 +# +initialization +2: for k = 1 : K do +3: +for i = 1 : N do +4: +˜ξi +k ∼ qk( · |ξi +k−1) +# +particles propagation +5: +˜ωi +k ← ψk(˜ξi +k|yk) +6: +˜ξi +k+1 = fk(˜ξi +k) +# +offspring generating +7: +˜ωi +k+1 ← ψk+1(˜ξi +k+1|yk+1) +# +offspring weighting +8: +ωi +k ← ˜ωi +k ˜ωi +k+1 +# +particles weighting +9: +end for +10: +ωi +k ← ωi +k/�N +i′=1 ωi′ +k , i = 1 : N +# +weights normalization +11: +ξ1:N +k +iid∼ �N +i′=1 ωi′ +k δ˜ξi′ +k +# +particles resampling +12: +return ξ1:N +k +13: end for +In many applications it is on the state model that the uncertainty concerns, so we will limit +ourselves to this case. In order to model this uncertainty, we assume that the state dynamics +corresponds to a transition kernel of the form: +qk(x|x′, m) +where m is a parameter that evolves dynamically in the finite set: +M = {m1, . . . , mL} . +Thus M represents the different possible regimes of the state model over time. +The principle of these robust filters consists in making the particles and the weights evolve +Algorithm III.3 Regime Switching predictive bootstrap particle smoother (RS-PBPS) +1: ξ1:N +0 +iid∼ ρ0 +# +initialization +2: return ξ1:N +0 +3: for k = 1 : K do +4: +µ1:N +k−1 +iid∼ U(M) +5: +˜ξi +k ∼ qk( · |ξi +k−1, µi +k−1) , i = 1 : N +# +particles propagation +6: +˜ωi +k ← ψk(˜ξi +k|yk) , i = 1 : N +7: +˜ξi +k+1 = fk(˜ξi +k) , i = 1 : N +# +offspring +8: +˜ωi +k+1 ← ψk+1(˜ξi +k+1|yk+1) , i = 1 : N +# +offspring weighting +9: +ωi +k ← ˜ωi +k ˜ωi +k+1 , i = 1 : N +# +particles weighting +10: +ωi +k ← ωi +k/�N +j=1 ωj +k , i = 1 : N +# +renormalization +11: +ξ1:N +k +iid∼ �N +j=1 ωj +k δ˜ξj +k +# +resampling +12: +return ξ1:N +k +13: end for +9 + +according to a mixture of the parameters M. The dynamic model averaging particle filter +(DMA-BPF) proposed in [25] is detailed in Algorithm A.2, we will not describe it here. +We now present one of the algorithms proposed by [15]. We use the simplest of them which +gives good results according to [15]. In the BPF, we replace the prediction step (Algorithm III.1 +line 4) by: +µ1:N +k−1 +iid∼ U(M) , +ξi +k ∼ qk( · |ξi +k−1, µi +k−1) , +i = 1 : N +where U(M) is the uniform distribution on M; the only difference lies in the fact that the +particles are sampled at random according to the various models represented by M. The re- +sulting method called RS-BPF is described in Algorithm A.3. The adaptation to PBPS and +APF is also immediate, leading to the RS-PBPS and the RS-APF; the RS-PBPS is detailed in +Algorithm III.3. +IV +SIMULATION STUDIES +We compare numerically the PBPS to other filters on two space-state models where observ- +ability is weakly conditioned. The performance of the filters is compared using an empiri- +cal evaluation of the root-mean-square error (RMSE). We simulate S independent trajectories +(X(s) +0:K, Y (s) +1:K)s=1:S of the state-space models. For each simulation s, we ran R times each filter +F ∈ {BPF, PBPS, APF, DMA-BPF, RS-BPF, RS-APF, RS-PBPS} and we compute the root +mean squared error at time k: +RMSEk(F) +def= +� +� +� +� 1 +S +S +� +s=1 +1 +R +R +� +r=1 +��� ˆXF(s,r) +k +− X(s) +k +��� +2 +, +(8) +where: +ˆXF(s,r) +k +def= +� +Rn x ηN,F(s,r) +k +(x) dx = 1 +N +N +� +i=1 +ξF(s,r),i +k +is the numerical approximation of ˆX(s) +k += E(X(s) +k |Y (s) +1:k ) by the filter F. We also compute the +global root mean squared error: +RMSE(F) +def= 1 +K +K +� +k=1 +RMSEk(F) . +Note that for the EKF and UKF filters the summation over r in (8) is useless. +All implementations are done in R language [24] using a 2.11 GHz core i7-8650U intel running +Windows 10 64bits with a 16 Go RAM. +10 + +0.00 +0.04 +0.08 +0.12 +0 +1000 +2000 +3000 +4000 +5000 +Number of particules +Time for filtering a trajectory (s) +APF +BPF +PBPS +0 +2 +4 +6 +0 +1000 +2000 +3000 +4000 +5000 +Number of particules +RMSE +Figure 1: One-dimensional case study (9) — We plot the average computation time for filtering a tra- +jectory (left) and the RMSE (right) of the filters PBPS, BPF and APF as a function of different values +of N (50, 100, 500, 1000, 3000, 5000) (with S = 100 and R = 40). In comparison, the EKF and UKF +have a respective RMSE of 16.83 and 6.88 (and a negligible computation time compared to the particle +approximations). +4.1 +First case study: a one-dimensional model +We consider the following one-dimensional nonlinear model [6, 11, 14]: +Xk = Xk−1 +2 ++ 25 Xk−1 +1 + X2 +k−1 ++ 8 cos +� +1.2 (k − 1) +� ++ Wk−1 , +Yk = X2 +k +20 + Vk , +(9) +with 1 ≤ k ≤ K (K = 50), X0 ∼ N(0, 1); Wk +iid∼ N(0, 32), Vk +iid∼ N(0, 1); X0, (Wk)k≥1 and +(Vk)k≥1 mutually independent. Note that the state process Xk is observed only through X2 +k so +the filters have difficulties to determine whether Xk is positive or negative, especially since the +state process Xk regularly changes it’s sign. Thus this model is regularly used as benchmark +for testing filters, as filters may easily lose track of Xk. +For this example we compare the filters PBPS, BPF, APF, EKF and UKF. +In Figure 1 we plot, as a function of different values of N (50, 100, 500, 1000, 3000, 5000), on +the left the average computation time for the filtering a trajectory, and on the right the RMSE +(with S = 100, R = 40). +The computation times of EKF and UKF are negligible compared to those of the particle ap- +proximations. On the other hand, the RMSE of EKF (16.83) are much higher than those of the +11 + +particle approximations; UKF with an RMSE of 6.88 behaves much better than EKF but is still +less accurate than the particle approximations. +The computation time of PBPS is naturally higher than that of BPF, but the latter is much less +accurate. For example, PBPS with N = 50 particles is 15 times faster than BPF with N = 5000 +particles while being 30% more accurate. +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +True Trajectory +APF +BPF +EKF +PBPS +UKF +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +Figure 2: One-dimensional case study (9) — On a single simulation, for each filter, we plot the true +state trajectory k → Xk and the approximation k → ˆXk with the associated 95% confidence region +(N = 5000). +In Figure 2, for each filter F ∈ {PBPS, BPF, APF, EKF, UKF} (with N = 5000), we plot the +true state trajectory k → Xk, the approximation k → ˆXF +k , and the associated 95% confidence +regions. For the particle approximations the confidence interval is given empirically by the +particles, for the EKF and UKF it’s given as an approximated Gaussian confidence interval. +The PBPS approximation has a smaller error k → | ˆXF +k − Xk| and a smaller confidence region +than the other filters. +In Figure 3, we compare k → RMSEk(F) for F ∈ {PBPS, BPF, APF, EKF, UKF} (with +N = 5000). Again, we see that the PBPS behaves better than the other filters and that the EKF +has a very erratic behavior. +The poor behavior of the EKF can be explained by the very nonlinear nature of this example and +by the observability problem around 0. Note the relatively good behavior of the UKF. However, +because of the observability problem at 0, both the UKF and the EKF cannot find the track of +Xk once in the wrong half-plane (Figure 2 around time k = 30). +12 + +0 +20 +40 +0 +10 +20 +30 +40 +50 +k +RMSEk +APF +BPF +EKF +PBPS +UKF +Figure 3: One-dimensional case study (9) — We compare k → RMSEk for each filter (S = 100, +R = 40, N = 5000). +In Figure 4, in the case of a single simulation of the state-space system and of the BPF and +PBPS filters (with N = 5000), we plot the true trajectory k → Xk as well as the set of particles +k → ξF,1:N +k +for F ∈ {BPF, PBPS}. Not surprisingly, we find that the PBPS particles are more +concentrated around the true value Xk and that BPF more often “loads” the symmetric part of +the state space. Indeed, the smoothing allows us to reduce the variance but also to compensate +the observability ambiguities; it is for this purpose that it was designed, and we reiterate that this +gain is obtained with a great improvement in the computation time (provided that less particles +are used). +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +−20 +−10 +0 +10 +20 +0 +10 +20 +30 +40 +50 +k +Xk +Figure 4: One-dimensional case study (9) — On a single simulation, we plot the true value k → Xk +with the evolution of the set of N = 500 particles k → ξ1:N +k +for the BPF (top) and the PBPS (bottom); +the particles are represented in transparency to better reflect their density. +13 + +4.2 +Second case study: a four-dimensional bearings-only tracking model +We consider the following four-dimensional model [3, 11, 23]: +Xk = +� 1 0 1 0 +0 1 0 1 +0 0 1 0 +0 0 0 1 +� +Xk−1 + σW +� 0.5 +0 +0 +0.5 +1 +0 +0 +1 +� +Wk−1 , +Yk ∼ wrapped Cauchy +� +arctan +�Xk[1] +Xk[2] +� +, ρ +� +, +(10) +with 1 ≤ k ≤ K = 20, X0 ∼ N( ¯X0, P0) with: +¯X0 = +� +−0.05, 0.2, 0.001, −0.055 +�∗ , +P0 = 0.01 diag +� +0.52, 0.32, 0.0052, 0.012� +, +σW = 0.001, Wk +iid∼ N(0, I2×2), ρ = 1 − 0.0052, X0 and (Wk)k≥1 mutually independent; +conditionally on (Xk)k≥0, (Yk)k≥1 and (Wk)k≥1 are independent. +The state equation corresponds to a target moving on a plane. The state vector is: +Xk = (Xk[1], Xk[2], Xk[3], Xk[4])∗ = (x1, x2, ˙x1, ˙x2)∗ , +where (x1, x2) are the Cartesian coordinates of the target in the plane and ( ˙x1, ˙x2) are the corre- +sponding velocities. The observer is located at the origin of the plane and accessed only to the +azimuth angle β = arctan(x1/x2) ∈ [−π, π] corrupted by noise. +It is known that in the situation where the observer follows a rectilinear trajectory at constant +speed, or zero speed as here, the filtering problem is poorly conditioned [20]. This is a situation +where it is known for example that the EKF diverges very strongly. +The conditional probability density function of the measured angle Yk given the state Xk is +assumed to be a wrapped Cauchy distribution with concentration parameter ρ [23]: +pYk|Xk=x(y) = +(2 π)−1 (1 − ρ2) +1 + ρ2 − 2 ρ cos +� +y − arctan(x[1] +x[2])) +� +(11) +for −π ≤ y < π, where ρ ∈ [0, 1] is the mean resultant length. Note that the state dynamics is +linear and Gaussian, but the observation dynamics is nonlinear and non-Gaussian. +Classically, in tracking studies, the trajectory model (the state equation) does not correspond +to reality. In reality, the target follows a uniform straight trajectory sometimes interrupted by +changes in heading and/or speed. The variance σW of the state equation (10) appears then as a +parameter of the filter: if it is too small, the filter will have trouble tracking the target, if it is +too large, the particles will spread out too much, the filter will then lose accuracy, and even lose +track of the target. +For bearings-only tracking applications, the EKF and UKF filters have poor performances and +will therefore not be considered in this example. On the other hand, the robust filters introduced +in Section 3.3 are relevant here because there is a mismatch model. +We compare the filters in two scenarios (with K = 40 time steps): +• the target follows a uniform rectilinear motion; +14 + +0.00 +0.05 +0.10 +0.15 +0 +1000 +2000 +3000 +4000 +5000 +Number of particules +Time for filtering a trajectory (s) +APF +BPF +PBPS +RS−APF +RS−BPF +RS−PBPS +DMA−BPF +Figure 5: Four-dimensional case study (11) — We plot the average computation time for filtering a +trajectory of the filters PBPS, BPF, APF, RS-PBPS, RS-BPF, RS-APF and DMA-BPF as a function of +different values of N (50, 100, 500, 1000, 3000, 5000) (with S = 20 and R = 50). +• the target follows a uniform rectilinear motion with change of heading at the middle of +the simulation. +As we will see, the first scenario is simpler than the second. For the filter we consider two +different values for σW: a small value σW = 0.001 and a large σW = 0.003. σW = 0.001 is +in fact too small to be consistent with the simulated trajectory, the second value σW = 0.003 is +more consistent with the simulated trajectory For the robust filter we consider: +M = {0.0005, 0.001, 0.003, 0.005} +as the set of possible models. +In Figure 5 we present the average computation times for filtering a trajectory as a function of +different values of N (50, 100, 500, 1000, 3000, 5000) (with S = 20 and R = 50). +In Figure 6, in agreement with Figure 5, we present the evolution of the RMSE as a function +of the number N of particles: the left column corresponds to the case without a turn, the right +column to the case with a turn. For PBPS, BPF and APF, the first row (top) presents the RMSE +for with σW = 0.001, the second (middle) with σW = 0.003. The third row (bottom) shows the +RMSE for the RS-PBPS, RS-BPF, RS-APF, and DMA-BPF filters. +In Figure 7 we plot k → RMSEk (for N = 5000, R = 50, S = 20) for the different filters: +the left column corresponds to the case without a turn, the right column to the case with a turn; +PBPS, BPF and APF for with σW = 0.001, on the first row; PBPS, BPF and APF for with +σW = 0.003, on the middle row; RS-PBPS, RS-BPF, RS-APF, and DMA-BPF, on the bottom +row. +In Figure 8, on a single simulation and for each filter, we plot the true trajectory of the target +with the estimates of each filter: without turn in the left column, with 1 turn in the right column; +PBPS, BPS and APF filters with small σW = 0.001 in the top row, PBPS, BPS and APF filters +with large σW = 0.003 in the middle row, RS-PBPS, RS-BPS, RS-APF, and DMA-BPF filters +in the bottom row. +According to Figure 5, at the same number of particles, BPF is slightly faster than PBPS, both +being faster than APF. The overheads for the robust version (RS) is also reasonable. In terms of +complexity, the computation time is linear with the number of particles. +15 + +In order to better understand the situation, we will first comment on Figures 7 and 8: with a too +small σW, the filters are able to track the target when it stays in a straight line (a) but they are +not able to adapt when a turn occurs and continue the tracking more or less in a straight line (b). +With a larger σW, the filters are still able to track the target when it remains in a straight line (c) +with a small and increasing inaccuracy; in contrast only PBPS behaves reasonably well when a +turn occurs (d). Except DMA-BPF which has a bad behavior, the RS robust filters behave better +and RS-PBPS remains better than all the others (e) and (f). +From Figures 5 and 6, in the case of a turn, which is the most complex situation, we see that in +order to reach PBPS/RS-PBPS accuracy with N = 1000 the other filters must use N = 5000 +particles, and even in this case PBPS/RS-PBPS is at least twice as fast as the others. +V +DISCUSSION AND CONCLUSION +Our goal was to propose a particle approximation algorithm slightly more complex than the BPF +but on the one hand allowing the variance of the error to be reduced and on the other hand being +more robust in poorly conditioned situations, especially when the observability conditions are +poor. We have therefore proposed an algorithm that deviates slightly from the “pure” filtering +method in the sense that it takes into account the next observation in addition to the current one. +We can thus consider PBPS as an approximation of the smoother with a fixed-lag interval of +one step. We have also proposed a more robust version of PBPS using the “regime switching” +strategy proposed in [15]. +In the considered examples, EKF shows very poor performance, which is perfectly understand- +able due to the very nonlinear nature and the poor observability conditions of the examples. It +should be noted that UKF performs much better than EKF while remaining far from the perfor- +mance of particle filters. Among the particle filters, APF gives slightly worse results than PBF +and PBPS. Finally, even if for a given number of particles, BPF is faster than PBPS, the latter +presents both a lower error variance and a much better behavior regarding observability prob- +lems. In terms of accuracy, PBPS with 1000 particles is still better than BPF with 5000 particles +while being much faster. This is also the case when comparing RS-PBPS with RS-BPF +These performances are due to the fact that PBPS is a very simplified approximation of the one +time step fixed-lag smoother. A strategy using a deeper fixed-lag interval of 2 or more time +steps would be too costly and would not allow us to recover such performances. +In view of the performance of PBPS it would be interesting to develop strategies where the +number N of particles adapts dynamically to the problem conditions. It would also be inter- +esting to test the PBPS strategy in other cases, such as when the time between two consecutive +observations is important and requires several successive prediction steps. +16 + +APPENDIX: ALGORITHMS +Algorithm A.1 Auxiliary particle filter (APF). +1: ξ1:N +0 +iid∼ ρ0 +# +initialization +2: return ξ1:N +0 +3: for k = 1 : K do +4: +¯ξi +k = fk(ξi +k−1) , i = 1 : N +# +high probability evolution +5: +¯ωi +k ← ψk(¯ξi +k|yk) , i = 1 : N +# +... +and weighting +6: +˜ξ1:N +k−1 +iid∼ �N +j=1 ¯ωi +k δξj +k−1 +# +initial particles resampling +7: +ξi +k− ∼ qk( · |˜ξi +k−1) , i = 1 : N +# +particles propagation +8: +ωi +k ← ψk(ξi +k−|yk)/¯ωi +k , i = 1 : N +# +likelihood +9: +ωi +k ← ωi +k/�N +j=1 ωj +k , i = 1 : N +# +renormalization +10: +ξ1:N +k +iid∼ �N +j=1 ωj +k δξj +k− +# +resampling +11: +return ξ1:N +k +12: end for +Algorithm A.2 Dynamic model averaging bootstrap particle filter (DMA-BPF) [25]. +1: ξ1:N +0 +iid∼ ρ0, µ1:N +0 +iid∼ U(M) +# +initialization +2: return ξ1:N +0 +3: for k = 1 : K do +4: +ξi +k∗ ∼ qk( · |ξi +k−1, µi +k−1), i = 1 : N +5: +ωl +k = �N +i=1 ψk(ξi +k∗|yk)1µi +k−1=ml, l = 1 : L # +likelihood of candidate models +6: +N 1:L +k +iid∼ Multinomial(N; m1, . . . , mL; ω1 +k, · · · , ωL +k ) +with N 1 +k + · · · + N L +k = N +# +number of particles per candidate model +7: +ˆξ1,1:L +k−1 +iid∼ �N +j=1 δξj +k−1 δµj +k−1=ml +# +initial particles resampling +# +per candidate model +8: +(˜ξ1:N +k−1, µ1:N +k +) ← (ˆξ +1:N1 +k,1 +k−1 +, m1), (ˆξ +1:N2 +k,2 +k−1 +, m2), . . . , (ˆξ +1:NL +k ,L +k−1 +, mL) +9: +ξi +k ∼ qk( · |˜ξi +k−1, µi +k) , i = 1 : N +# +particles propagation +10: +return ξ1:N +k +11: end for +17 + +Algorithm A.3 Regime Switching Bootstrap Particle Filter (RS-BPF) [15]. +1: ξ1:N +0 +iid∼ ρ0 +# +initialization +2: return ξ1:N +0 +3: for k = 1 : K do +4: +µ1:N +k−1 +iid∼ U(M) +5: +ξi +k− ∼ qk( · |ξi +k−1, µi +k−1), i = 1 : N +# +particles propagation +6: +ωi +k ← ψk(ξi +k−|yk) , i = 1 : N +# +likelihood +7: +ωi +k ← ωi +k/�N +j=1 ωj +k, i = 1 : N +# +renormalization +8: +ξ1:N +k +iid∼ �N +j=1 ωj +k δξj +k− +# +resampling +9: +return ξ1:N +k +10: end for +REFERENCES +[1] +B. 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Zuo. “Dynamic resampling for alleviating sample impoverishment of particle filter”. +In: IET Radar, Sonar Navigation 7.9 (2013), pages 968–977. +19 + +0.0 +0.1 +0.2 +0.3 +0 +1000 +2000 +3000 +4000 +5000 +RMSE +APF +BPF +PBPS +(a) +0.0 +0.1 +0.2 +0.3 +0 +1000 +2000 +3000 +4000 +5000 +(b) +0.0 +0.1 +0.2 +0.3 +0 +1000 +2000 +3000 +4000 +5000 +RMSE +(c) +0.0 +0.1 +0.2 +0.3 +0 +1000 +2000 +3000 +4000 +5000 +(d) +0.0 +0.1 +0.2 +0.3 +0 +1000 +2000 +3000 +4000 +5000 +Number of particules +RMSE +RS−APF +RS−BPF +RS−PBPS +DMA−BPF +(e) +0.0 +0.1 +0.2 +0.3 +0 +1000 +2000 +3000 +4000 +5000 +Number of particules +(f) +Figure 6: Four-dimensional case study (11) — Evolution of the RMSE as a function of the number N +of particles: the left column corresponds to the case without a turn, the right column to the case with +a turn. For PBPS, BPF and APF, the first row (top) presents the RMSE for with σW = 0.001, the +second (middle) with σW = 0.003. The third row (bottom) shows the RMSE for the RS-PBPS, RS-BPF, +RS-APF, and DMA-BPF filters. +20 + +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +k +RMSEk +APF +BPF +PBFS +(a) +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +k +RMSEk +(b) +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +k +RMSEk +(c) +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +k +RMSEk +(d) +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +k +RMSEk +RS−APF +RS−BPF +RS−PBPS +DMA−BPF +(e) +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +k +RMSEk +(f) +Figure 7: Four-dimensional case study (11) — k →RMSEk (for N = 5000, R = 50, S = 20) for +the different filters: the left column corresponds to the case without a turn, the right column to the case +with a turn; PBPS, BPF and APF for with σW = 0.001, on the first row; PBPS, BPF and APF for with +σW = 0.003, on the middle row; RS-PBPS, RS-BPF, RS-APF, and DMA-BPF, on the bottom row. +21 + +0.0 +0.5 +1.0 +1.5 +2.0 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +x1 +x2 +APF +BPF +PBPS +Trajectory +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +x1 +x2 +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +x1 +x2 +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +x1 +x2 +(d) +0.0 +0.5 +1.0 +1.5 +2.0 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +x1 +x2 +RS−APF +RS−BPF +RS−PBPS +DMA−BPF +Trajectory +(e) +0.0 +0.5 +1.0 +1.5 +2.0 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +x1 +x2 +(f) +Figure 8: Four-dimensional case study (11) — On a single simulation and for each filter, we plot the true +trajectory of the target with the estimates of each filter: without turn in the left column, with 1 turn in the +right column; PBPS, BPS and APF filters with small σW = 0.001 in the top row, PBPS, BPS and APF +filters with large σW = 0.003 in the middle row, RS-PBPS, RS-BPS, RS-APF, and DMA-BPF filters in +the bottom row. The observer is in position (0, 0). +22 + diff --git a/iNE0T4oBgHgl3EQfpgG_/content/tmp_files/load_file.txt b/iNE0T4oBgHgl3EQfpgG_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..da33a7854ac3b68b3af1def0cb5b1c3254c3e863 --- /dev/null +++ b/iNE0T4oBgHgl3EQfpgG_/content/tmp_files/load_file.txt @@ -0,0 +1,694 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf,len=693 +page_content='The one step fixed-lag particle smoother as a strategy to improve the prediction step of particle filtering S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Nyobe*1,2, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Campillo3, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Moto1,2, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Rossi4,5 1MIBA research Unity, Faculty of Science, University of Yaound´e, Cameroon 2UMI 209, UMMISCO-Cameroon, IRD, Sorbonne University, France 3Inria, MathNeuro Team, Montpellier, France 4RU Forets and Societies, CIRAD, Yaound´e, Cameroon 5National Advanced School of Engineering, University of Yaound´e, Cameroon E-mail : samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='nyobe@facsciences-uy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='cm Abstract Sequential Monte Carlo methods have been a major advance in the field of numerical filtering for stochastic dynamical state-space systems with partial and noisy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' However, these methods still have some weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' One of its main weaknesses concerns the degeneracy of these particle filters due to the impoverishment of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Indeed, during the prediction step of these filters, the particles explore the state space, and if this exploration phase is not done correctly, a large part of the particles will end up in areas that are weakly weighted by the new measurement and will be mostly eliminated, only a few particles will be kept, leading to a degeneracy of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In order to improve this last step within the framework of the classic bootstrap particle filter, we propose a simple approximation of the one step fixed-lag smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' At each time iteration, we propose to perform additional simulations during the prediction step in order to improve the likelihood of the selected particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note that we aim to propose an algorithm that is almost as fast and of the same order of complexity as the bootstrap particle filter, and which is robust in poorly conditioned filtering situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We also investigate a robust version of this smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Keywords particle filter, robust particle filter, regime switching particle filter, one step fixed-lag particle smoother, extended Kalman filter, unscented Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' I INTRODUCTION Since the 1980s sequential Monte Carlo (SMC) methods, also called particle filter (PF) methods [2, 4, 6, 7, 11], have met with vast success and broad usage in the context of nonlinear filtering for state-space models with partial and noisy observations, also known as hidden Markov mod- els (HMM) for state-space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The success of these methods is due to their ability to take into account, in a numerically realistic and efficient way, the non-linearity of the dynamics and the non-Gaussianity of the underlying conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The exact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=', non approximated) dynamics of these HMMs takes the form of a sequential Bayes formula represented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' (3)-(4) also called Bayesian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In the case of linear 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='02541v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='AP] 6 Jan 2023 models with additive Gaussian noise, the Kalman filter gives an exact solution of this problem as well as an efficient algorithm to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' There are a few cases where this filter can be computed explicitly in a finite-dimensional way, but in the majority of cases it is necessary to use numerical approximation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Historically, the first method proposed is the extended Kalman filter (EKF) [1] and its variants, which consist in linearizing the model around the current estimate and applying the Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' But in the case of strongly nonlinear models, the EKF often diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Since the 1980s, Monte Carlo methods have become very effective alternatives, they are now recognized as a powerful tool in the estimation with Bayesian filters in nonlinear/non-Gaussian hidden Markov models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Among Monte Carlo methods, the PF techniques rely on an online importance sampling ap- proximation of the sequential Bayes formula (3)-(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' At each time iteration, the PF builds a set of particles i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' an independently and identically distributed sample from an approximation of the theoretical solution of the Bayesian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' A PF time iteration classically includes two steps: a prediction step consisting in exploring the state space by moving the particles according to the state equation, followed by a correction step consisting in first weighting the particles according to their correspondence with the new observation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' according to their likelihood, then resampling these particles according to their weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' These two steps can be understood respectively as the mutation and selection steps of genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The first really efficient PF algorithm, namely the bootstrap particle filter (BPF) or sampling- importance-resampling filter proposed in 1993 [11], follows exactly these two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The resampling step is essential, without it we notice after a few time iterations an impoverish- ment of the particles: very few particles, even one, will concentrate all the likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the other particles then become useless, the filter then loses track of the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This resampling step is therefore essential, but still it does not prevent another type of particle degeneracy [2, 7, 17, 21, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The latter appears when the particles explore areas of the state space that are not in agreement with the new observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This is due to the fact that in its classical version, especially in the case of BPF, the prediction step propagates the particles in the state space without taking into account the next observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Thus, when weighting via likelihood, a large proportion of the particles may have almost zero weight, this phenomenon is aggravated by the machine epsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In the worst case, all the particles can have a zero weight, the filter has then completely lost track of the state evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' To overcome this problem, it is relevant to take into account the following observation during the prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' To overcome this problem it is relevant to take into account the observations in the prediction step consisting in exploring the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The auxiliary particle filter (APF) [22, 23] is one of the first attempts to take into account the observation in the prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' One can consult the review articles [9, 10] concerning the possibilities of improvement of particle filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Another possibility to overcome this difficulty, without using the next observation, is to use robust versions of particle filters, such as the regime switching method [15] or the averaging- model approach [19, 25], which allow model uncertainties to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The present paper aims to propose a new PF algorithm, called predictive bootstrap particle smoother (PBPS), to further improve the prediction step but keeping the simplicity of the BPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 2 The principle of the PBPS is to take into account the current and the next observations for the prediction and correction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' At each prediction step, we make additional explorations in order to better determine the likelihood according to the next observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' It can be seen as a simple approximation of the one step fixed-lag smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The additional explorations contribute to the correction step in order to improve the likelihood of the selected particles with the current and next observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We want to propose an algorithm that is only a little more complex than the BPF, this is why we do not consider more complex algorithms, like the smoother with a deeper time-lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We also want to propose an efficient algorithm in case of unfavorable filtering situations such as when observability is poorly conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Section II state-space models and the (exact) formulas for nonlinear filters and smoothers are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Section III, we first present the classical bootstrap particle filter (BPF), then the predictive bootstrap particle smoother (PBPS) and its “robustfified” version called regime switching predictive bootstrap particle smoother (RS-PBPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Section IV we present simulations in two test cases where observability is weakly condi- tioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The first test case is a classical example in state space dimension 1 for which we com- pare the PBPS to the extended Kalman filter (EKF) [1], the unscented Kalman filter (UKF) [13, 26], the bootstrap particle filter (BPF) [11], and the the auxiliary particle filter (APF) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The second is more realistic, it is a classical problem of bearing-only 2D tracking with a state space of dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In this case we first compare the PBPS to the BPF and the APF filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In this example there are often model uncertainties, so it is relevant to use robust filters, and we compare the RS-PBPS to “robustified” versions of the particle filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' II NONLINEAR FILTERING AND SMOOTHING 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 The state space model We consider a Markovian state-space model with state process (Xk)k≥0 taking values in Rn and observation process (Yk)k≥1 taking values in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We suppose that conditionally on (Xk)k≥0, the observations Yk are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The ingredients of the state-space model are: qk(x|x′) def= pXk|Xk−1=x′(x) , (state transition kernel) ψk(x|y) def= pYk|Xk=x(y) , (local likelihood function) ρ0(x) def= pX0(x) , (initial distribution) for any x, x′ ∈ Rn, y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' These ingredients can be made more explicit in the case of the following state space model: Xk = fk−1(Xk−1) + gk−1(Xk−1) Wk−1 , (1) Yk = hk(Xk) + Vk , (2) for 1 ≤ k ≤ K, where Xk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Yk, Wk, Vk) takes values in Rn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Rd, Rm, Rd), fk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' gk, hk) is a differentiable and at most linear growth, uniformly in k, from Rn to Rn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Rd, Rn×m, Rd), gk being also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Random sequences Wk and Vk are independently and 3 identically distributed centered white Gaussian noises with respective variances Qk and Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Wk, Vk, X0 being independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In this case: qk(x|x′) = 1 � (2 π)n det Qk−1 exp � −1 2 � x − fk−1(x′) �∗ � gk(x′) Qk−1 gk(x′)∗�−1 � x − fk−1(x′) �� and ψk(x|y) ∝ exp � −1 2 � y − hk(x) �∗ R−1 k � y − hk(x) �� (ψk has to be known up to a multiplicative constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 Nonlinear filtering Nonlinear filtering aims at determining the conditional distribution ηk of the current state Xk given the past observations Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' , Yk, namely: ηk(x) def= pXk|Y1:k=y1:k(x) , x ∈ Rn for any k ≥ 1 and y1:k ∈ (Rd)k, here we use the notation: “Y1:k” for (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' , Yk) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Y1:k = y1:k means Yℓ = yℓ for all ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' , k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The nonlinear filter allows us to determine ηk from ηk−1 using the classical two-step recursive Bayes formula: Prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The predicted distribution ηk−(x) def= pXk|Y1:k−1=y1:k−1(x) of Xk given Y1:k−1 = y1:k−1 is given by: ηk−(x) = � Rn qk(x|x′) ηk−1(x′) dx′ , (3) for x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Correction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The new observation Yk = yk allows the predicted distribution to be updated in order to obtain ηk according to the Bayes formula: ηk(x) = ψk(x|yk) ηk−(x) � Rn ψk(x′|yk) ηk−(x′) dx′ , (4) for x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note that in the correction step (4) , ηk is proportional to the product of ηk−1 and the local likelihood function, that is: ηk(x) ∝ ψk(x|yk) ηk−(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 Nonlinear smoothing To get ¯ηk−1(x) the conditional distribution of Xk−1 given Y0:k = y0:k, we consider the extended state vector Xk = (Xk, Xk−1), it’s a Markov with transition: Qk(x′, x′′|xk−1, xk−2) = = pXk,Xk−1|Xk−1=xk−1,Xk−2=xk−2(x′, x′′) = pXk|Xk−1=x′′,Xk−1=xk−1,Xk−2=xk−2(x′) pXk−1|Xk−1=xk−1,Xk−2=xk−2(x′′) = pXk|Xk−1=xk−1,Xk−2=xk−2(x′) δxk−1(x′′) = pXk|Xk−1=xk−1(x′) δxk−1(x′′) = qk(x′|xk−1) δxk−1(x′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The conditional distribution ηk(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' x′′) of Xk = (Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Xk−1) given Y0:k = y0:k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' we apply the previous filter formula: ηk−(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' x′′) = �� Qk(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' x′′|xk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' xk−2) ηk−1(xk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' xk−2) dxk−1 dxk−2 = �� qk(x′|xk−1) δxk−1(x′′) ηk−1(xk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' xk−2) dxk−1 dxk−2 = �� qk(x′|xk−1) δxk−1(x′′) ηk−1(x′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' xk−2) dxk−1 dxk−2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' and the distribution of Yk given (Xk = xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Xk−1 = xk−1) is the distribution of Yk given Xk = xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' hence: ηk(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' x′′) ∝ ψk(x′|yk) ηk−(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' x′′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' (5) which the x′′-marginal distribution gives the conditional distribution of Xk−1 given Y0:k = y0:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 Approximations The main difficulty encountered by the nonlinear filter (3)-(4) lies in the two integrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' These integrations can be solved explicitly only in the linear/Gaussian case, leading to the Kalman fil- ter, and in a very few other specific nonlinear/non-Gaussian cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In the latter cases, the optimal filter can be solved explicitly in the form of a finite dimensional filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' hence in the vast major- ity of cases, it is necessary to use approximation techniques [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Among the approximation techniques, we will consider the extended Kalman filter (EKF) [1], the unscented Kalman filter (UKF) [13, 26] and particle filter techniques, also called sequential Monte Carlo techniques [6, 11], see [8] for a recent overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Concerning the particle filters, as we will see in the next section, it is important to notice that on the one hand we do not need to know the analytical expressions of the state transition kernel and of the initial distribution, we just need to be able to sample (efficiently) from them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' on the other hand, we do need the analytical expression of the local likelihood function (up to a multiplicative constant), indeed, for a given y, we need to compute ψk(x|y) for a very large number of x values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 5 III PARTICLE APPROXIMATIONS The particle approximation ηN k of ηk is a Monte Carlo empirical approximation of the form: ηN k (x) = N � i=1 ωi k δξi k(x) , x ∈ Rn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' composed of N particles ξi k in Rn and weights ωi k, the weights are positive and sum to one [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Ideally the particles are sampled from ηk and the importance weight are all equal to 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 The bootstrap particle filter The bootstrap particle filter (BPF) filter is a classical sequential importance resampling method: suppose we have a good approximation ηN k−1 = 1 N �N i=1 δξi k−1 of ηN k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We can apply the predic- tion step (3) to ηN k−1 and get: � Rn qk(x|x′) ηN k−1(x′) dx′ = N � i=1 ωi k−1 qk(x|ξi k−1) and then apply the correction step (4) and get: �N i=1 ωi k−1 ψk(x|yk) qk(x|ξi k−1) � Rn �N i=1 ωi k−1 ψk(x′|yk) qk(x′|ξi k−1) dx′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Both the two last expressions are mixture of the densities qk( · |ξi k−1) and therefor not of the particle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The bootstrap particle filter (BPF) proposed by [11] is the simplest method to propose a particle approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For the prediction step we use the sampling technique: ξi k− ∼ qk( · |ξi k−1) , i = 1 : N (independently) which is: ξi k− = fk−1(ξi k−1) + gk−1(ξi k−1) wi where wi are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' N(0, Rk−1) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Then we let ηN k−(x) def= N � i=1 ωi k−1 δξi k−(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Through the correction step (4), this approximation ηN k− gives �N i=1 ωi k δξi k− where: ωi k def= ψk(ξi k−|yk) ωi k− �N j=1 ψk(ξj k−|yk) ωj k− (6) are the updated weights taking account of the new observation yk through the likelihood func- tion ψk,yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This correction step must be completed by a resampling of the particles ξi k− according to the importance weights ωi k: ξi k ∼ N � j=1 ωj k δξj k− , i = 1 : N (independently) (7) 6 leading the to the particle approximation: ηN k (x) def= N � i=1 ωi k δξi k(x) , with ωi k = 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 gives a summary of the BPF, in this version a resampling is performed at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The multinomial resampling step (7), the basic idea proposed in [11], could be deeply improved, there are several resampling techniques, see [5, 16] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 Bootstrap particle filter (BPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 1: ξ1:N 0 iid∼ ρ0 # initialization 2: return ξ1:N 0 3: for k = 1 : K do 4: ξi k− ∼ qk( · |ξi k−1) , i = 1 : N # particles evolution 5: ωi k ← ψk(ξi k−|yk) , i = 1 : N # likelihood 6: ωi k ← ωi k/�N j=1 ωj k , i = 1 : N # renormalization 7: ξ1:N k iid∼ �N j=1 ωj k δξj k− # resampling 8: return ξ1:N k 9: end for It is not necessary to resample the particles systematically at each time iteration like in Algo- rithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' However, this resampling step should be done regularly in terms of time iterations to avoid degeneracy of the weights [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We will consider another degeneracy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Suppose that in step (6), the local likelihood values ψk(ξi k−|yk) associated with the predicted particles ξi k− are all very small, or even equal to zero due to rounding in floating point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This normalization step (6) is then impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This problem occurs when the filter loses track of the true state Xk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the likelihood of the predicted particles ξi k− w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the observation yk are all negligible, in this case the observation yk appears as an outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This occurs especially when the period of time between two successive instants of observation is very large compared to the dynamics of the state process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Strategies to overcome this weakness encompass the iterated extended Kalman particle filter [18] or the iterated unscented Kalman particle filter [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' see [10] and [17] for reviews of the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 The predictive bootstrap particle smoother The purpose of the predictive bootstrap filter (PBPS) is to improve the correction step (6) of the BPF by using both observations Yk = yk and Yk+1 = yk+1 at each iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In a way, the PBPS can be seen as an approximation of the distribution of Xk given Y1:k+1 = y1:k+1, the one step fixed-lag smoother presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In the proposed algorithm, the prediction step consists in a first step to propagate at time k the ηN k−1 particles by simulating the transition kernel sampler, then in a second step to extend these particles according to a one-step-ahead sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Then the correction step consists in updating the weights of the N particles of ηN k− according to their likelihood with yk and the likelihood of their one-step ahead offspring particles with yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The iteration k − 1 → k of the filter is more precisely: 7 Prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Like for the BPF we sample N particles and compute N normalized likelihood weights: ˜ξi k ∼ qk( · |ξi k−1) , ˜ωi k ∝ ψk(ξi k−|yk) ωi k− , i = 1 : N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Again we will resample the particles ˜ξ1:N k , but instead of doing so according to the weights ˜ω1:N k , we will first modify the latter ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For each index i, we generate a one-step-ahead offspring particles and compute its weight: ˜ξi k+1 ∼ ˜qk+1( · |˜ξi k) , ˜ωi k+1 def= ψk+1(˜ξi k+1|yk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note that the likelihood weights ˜ωi k+1 depend on the next observation yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' See later for the choice of one-step-ahead sampler ˜qk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Correction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We compute the weights at time k according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' (5): ωi k def= ˜ωi k ˜ωi k+1 for i = 1 : N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Then, N particles ˜ξ1:N k are resampled according to the weights ω1:N k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The PBPS is depicted in Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Choice of one-step-ahead sampler ˜qk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For the special case of the system (1)-(2) we can choose a simpler “deterministic sampler”: ˜ξi k+1 = fk(˜ξi k) which corresponds to the one-step-ahead mean: ˜ξi k+1 = E � Xk+1 ��Xk = ˜ξi k � = E � fk(Xk) + gk(Xk) Wk ��Xk = ˜ξi k � = fk(˜ξi k) + gk(˜ξi k) E(Wk|Xk = ˜ξi k) � �� � =0 = fk(˜ξi k) This choice greatly reduces the computation burden and it is sufficient, as we will see, in lin- ear state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For highly nonlinear state equation, we can choose ˜qk+1 = qk+1, which corresponds to simulate the state dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 The regime switching predictive bootstrap particle smoother In most applications, both the state model and the observation model are only partially known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' As we will see in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2, applying a filter without care in such situations can be delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' There are several techniques to make these filters more robust to model mismatch such as the dynamic model averaging (DMA) [19, 25] or the regime switching (RS) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We will use both DMA and RS techniques in the simulations of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2, but we introduce only the RS technique in the present section by proposing a “robustified” version of PBPS, called regime switching PBPS (RS-PBPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The partial knowledge on the state-space model introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 can be on the state model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' on the transition kernel qk, as well as on the observation model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' on the likelihood function Ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 8 Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 predictive bootstrap particle smoother (PBPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 1: ξ1:N 0 iid∼ ρ0 # initialization 2: for k = 1 : K do 3: for i = 1 : N do 4: ˜ξi k ∼ qk( · |ξi k−1) # particles propagation 5: ˜ωi k ← ψk(˜ξi k|yk) 6: ˜ξi k+1 = fk(˜ξi k) # offspring generating 7: ˜ωi k+1 ← ψk+1(˜ξi k+1|yk+1) # offspring weighting 8: ωi k ← ˜ωi k ˜ωi k+1 # particles weighting 9: end for 10: ωi k ← ωi k/�N i′=1 ωi′ k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N # weights normalization 11: ξ1:N k iid∼ �N i′=1 ωi′ k δ˜ξi′ k # particles resampling 12: return ξ1:N k 13: end for In many applications it is on the state model that the uncertainty concerns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' so we will limit ourselves to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In order to model this uncertainty, we assume that the state dynamics corresponds to a transition kernel of the form: qk(x|x′, m) where m is a parameter that evolves dynamically in the finite set: M = {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' , mL} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Thus M represents the different possible regimes of the state model over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The principle of these robust filters consists in making the particles and the weights evolve Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 Regime Switching predictive bootstrap particle smoother (RS-PBPS) 1: ξ1:N 0 iid∼ ρ0 # initialization 2: return ξ1:N 0 3: for k = 1 : K do 4: µ1:N k−1 iid∼ U(M) 5: ˜ξi k ∼ qk( · |ξi k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' µi k−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N # particles propagation 6: ˜ωi k ← ψk(˜ξi k|yk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N 7: ˜ξi k+1 = fk(˜ξi k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N # offspring 8: ˜ωi k+1 ← ψk+1(˜ξi k+1|yk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N # offspring weighting 9: ωi k ← ˜ωi k ˜ωi k+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N # particles weighting 10: ωi k ← ωi k/�N j=1 ωj k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' i = 1 : N # renormalization 11: ξ1:N k iid∼ �N j=1 ωj k δ˜ξj k # resampling 12: return ξ1:N k 13: end for 9 according to a mixture of the parameters M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The dynamic model averaging particle filter (DMA-BPF) proposed in [25] is detailed in Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2, we will not describe it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We now present one of the algorithms proposed by [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We use the simplest of them which gives good results according to [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In the BPF, we replace the prediction step (Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 line 4) by: µ1:N k−1 iid∼ U(M) , ξi k ∼ qk( · |ξi k−1, µi k−1) , i = 1 : N where U(M) is the uniform distribution on M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the only difference lies in the fact that the particles are sampled at random according to the various models represented by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The re- sulting method called RS-BPF is described in Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The adaptation to PBPS and APF is also immediate, leading to the RS-PBPS and the RS-APF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the RS-PBPS is detailed in Algorithm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' IV SIMULATION STUDIES We compare numerically the PBPS to other filters on two space-state models where observ- ability is weakly conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The performance of the filters is compared using an empiri- cal evaluation of the root-mean-square error (RMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We simulate S independent trajectories (X(s) 0:K, Y (s) 1:K)s=1:S of the state-space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For each simulation s, we ran R times each filter F ∈ {BPF, PBPS, APF, DMA-BPF, RS-BPF, RS-APF, RS-PBPS} and we compute the root mean squared error at time k: RMSEk(F) def= � � � � 1 S S � s=1 1 R R � r=1 ��� ˆXF(s,r) k − X(s) k ��� 2 , (8) where: ˆXF(s,r) k def= � Rn x ηN,F(s,r) k (x) dx = 1 N N � i=1 ξF(s,r),i k is the numerical approximation of ˆX(s) k = E(X(s) k |Y (s) 1:k ) by the filter F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We also compute the global root mean squared error: RMSE(F) def= 1 K K � k=1 RMSEk(F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note that for the EKF and UKF filters the summation over r in (8) is useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' All implementations are done in R language [24] using a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='11 GHz core i7-8650U intel running Windows 10 64bits with a 16 Go RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='12 0 1000 2000 3000 4000 5000 Number of particules Time for filtering a trajectory (s) APF BPF PBPS 0 2 4 6 0 1000 2000 3000 4000 5000 Number of particules RMSE Figure 1: One-dimensional case study (9) — We plot the average computation time for filtering a tra- jectory (left) and the RMSE (right) of the filters PBPS, BPF and APF as a function of different values of N (50, 100, 500, 1000, 3000, 5000) (with S = 100 and R = 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In comparison, the EKF and UKF have a respective RMSE of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='83 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='88 (and a negligible computation time compared to the particle approximations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 First case study: a one-dimensional model We consider the following one-dimensional nonlinear model [6, 11, 14]: Xk = Xk−1 2 + 25 Xk−1 1 + X2 k−1 + 8 cos � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 (k − 1) � + Wk−1 , Yk = X2 k 20 + Vk , (9) with 1 ≤ k ≤ K (K = 50), X0 ∼ N(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Wk iid∼ N(0, 32), Vk iid∼ N(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' X0, (Wk)k≥1 and (Vk)k≥1 mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note that the state process Xk is observed only through X2 k so the filters have difficulties to determine whether Xk is positive or negative, especially since the state process Xk regularly changes it’s sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Thus this model is regularly used as benchmark for testing filters, as filters may easily lose track of Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For this example we compare the filters PBPS, BPF, APF, EKF and UKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 1 we plot, as a function of different values of N (50, 100, 500, 1000, 3000, 5000), on the left the average computation time for the filtering a trajectory, and on the right the RMSE (with S = 100, R = 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The computation times of EKF and UKF are negligible compared to those of the particle ap- proximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' On the other hand, the RMSE of EKF (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='83) are much higher than those of the 11 particle approximations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' UKF with an RMSE of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='88 behaves much better than EKF but is still less accurate than the particle approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The computation time of PBPS is naturally higher than that of BPF, but the latter is much less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For example, PBPS with N = 50 particles is 15 times faster than BPF with N = 5000 particles while being 30% more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' −20 −10 0 10 20 0 10 20 30 40 50 k Xk True Trajectory APF BPF EKF PBPS UKF −20 −10 0 10 20 0 10 20 30 40 50 k Xk −20 −10 0 10 20 0 10 20 30 40 50 k Xk −20 −10 0 10 20 0 10 20 30 40 50 k Xk −20 −10 0 10 20 0 10 20 30 40 50 k Xk Figure 2: One-dimensional case study (9) — On a single simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' for each filter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' we plot the true state trajectory k → Xk and the approximation k → ˆXk with the associated 95% confidence region (N = 5000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 2, for each filter F ∈ {PBPS, BPF, APF, EKF, UKF} (with N = 5000), we plot the true state trajectory k → Xk, the approximation k → ˆXF k , and the associated 95% confidence regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For the particle approximations the confidence interval is given empirically by the particles, for the EKF and UKF it’s given as an approximated Gaussian confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The PBPS approximation has a smaller error k → | ˆXF k − Xk| and a smaller confidence region than the other filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 3, we compare k → RMSEk(F) for F ∈ {PBPS, BPF, APF, EKF, UKF} (with N = 5000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Again, we see that the PBPS behaves better than the other filters and that the EKF has a very erratic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The poor behavior of the EKF can be explained by the very nonlinear nature of this example and by the observability problem around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note the relatively good behavior of the UKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' However, because of the observability problem at 0, both the UKF and the EKF cannot find the track of Xk once in the wrong half-plane (Figure 2 around time k = 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 12 0 20 40 0 10 20 30 40 50 k RMSEk APF BPF EKF PBPS UKF Figure 3: One-dimensional case study (9) — We compare k → RMSEk for each filter (S = 100, R = 40, N = 5000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 4, in the case of a single simulation of the state-space system and of the BPF and PBPS filters (with N = 5000), we plot the true trajectory k → Xk as well as the set of particles k → ξF,1:N k for F ∈ {BPF, PBPS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Not surprisingly, we find that the PBPS particles are more concentrated around the true value Xk and that BPF more often “loads” the symmetric part of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Indeed, the smoothing allows us to reduce the variance but also to compensate the observability ambiguities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' it is for this purpose that it was designed, and we reiterate that this gain is obtained with a great improvement in the computation time (provided that less particles are used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' −20 −10 0 10 20 0 10 20 30 40 50 k Xk −20 −10 0 10 20 0 10 20 30 40 50 k Xk Figure 4: One-dimensional case study (9) — On a single simulation, we plot the true value k → Xk with the evolution of the set of N = 500 particles k → ξ1:N k for the BPF (top) and the PBPS (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the particles are represented in transparency to better reflect their density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 Second case study: a four-dimensional bearings-only tracking model We consider the following four-dimensional model [3, 11, 23]: Xk = � 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 � Xk−1 + σW � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='5 1 0 0 1 � Wk−1 , Yk ∼ wrapped Cauchy � arctan �Xk[1] Xk[2] � , ρ � , (10) with 1 ≤ k ≤ K = 20, X0 ∼ N( ¯X0, P0) with: ¯X0 = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='055 �∗ , P0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='01 diag � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='32, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0052, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='012� , σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, Wk iid∼ N(0, I2×2), ρ = 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0052, X0 and (Wk)k≥1 mutually independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' conditionally on (Xk)k≥0, (Yk)k≥1 and (Wk)k≥1 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The state equation corresponds to a target moving on a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The state vector is: Xk = (Xk[1], Xk[2], Xk[3], Xk[4])∗ = (x1, x2, ˙x1, ˙x2)∗ , where (x1, x2) are the Cartesian coordinates of the target in the plane and ( ˙x1, ˙x2) are the corre- sponding velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The observer is located at the origin of the plane and accessed only to the azimuth angle β = arctan(x1/x2) ∈ [−π, π] corrupted by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' It is known that in the situation where the observer follows a rectilinear trajectory at constant speed, or zero speed as here, the filtering problem is poorly conditioned [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This is a situation where it is known for example that the EKF diverges very strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The conditional probability density function of the measured angle Yk given the state Xk is assumed to be a wrapped Cauchy distribution with concentration parameter ρ [23]: pYk|Xk=x(y) = (2 π)−1 (1 − ρ2) 1 + ρ2 − 2 ρ cos � y − arctan(x[1] x[2])) � (11) for −π ≤ y < π, where ρ ∈ [0, 1] is the mean resultant length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Note that the state dynamics is linear and Gaussian, but the observation dynamics is nonlinear and non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Classically, in tracking studies, the trajectory model (the state equation) does not correspond to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In reality, the target follows a uniform straight trajectory sometimes interrupted by changes in heading and/or speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The variance σW of the state equation (10) appears then as a parameter of the filter: if it is too small, the filter will have trouble tracking the target, if it is too large, the particles will spread out too much, the filter will then lose accuracy, and even lose track of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For bearings-only tracking applications, the EKF and UKF filters have poor performances and will therefore not be considered in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' On the other hand, the robust filters introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 are relevant here because there is a mismatch model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We compare the filters in two scenarios (with K = 40 time steps): the target follows a uniform rectilinear motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='15 0 1000 2000 3000 4000 5000 Number of particules Time for filtering a trajectory (s) APF BPF PBPS RS−APF RS−BPF RS−PBPS DMA−BPF Figure 5: Four-dimensional case study (11) — We plot the average computation time for filtering a trajectory of the filters PBPS, BPF, APF, RS-PBPS, RS-BPF, RS-APF and DMA-BPF as a function of different values of N (50, 100, 500, 1000, 3000, 5000) (with S = 20 and R = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' the target follows a uniform rectilinear motion with change of heading at the middle of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' As we will see, the first scenario is simpler than the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For the filter we consider two different values for σW: a small value σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001 and a large σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001 is in fact too small to be consistent with the simulated trajectory, the second value σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003 is more consistent with the simulated trajectory For the robust filter we consider: M = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='005} as the set of possible models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 5 we present the average computation times for filtering a trajectory as a function of different values of N (50, 100, 500, 1000, 3000, 5000) (with S = 20 and R = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 6, in agreement with Figure 5, we present the evolution of the RMSE as a function of the number N of particles: the left column corresponds to the case without a turn, the right column to the case with a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For PBPS, BPF and APF, the first row (top) presents the RMSE for with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, the second (middle) with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The third row (bottom) shows the RMSE for the RS-PBPS, RS-BPF, RS-APF, and DMA-BPF filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 7 we plot k → RMSEk (for N = 5000, R = 50, S = 20) for the different filters: the left column corresponds to the case without a turn, the right column to the case with a turn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' PBPS, BPF and APF for with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, on the first row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' PBPS, BPF and APF for with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003, on the middle row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' RS-PBPS, RS-BPF, RS-APF, and DMA-BPF, on the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In Figure 8, on a single simulation and for each filter, we plot the true trajectory of the target with the estimates of each filter: without turn in the left column, with 1 turn in the right column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' PBPS, BPS and APF filters with small σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001 in the top row, PBPS, BPS and APF filters with large σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003 in the middle row, RS-PBPS, RS-BPS, RS-APF, and DMA-BPF filters in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' According to Figure 5, at the same number of particles, BPF is slightly faster than PBPS, both being faster than APF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The overheads for the robust version (RS) is also reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In terms of complexity, the computation time is linear with the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 15 In order to better understand the situation, we will first comment on Figures 7 and 8: with a too small σW, the filters are able to track the target when it stays in a straight line (a) but they are not able to adapt when a turn occurs and continue the tracking more or less in a straight line (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' With a larger σW, the filters are still able to track the target when it remains in a straight line (c) with a small and increasing inaccuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' in contrast only PBPS behaves reasonably well when a turn occurs (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Except DMA-BPF which has a bad behavior, the RS robust filters behave better and RS-PBPS remains better than all the others (e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' From Figures 5 and 6, in the case of a turn, which is the most complex situation, we see that in order to reach PBPS/RS-PBPS accuracy with N = 1000 the other filters must use N = 5000 particles, and even in this case PBPS/RS-PBPS is at least twice as fast as the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' V DISCUSSION AND CONCLUSION Our goal was to propose a particle approximation algorithm slightly more complex than the BPF but on the one hand allowing the variance of the error to be reduced and on the other hand being more robust in poorly conditioned situations, especially when the observability conditions are poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We have therefore proposed an algorithm that deviates slightly from the “pure” filtering method in the sense that it takes into account the next observation in addition to the current one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We can thus consider PBPS as an approximation of the smoother with a fixed-lag interval of one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' We have also proposed a more robust version of PBPS using the “regime switching” strategy proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In the considered examples, EKF shows very poor performance, which is perfectly understand- able due to the very nonlinear nature and the poor observability conditions of the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' It should be noted that UKF performs much better than EKF while remaining far from the perfor- mance of particle filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Among the particle filters, APF gives slightly worse results than PBF and PBPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Finally, even if for a given number of particles, BPF is faster than PBPS, the latter presents both a lower error variance and a much better behavior regarding observability prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In terms of accuracy, PBPS with 1000 particles is still better than BPF with 5000 particles while being much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' This is also the case when comparing RS-PBPS with RS-BPF These performances are due to the fact that PBPS is a very simplified approximation of the one time step fixed-lag smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' A strategy using a deeper fixed-lag interval of 2 or more time steps would be too costly and would not allow us to recover such performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In view of the performance of PBPS it would be interesting to develop strategies where the number N of particles adapts dynamically to the problem conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' It would also be inter- esting to test the PBPS strategy in other cases, such as when the time between two consecutive observations is important and requires several successive prediction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 16 APPENDIX: ALGORITHMS Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 Auxiliary particle filter (APF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 1: ξ1:N 0 iid∼ ρ0 # initialization 2: return ξ1:N 0 3: for k = 1 : K do 4: ¯ξi k = fk(ξi k−1) , i = 1 : N # high probability evolution 5: ¯ωi k ← ψk(¯ξi k|yk) , i = 1 : N # .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' and weighting 6: ˜ξ1:N k−1 iid∼ �N j=1 ¯ωi k δξj k−1 # initial particles resampling 7: ξi k− ∼ qk( · |˜ξi k−1) , i = 1 : N # particles propagation 8: ωi k ← ψk(ξi k−|yk)/¯ωi k , i = 1 : N # likelihood 9: ωi k ← ωi k/�N j=1 ωj k , i = 1 : N # renormalization 10: ξ1:N k iid∼ �N j=1 ωj k δξj k− # resampling 11: return ξ1:N k 12: end for Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 Dynamic model averaging bootstrap particle filter (DMA-BPF) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 1: ξ1:N 0 iid∼ ρ0, µ1:N 0 iid∼ U(M) # initialization 2: return ξ1:N 0 3: for k = 1 : K do 4: ξi k∗ ∼ qk( · |ξi k−1, µi k−1), i = 1 : N 5: ωl k = �N i=1 ψk(ξi k∗|yk)1µi k−1=ml, l = 1 : L # likelihood of candidate models 6: N 1:L k iid∼ Multinomial(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' , mL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' ω1 k, · · · , ωL k ) with N 1 k + · · · + N L k = N # number of particles per candidate model 7: ˆξ1,1:L k−1 iid∼ �N j=1 δξj k−1 δµj k−1=ml # initial particles resampling # per candidate model 8: (˜ξ1:N k−1, µ1:N k ) ← (ˆξ 1:N1 k,1 k−1 , m1), (ˆξ 1:N2 k,2 k−1 , m2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' , (ˆξ 1:NL k ,L k−1 , mL) 9: ξi k ∼ qk( · |˜ξi k−1, µi k) , i = 1 : N # particles propagation 10: return ξ1:N k 11: end for 17 Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 Regime Switching Bootstrap Particle Filter (RS-BPF) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 1: ξ1:N 0 iid∼ ρ0 # initialization 2: return ξ1:N 0 3: for k = 1 : K do 4: µ1:N k−1 iid∼ U(M) 5: ξi k− ∼ qk( · |ξi k−1, µi k−1), i = 1 : N # particles propagation 6: ωi k ← ψk(ξi k−|yk) , i = 1 : N # likelihood 7: ωi k ← ωi k/�N j=1 ωj k, i = 1 : N # renormalization 8: ξ1:N k iid∼ �N j=1 ωj k δξj k− # resampling 9: return ξ1:N k 10: end for REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Anderson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Optimal Filtering.' metadata={'source': 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153–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' Zuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' “Dynamic resampling for alleviating sample impoverishment of particle filter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' In: IET Radar, Sonar Navigation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='9 (2013), pages 968–977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 RMSE APF BPF PBPS (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 RMSE (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 Number of particules RMSE RS−APF RS−BPF RS−PBPS DMA−BPF (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 Number of particules (f) Figure 6: Four-dimensional case study (11) — Evolution of the RMSE as a function of the number N of particles: the left column corresponds to the case without a turn, the right column to the case with a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' For PBPS, BPF and APF, the first row (top) presents the RMSE for with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, the second (middle) with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' The third row (bottom) shows the RMSE for the RS-PBPS, RS-BPF, RS-APF, and DMA-BPF filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='6 0 10 20 30 40 k RMSEk APF BPF PBFS (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='6 0 10 20 30 40 k RMSEk (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='6 0 10 20 30 40 k RMSEk (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='6 0 10 20 30 40 k RMSEk (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='6 0 10 20 30 40 k RMSEk RS−APF RS−BPF RS−PBPS DMA−BPF (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='6 0 10 20 30 40 k RMSEk (f) Figure 7: Four-dimensional case study (11) — k →RMSEk (for N = 5000, R = 50, S = 20) for the different filters: the left column corresponds to the case without a turn, the right column to the case with a turn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' PBPS, BPF and APF for with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='001, on the first row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' PBPS, BPF and APF for with σW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='003, on the middle row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' RS-PBPS, RS-BPF, RS-APF, and DMA-BPF, on the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content=' 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfpgG_/content/2301.02541v1.pdf'} +page_content='5 1.' metadata={'source': 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FILES, VOL. 14, NO. 8, AUGUST 2015 +1 +Audio-Visual Segmentation with Semantics +Jinxing Zhou∗, Xuyang Shen∗, Jianyuan Wang∗, Jiayi Zhang, Weixuan Sun, Jing Zhang, +Stan Birchfield, Dan Guo, Lingpeng Kong, Meng Wang†, Fellow, IEEE, and Yiran Zhong† +Abstract—We propose a new problem called audio-visual segmentation (AVS), in which the goal is to output a pixel-level map of the +object(s) that produce sound at the time of the image frame. To facilitate this research, we construct the first audio-visual segmentation +benchmark, i.e., AVSBench, providing pixel-wise annotations for sounding objects in audible videos. It contains three subsets: +AVSBench-object (Single-source subset, Multi-sources subset) and AVSBench-semantic (Semantic-labels subset). Accordingly, three +settings are studied: 1) semi-supervised audio-visual segmentation with a single sound source; 2) fully-supervised audio-visual +segmentation with multiple sound sources, and 3) fully-supervised audio-visual semantic segmentation. The first two settings need to +generate binary masks of sounding objects indicating pixels corresponding to the audio, while the third setting further requires generating +semantic maps indicating the object category. To deal with these problems, we propose a new baseline method that uses a temporal +pixel-wise audio-visual interaction module to inject audio semantics as guidance for the visual segmentation process. We also design a +regularization loss to encourage audio-visual mapping during training. Quantitative and qualitative experiments on AVSBench compare +our approach to several existing methods for related tasks, demonstrating that the proposed method is promising for building a bridge +between the audio and pixel-wise visual semantics. Code is available at https://github.com/OpenNLPLab/AVSBench. Online benchmark is +available at http://www.avlbench.opennlplab.cn. +Index Terms—Audio-visual segmentation, Multi-modal segmentation, Audio-visual learning, AVSBench, Semantic segmentation, Video +segmentation. +! +1 +INTRODUCTION +H +UMANS largely rely on visual and auditory cues to +understand their environmental surroundings. For +example, a dog barking can be distinguished from a bird +calling based on both their sound and appearance. Such +audio-visual information is integrated with the brain in a +synthesis process [1], crucial for comprehensively perceiving +the world. Inspired by this cognitive ability of humans, we +explore audio-visual learning with deep models via the +integration of multi-modal signals. +Over the years, researchers have studied various prob- +lems within audio-visual artificial perception. For instance, +some researchers investigate the audio-visual correspon- +dence (AVC) problem [2], [3], [4], which aims to determine +whether an audio signal and a visual image describe the +same scene. AVC is based on the phenomenon that these two +signals usually occur simultaneously, such as a barking dog, +a singing person, and a humming car. Others study the audio- +• +Jinxing Zhou, Dan Guo and Meng Wang are with Key Laboratory of +Knowledge Engineering with Big Data (HFUT), Ministry of Education +and School of Computer Science and Information Engineering, Hefei +University of Technology, Hefei, China. +• +Xuyang Shen is with the Sensetime Research, Shanghai, China. +• +Jianyuan Wang is with Visual Geometry Group, University of Oxford, +Oxford, United Kingdom. +• +Jiayi Zhang is with School of Computer Science and Engineering, the +Beihang University, Beijing, China. +• +Weixuan Sun and Jing Zhang are with School of Computing, the Australian +National University, Canberra, Australia. +• +Stan Birchfield is with Nvidia, Redmond, WA, USA. +• +Lingpeng Kong is with the University of Hong Kong, Hong Kong, China +and Shanghai AI Lab, Shanghai, China. +• +Yiran Zhong is with Shanghai AI Lab, Shanghai, China. +• +∗: These authors have equal contributions. +• +†: Meng Wang and Yiran Zhong are corresponding authors (e-mail: +eric.mengwang@gmail.com, zhongyiran@gmail.com). +visual event localization (AVEL) [5], [6], [7], [8], [9], [10], [11], +[12], [13], [14], which classifies the segments of a video using +a set of pre-defined event labels. Similarly, some research +explores audio-visual video parsing (AVVP) [15], [16], [17], +[18], [19], [20], [21], whose goal is to divide a video into +several events and classify them as audible, visible, or both. +Due to a lack of pixel-level annotations, all these scenarios +are restricted to the frame/temporal level, thus reducing the +problem to audible image classification. +A related problem, known as sound source localization +(SSL), aims to locate the visual regions within the image +frames that correspond to the sound [2], [3], [22], [23], [24], +[25], [26], [27], [28]. Compared to AVC/AVEL/AVVP, SSL +seeks patch-level scene understanding, i.e., the results are +usually presented by a heat map that is obtained either +by visualizing the similarity matrix of the audio feature +and the visual feature map, or by class activation mapping +(CAM) [29]—without considering the actual shape of the +sounding objects. +Building on this research, in this work we propose the +pixel-level audio-visual segmentation (AVS) problem. This +problem requires the network to densely predict whether +each pixel corresponds to the given audio, so that a mask +of the sounding object(s) is generated. Fig. 1 illustrates the +differences between SSL and AVS. As can be seen, the AVS +task is more challenging as it requires the network to not +only locate the audible frames but also delineate the shape +of the sounding objects. Moreover, the AVS finally needs to +classify the category semantics of different sounding objects. +As shown in Fig. 1, each type of sounding object is assigned +a specific color indicating its unique semantic category. +To facilitate this research, we release the AVSBench +dataset, which is the first pixel-level audio-visual segmen- +tation benchmark that provides ground truth labels for +arXiv:2301.13190v1 [cs.CV] 30 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +Video +SSL +AVS- +object +Audio +28XJmUz906g_0 +dog barking +lawn mower +UsTf7brftGg_0 +man talking +man talking +man talking, +playing piano +playing piano +playing piano +(a) AVS-S4 +dog barking +lawn mower +dog barking +lawn mower +dog barking +lawn mower +dog barking +lawn mower +AVS- +semantic +Fig. 1. Comparison of the proposed AVS task with the Sound source localization (SSL) task. SSL aims to estimate an approximate location of +the sounding objects in the visual frame, at a patch level. In contrast, AVS estimates pixel-wise masks for all the sounding objects, regardless of the +number of visible sounding objects. The segmentation masks can be binary or semantic under different task settings. The binary masks indicate +objects making sounds while the semantic masks further distinguish the object category. In the last row, the ground truths are displayed with the +semantic masks. +sounding objects. The dataset is divided into three subsets. +In the first subset, there is a single sound source in the +video, leading to the task we call semi-supervised Single Sound +Source Segmentation (S4). In the second subset, there are +multiple sound sources, leading to the task of fully-supervised +Multiple Sound Source Segmentation (MS3). For these two +subsets, the ground truths are binary masks indicating pixels +emitting the sounds. We study these two settings to have +a basic perception of audio-visual segmentation from pixel- +level. While, the third subset is a Semantic-labels subset +that introduces semantic labels of the sounding objects, +exploring the task of fully-supervised Audio-Visual Semantic +Segmentation (AVSS). Compared to the S4 and MS3 settings, +AVSS requires generating semantic maps that further tell the +category information of the masked sounding objects. As +shown on the left example of Fig. 1, pixels of the dog and +lawn mower are assigned with different colors indicating the +unique semantic categories. For convenience, we denote that +the first two subsets, i.e., Single-source and Multi-sources +subsets, constitute the AVSBench-object dataset, and the third +Semantic-labels subset is also called the AVSBench-semantic +dataset. For all the settings, the goal is to segment the object(s) +from the visual frames that are producing sounds. Compared +with traditional semantic segmentation [30], [31], [32], [33], +[34] task or video object segmentation [35], [36], [37] task, +AVS is a multi-modal segmentation problem that necessitates +the alignment of visual and audio semantics rather than +classifying each pixel solely based on visual cues. +To deal with the aforementioned three settings, we test +several methods from related tasks on AVSBench dataset +and provide a new AVS method as a strong baseline. The +framework is shown in Fig. 4. It utilizes a standard encoder- +decoder architecture but with a novel temporal pixel-wise +audio-visual interaction (TPAVI) module to better introduce +the audio semantics for guiding visual segmentation. We +also propose a loss function to utilize the correlation of +audio-visual signals, which further enhances segmentation +performance. +At last, we remind that the audio-visual segmentation +problem is first introduced in our previous work [38] that has +been published in ECCV 2022. Compared with the conference +version, we add the following new extensions in this paper. +Firstly, we expand upon our previous work by incorporating +a new and challenging setting, i.e., the fully-supervised +AVSS, which can be viewed as an independent task by the +research community. We also conduct extensive ablation +studies in this setting. These explorations help us gain a +deeper understanding of audio-visual scenarios and design +a more realistic model that enables us to perceive pixel-wise +semantics. Secondly, we propose the AVSBench-semantic +dataset containing a Semantic-labels subset that newly pro- +vides pixel-wise semantic labels, as a significant complement +of the original AVSBench dataset. The AVSBench-semantic +dataset includes significantly more event categories (70 vs. 23) +and frames (80k vs. 10k) compared to the original AVSBench +dataset. More extending details of the video statistic and +annotation are introduced in Sec. 3. Thirdly, we update the +previous AVS model [38] to predict the semantic maps and +add extensive experiments on the new AVSBench dataset. +For the convenience of the community, we build an online +benchmark suite at http://www.avlbench.opennlplab.cn. +2 +RELATED WORK +Sound Source Localization (SSL). Perhaps the most closely +related problem to ours is SSL, which aims to locate the +regions in the visual frames responsible for the sounds. The +prediction of SSL is usually computed from the similarity +matrix of the learned audio feature and the visual feature +map [2], [3], [22], [23], [24], [25], displayed as a heat map. +SSL can also be divided into two settings according to the +complexity of sound sources, viz., single and multiple sound + +20000 +10000 +0 +-10000 +-20000 +0 +1 +2 +3 +5 +20000 +10000 +0 +-10000 +-20000 +0 +1 +4 +5 +time (seconds)20000 ++++++ +10000 +0 +-10000 +-20000 +0 +2 +3 +5 +20000 +10000 +0 +-10000 +-20000 +0 +Y +2 +3 +4 +5 +time (seconds)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +source(s) localization. Here we focus on the challenging +setting of multiple sources, which requires accurately lo- +calizing the true sound source among multiple potential +candidates [26], [27], [28], [39]. In pioneering work, Hu et +al. [26] divide the audio and visual features into multiple +cluster centers and take the center distance as a supervision +signal to rank the paired audio-visual information. Qian et +al. [27] first train an audio-visual correspondence model to +extract coarse feature representations of audio and visual +signals, and then use Grad-CAM [40] to visualize the class- +specific features for localization. Furthermore, Hu et al. [28] +adopt a two-stage method, which first learns audio-visual +semantics in the single sound source condition, using such +learned knowledge to help with multiple sound sources +localization. Rouditchenko et al. [41] tackles this problem +by disentangling category concepts in the neural networks. +This method is actually more related to the task of sound +source separation [42], [43], [44], [45] and shows sub-optimal +performance regarding visual localization. Although these +existing SSL methods indicate which regions in the image +are making sound, the results do not clearly delineate the +shape of the objects. Rather, the location map is computed +by up-sampling the audio-visual similarity matrix from +a low resolution. Moreover, the methods above all rely +on unsupervised learning when capturing the shape of +sounding objects, which partly suffers from the lack of an +annotated dataset. To overcome these limitations, this paper +provides an audio-visual segmentation dataset with pixel- +level ground truth labels, which enables to achieve more +accurate segmentation predictions. +Video Object Segmentation (VOS). The VOS task aims to +segment the object of interest throughout the entire video +sequence. It is divided into two settings: the semi-supervised +and unsupervised. For the semi-supervised VOS, the target +object is decided given a one-shot mask of the first sampled +video frame [35], [36], [37]. As for unsupervised VOS, it +needs to automatically segment all the primary objects [46], +[47], [48]. Many excellent works are proposed and proven +to achieve impressive segmentation performance [49], [50], +[51], [52], [53], [54]. However, these fancy designs are limited +to a single visual modality. Recently, referring video object +segmentation (R-VOS) attracts more attention [55], [56], [57], +[58]. The target object in R-VOS task is referred by a short +language expression, whereas the proposed AVS task focuses +on the audio-aligned visual objects, i.e., the object of interest +is determined by the audio. Unlike the language used in R- +VOS has clear semantics, the proposed AVSS requires a joint +semantic classification for both audio and visual information, +which makes it more challenging than the R-VOS task. +Audio-Visual Dataset. To the best of our knowledge, there +are no publicly available datasets that provide segmentation +masks for the sounding visual objects with audio signals. +Here we briefly introduce the popular datasets in the audio- +visual community. For example, the AVE [7] and LLP [15] +datasets are respectively collected for audio-visual event +localization and video parsing tasks. They only have category +annotations for video frames, and hence cannot be used for +pixel-level segmentation. For the sound source localization +problem, researchers usually use the Flickr-SoundNet [22] +and VGG-SS [25] datasets, where the videos are sampled +from the large-scale Flickr [4] and VGGSound [59] datasets, +TABLE 1 +AVSBench statistics. The videos are split into train/valid/test. The +asterisk (∗) indicates one annotation per video whereas others are one +annotation per second. ⋄ in the last row indicates that 1,000 videos are +withheld for online benchmarking. +subsets +classes +videos +train/valid/test +labeled frames +single-source +23 +4,932 +3,452∗/740/740 +10,852 +multi-source +23 +424 +296/64/64 +2,120 +semantic-labels +70 +12,356⋄ +8,498/1,304/1,554 +82,972 +TABLE 2 +Existing audio-visual dataset statistics. Each benchmark is shown +with the number of videos and the annotated frames. The final column +indicates whether the frames are labeled by category, bounding boxes, +or pixel-level masks. AVSBench extension provides pixel-level semantic +labels with object category information. +benchmark +videos frames classes types +annotations +AVE [7] +4,143 41,430 +28 +video +category +LLP [15] +11,849 11,849 +25 +video +category +Flickr-SoundNet [22] +5,000 +5,000 +50 +image +bbox +VGG-SS [25] +5,158 +5,158 +220 +image +bbox +AVSBench-object [38] +5,356 12,972 +23 +video +pixel +AVSBench-semantic +12,356 82,972 +70 +video pixel & category +respectively. The authors provide bounding boxes to outline +the location of the target sound source, which could serve as +patch-level supervision. However, this still inevitably suffers +from incorrect evaluation results since the sounding objects +are usually irregular in shape and some regions within the +bounding box actually do not correspond to the real sound +source. The proposed AVSBench dataset provides pixel-wise +semantic masks that accurately outline the shape of sounding +objects. This is beneficial for the research of pixel-level audio- +visual learning. +3 +THE AVSBENCH DATASET +The AVSBench dataset is first proposed in our previous +work [38]. It contains a Single-source and a Multi-sources +subset. Ground truths of these two subsets are binary +segmentation maps indicating pixels of the sounding objects. +Recently, we collected a new Semantic-labels subset that +provides semantic segmentation maps as labels. We add it +to the original AVSBench dataset as the third subset. For +convenience, we denote the original AVSBench dataset as +AVSBench-object, and the newly added Semantic-labels subset +as AVSBench-semantic. In this section, we first introduce the +video statistics and annotations of the AVSBench-object and +then provide the extending details of AVSBench-semantic. +Lastly, we introduce three benchmark settings on the updated +AVSBench dataset. +3.1 +Dataset Statistics +AVSBench-object. We collected the videos using the tech- +niques introduced in VGGSound [59] to ensure the audio and +visual clips correspond to the intended semantics. AVSBench- +object [38] contains two subsets—Single-source and Multi- +sources—depending on the number of sounding objects. All + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +296312 +57 +406 +260 +5310743 +225 +47 52 49 47 64 55 54 32 +14277 56 +187 +52 91 86 +168 +39 52 23 +332 332 +23 93 +361 +50 57 41 +12111693 47 +278 +149 +265 +1021 +1370 +169 +78 +219 +388 +560 +99136 +310 +170 +93102 +1309 +1107 +99 84107 +142 +94 92 92 85 +210 +60 57 +310 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +bird +lion +squirrel +dog +cat +tiger +wolf +leopard +horse +duck +hen +pig +donkey +sheep +goose +parrot +elephant +mower +bell +clipper +gun +clock +saxophone +hair-dryer +handpan +frying-food +cacuum-cleaner +axe +keyboard +saw +missile-rocket +airplane +helicopter +truck +tank +motorcycle +tractor +boat +train +utv +emergency-car +bus +car +woman +man +girl +boy +baby +tabla +violin +guzheng +clarinet +marimba +cello +pipa +accordion +guitar +piano +trombone +tuba +bassoon +flute +harp +sitar +trumpet +erhu +drum +suona +harmonica +ukulele +Fig. 2. Statistics of the AVSBench dataset extension, i.e., the AVSBench-semantic dataset. There are 70 categories in the extension and the +video number of each category is given. +videos were downloaded from YouTube under the Creative +Commons license, and each video was trimmed to 5 seconds. +The Single-source subset contains 4, 932 videos over 23 +categories, covering sounds from humans, animals, vehicles, +and musical instruments. To collect the Multi-sources subset, +we selected the videos that contain multiple sounding objects, +e.g., a video of baby laughing, man speaking, and then +woman singing. To be specific, we randomly chose two +or three category names from the Single-source subset as +keywords to search for online videos, then manually filtered +out videos to ensure 1) each video has multiple sound +sources, 2) the sounding objects are visible in the frames, +and 3) there is no deceptive sound, e.g., canned laughter. In +total, this process yielded 424 videos for the Multi-sources +subset out of more than six thousand candidates. The ratio of +train/validation/test split percentages is set as 70/15/15 for +both subsets, as shown in Table 1. Several video examples +are visualized in Fig. 3, where the red text indicates the name +of sounding objects. +AVSBench-semantic. Since the original version of AVSBench +dataset, i.e., AVSBench-object, we have extended the dataset +by adding a third Semantic-labels subset that provides +semantic segmentation maps as labels. AVSBench-semantic is +enriched in video amount and audio-visual scene categories. +In total, it contains 12,356 videos covering 70 categories. In +Fig. 2, we show the category names and the video number +for each category. This extension reserves all 5,356 videos +from the original and upgrades them to 720p resolution. In +addition, we further collect another 7,000 multi-source videos +following the principle of collecting multi-sources subset +of the original dataset. We reserve 1,000 videos for online +evaluation and it will only be available for contestants in the +future AVS Benchmark competition. These newly collected +videos are trimmed to 10 seconds which helps to train a +segmentation model with the ability to encode long-range +audio-visual sequences. Except for the 1,000 withheld videos, +the rest of the videos are split into 8,498 for training, 1,304 +for validation, 1,554 for testing. We also display some video +examples in Fig. 3. +The AVSBench-object and the AVSBench-semantic to- +gether form the updated AVSBench dataset. We make a com- +parison between AVSBench with other popular audio-visual +benchmarks in Table 2. The AVE [7] dataset contains 4,143 +videos covering 28 event categories. The LLP [15] dataset +consists of 11,849 YouTube video clips spanning 25 cate- +gories, collected from AudioSet [60]. Both the AVE and LLP +datasets are labeled at a frame level, through audio-visual +event boundaries. Meanwhile, the Flickr-SoundNet [22] +dataset and VGG-SS [25] dataset are proposed for sound +source localization (SSL), labeled at a patch level through +bounding boxes. The AVSBench-object (original AVSBench +dataset [38]) contains 5,356 videos with 12,972 pixel-wise +annotated frames which is designed to facilitate research on +fine-grained audio-visual segmentation. AVSBench-semantic +further extends it from three aspects: 1) the video quantity +is expanded to 12,356 and focuses more on the multi-source +case; 2) the number of object categories is enlarged from 23 +to 70; 3) annotations are updated from pixel-wise binary +mask to semantic masks. The recent AVSBench dataset +provides accurate semantic maps as ground truth. This +makes it beneficial not only for the proposed audio-visual +segmentation but also for sound source localization, which +could help the training of SSL methods and serve as an +evaluation benchmark. +3.2 +Annotation +AVSBench-object. Videos in AVSBench-object are trimmed +to 5 seconds. We divide each 5-second video into five equal 1- +second clips, and we provide manual pixel-level annotations +for the 1-second clips. The ground truth label is a binary +mask indicating the pixels of sounding objects, according +to the audio at the corresponding time. For example, in the +Multi-sources subset, even though a dancing person shows +drastic movement spatially, it would not be labeled as long +as no sound was made. In clips where objects do not make +sound, the object should not be masked, e.g., the piano in +the first two clips of the last row of Fig. 3b. Similarly, when +more than one object emits sound, all the emitting objects +are annotated, e.g., the guitar and ukulele in the first row +in Fig. 3b. Also, when the sounding objects in the video are +changing dynamically, the difficulty is further increased, e.g., +the second, third, and fourth rows in Fig. 3b. +We use two types of labeling strategies, based on the +different difficulties between the Single-source and the Multi- +sources subsets. For the videos in the training split of Single- +source, we only annotate the first sampled frame (with the +assumption that the information from one-shot annotation +is sufficient, as the Single-source subset has a single and +consistent sounding object over time). This assumption is +verified by the quantitative experimental results shown in +Table 3. For the more challenging Multi-sources subset, all + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +mynah bird singing +ambulance siren +horse clip-clop +man with violin and piano +guitar with ukulele +man with piano +(a) Video examples in Single-source subset of AVSBench +(b) Video examples in Multi-sources subset of AVSBench +man with vacuum-cleaner +(c) Video examples in Semantic-labels subset of AVSBench +-SPr6nTlav0_181000_191000 +-u5tbtyhBMA_35000_45000 +024k2nxvSRI_144000_154000 +donkey with woman +girl with guitar +68Mm2Nc-W4w_50000_60000 +Fig. 3. AVSBench samples. The AVSBench dataset contains the Single-source subset (a), Multi-sources subset (b), and Semantic-labels subset +which mainly contains the multi-source videos (c). Each video is divided into 5 clips for the first two, while 10 clips for the latter, as shown. Annotated +clips are indicated by brown framing rectangles while the green rectangles represent there are no sounding objects in those frames; the name of +sounding objects is indicated by red text. Binary masks of the sounding objects are annotated in the first two, reflected by the orange masks in (a) and +(b). The third subset provides colorful semantic masks indicating different object categories. Note that for the Single-source training set of AVSBench, +only the first frame of each video is annotated, whereas all of the extracted frames are annotated for all other sets. +clips are annotated for training, since the sounding objects +may change over time. Note that for validation and test splits, +all clips are annotated, as shown in Table 1. +AVSBench-semantic. The AVSBench-semantic subset uses +the videos from AVSBench-object. For these videos, we +update the annotated binary masks to semantic masks by +adding category information of the sounding objects. As for +the newly collected 10-second videos, we sample ten video +frames and provide their semantic annotations, similar to +the annotation process of AVSBench-object. We show some +annotation examples in Fig. 3(c). As shown, the sounding +object is highlighted with unique color indicating its category. +Also, when there is no sound or the sounding object is out of +the screen (green boxes in the second row of Fig. 3(c)), that +video frame will not be annotated. +3.3 +Benchmark Setting +We provide three benchmark settings: the semi-supervised +Single Sound Source Segmentation (S4), the fully supervised +Multiple Sound Source Segmentation (MS3), and the fully +supervised audio-visual semantic segmentation (AVSS). The +former two settings are based on the AVSBench-object dataset +while the AVSS is conducted on the AVSBench-semantic. +For ease of expression, we denote the video sequence as S, +which consists of T non-overlapping yet continuous clips +{Sv +t , Sa +t }T +t=1, where Sv and Sa are the visual and audio +components, T is equal to 5 for AVSBench-object while 10 for +AVSBench-semantic. In practice, we extract the video frame +at the end of each second. +Semi-supervised S4 corresponds to the Single-source subset +of AVSBench-object. It is termed as semi-supervised because +only part of the ground truth is given during training (i.e., +the first sampled frame of the videos) but all the video +frames require a prediction during evaluation. We denote the +pixel-wise label as Y s +t=1 ∈ RH×W , where H and W are the +frame height and width, respectively. Y s +t=1 is a binary matrix +where 1 indicates sounding objects while 0 corresponds to +background or silent objects. +Fully-supervised MS3 deals with the Multi-sources subset +of AVSBench-object, where the labels of all five frames of +each video are available for training. The ground truth is +denoted as {Y m +t }T +t=1, where Y m +t +∈ RH×W is the binary +label for the t-th video clip. + +2bestathire +NatonalTool&EquipmentHire +0F-0723bestathire +Naonal Tool&EquipmentHire +6723bestathire +National Tool&EquipmentHire +-0323bestathire +632ebestataire +Natonalbestathire +Na +Carpet Cieaner Hirebestathirebestathire +Nason +Tool8FiinJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +stage 1 +stage 2 +stage 3 +stage 4 +𝑇×𝐻×𝑊×3 +𝑭!: 𝑇× 𝐻 +4 × 𝑊 +4 ×𝐶! +𝑭": 𝑇× 𝐻 +8 × 𝑊 +8 ×𝐶" +𝑭#: 𝑇× 𝐻 +16 × 𝑊 +16 ×𝐶# +𝑭$: 𝑇× 𝐻 +32 × 𝑊 +32 ×𝐶$ +ASPP +TPAVI +visual encoder +visual encoder +visual encoder +visual encoder +𝑨: 𝑇×𝑑 +𝑇 seconds +audio encoder +ASPP +ASPP +ASPP +TPAVI +TPAVI +TPAVI +𝑴: 𝑇×𝐻×𝑊×𝐾 +𝑷$: 𝑇× 𝐻 +2 × 𝑊 +2 ×𝐶 +𝑷#: 𝑇× 𝐻 +4 × 𝑊 +4 ×𝐶 +𝑷": 𝑇× 𝐻 +8 × 𝑊 +8 ×𝐶 +𝑷!: 𝑇× 𝐻 +16 × 𝑊 +16 ×𝐶 +stage 4 +stage 3 +stage 2 +stage 1 +Encoder +Decoder +𝑽! +𝑽" +𝑽# +ASPP +𝑽$ +𝒁! +𝒁" +𝒁# +𝒁$ +Fig. 4. Overview of the Baseline, which follows a hierarchical Encoder-Decoder pipeline. The encoder takes the video frames and the entire audio +clip as inputs, and outputs visual and audio features, respectively denoted as Fi and A. The visual feature map Fi at each stage is further sent to +the ASPP [61] module and then our TPAVI module (introduced in Sec. 4). ASPP provides different receptive fields for recognizing visual objects, +while TPAVI focuses on the temporal pixel-wise audio-visual interaction. The decoder progressively enlarges the fused feature maps by four stages +and finally generates the output mask M for sounding objects. +Fully-supervised AVSS deals with the Semantic-labels +subset of AVSBench-semantic, where the semantic masks +of all ten frames of each video are known during train- +ing. The ground truth can be denoted as {Yt}T +t=1, where +Yt ∈ RH×W ×K is the semantic label for the t-th video clip, +K is the total category number of the sounding objects in the +dataset. +The goal for all the settings is to correctly segment the +sounding object(s) for each video clip by utilizing the audio +and visual cues, i.e., Sa and Sv. Different from S4 and MS3 +settings, AVSS setting needs to further output the category +of the sounding objects. Generally, it is expected Sa to +indicate the target object, while Sv provides information +for fine-grained segmentation. The predictions are denoted +as {Mt}T +t=1, Mt ∈ RH×W ×K, where K = 1 under S4 and +MS3 settings. +4 +A BASELINE +We propose a new baseline method for the pixel-level audio- +visual segmentation problem as shown in Fig. 4. Following +the convention of semantic segmentation methods [30], +[31], [33], [34], our method adopts an encoder–decoder +architecture. +The Encoder: We extract audio and visual features inde- +pendently. Given an audio clip Sa, we first process it to a +spectrogram via the short-time Fourier transform and then +send it to a convolutional neural network, VGGish [62]. +We use the weights that are pretrained on AudioSet [60] +to extract audio features A ∈ RT ×d, where d = 128 is +the feature dimension. For a video frame Sv, we extract +visual features with popular convolution-based or vision +transformer-based backbones. We try both two options +in the experiments and they show similar performance +trends. These backbones produce hierarchical visual feature +maps during the encoding process, as shown in Fig. 4. +We denote the features as Fi +∈ RT ×hi×wi×Ci, where +(hi, wi) = (H, W)/2i+1, i = 1, . . . , n. The number of levels +is set to n = 4 in all experiments. +Cross-Modal Fusion: We use Atrous Spatial Pyramid Pool- +ing (ASPP) modules [61] to further post-process the visual +features Fi to Vi ∈ RT ×hi×wi×C, where C = 256. These +modules employ multiple parallel filters with different rates +and hence help to recognize visual objects with different +receptive fields, e.g., different-sized moving objects. +Then, we consider introducing the audio information to +build the audio-visual mapping to assist with identifying +the sounding object. This is particularly essential for the +MS3 and AVSS settings where there are multiple dynamic +sound sources. Our intuition is that, although the auditory +and visual signals of the sound sources may not appear +simultaneously, they usually exist in more than one video +frame. Therefore, integrating the audio and visual signals of +the whole video should be beneficial. Motivated by [63] that +uses the non-local block to encode space-time relation, we +adopt a similar module to encode the temporal pixel-wise +audio-visual interaction (TPAVI). As illustrated in Fig. 5, the +current visual feature map Vi and the audio feature A of the +entire video are sent into the TPAVI module. Specifically, the +audio feature A is first transformed to a feature space with +the same dimension as the visual feature Vi, by a linear layer. +Then it is spatially duplicated hiwi times and reshaped to +the same size as Vi. We denote such processed audio feature +as ˆ +A. Next, it is expected to find those pixels of visual feature +map Vi that have a high response to the audio counterpart +ˆ +A through the entire video. + +2000 +1000 +0 +-1000 +-2000 +0 +3 +4 +1000 +0 +-1000 +0 +2 +3 +4 +5 +time (seconds)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +Fig. 5. The TPAVI module takes the i-th stage visual feature Vi and +the audio feature A as inputs. The colored boxes represent 1 × 1 × 1 +convolutions, while the yellow boxes indicate reshaping operations. The +symbols “⊗” and “⊕” denote matrix multiplication and element-wise +addition, respectively. +Such an audio-visual interaction can be measured by dot- +product, then the updated feature maps Zi at the i-th stage +can be computed as, +Zi = Vi + µ(αi g(Vi)), where αi = θ(Vi) φ( ˆ +A) +⊤ +N +(1) +where θ, φ, g and µ are 1×1×1 convolutions, N = T ×hi×wi +is a normalization factor, αi denotes the audio-visual sim- +ilarity, and Zi ∈ RT ×hi×wi×C. Each visual pixel interacts +with all the audio through the TPAVI module. We provide a +visualization of the audio-visual attention in TPAVI later +in Fig. 12, which shows a similar “appearance” to the +prediction of SSL methods because it constructs a pixel-to- +audio mapping. +The Decoder: We adopt the decoder of Panoptic-FPN [64] +in this work for its flexibility and effectiveness, though any +valid decoder architecture could be used. In short, at the +j-th stage, where j = 2, 3, 4, both the outputs from stage +Z5−j and the last stage Z6−j of the encoder are utilized +for the decoding process. The decoded features are then +upsampled to the next stage. The final output of the decoder +is M ∈ RT ×H×W ×K. For S4 and MS3 settings, K = 1, and +the output is then activated by sigmoid function. During +inference of the AVSS setting, the output is further processed +by a softmax operation along the K channel, and the index +with the highest probability represents the category of the +sounding object. +Objective function: Given the prediction M and the pixel- +wise label Y , we adapt the binary cross entropy (BCE) loss as +the main supervision function. Besides, we use an additional +regularization term LAVM to force the audio-visual mapping. +Specifically, we use the Kullback–Leibler (KL) divergence to +ensure the masked visual features have similar distributions +with the corresponding audio features. In other words, if the +audio features of some frames are close in the feature space, +the corresponding sounding objects are expected to be close +in the feature space. The total objective function L can be +computed as follows: +L = BCE(M, Y ) + λLAVM(M, Z, A), +(2) +LAVM = +n +� +i=1 +(KL(avg (Mi ⊙ Zi), Ai), +(3) +where λ is a balance weight, ⊙ denotes element-wise mul- +tiplication, and avg denotes the average pooling operation. +At each stage, we down-sample the prediction M to Mi via +average pooling to have the same shape as Zi. The vector +Ai is a linear transformation of A that has the same feature +dimension with Zi. For the semi-supervised S4 setting, we +found that the audio-visual regularization loss does not help, +so we set λ = 0 in this setting. +5 +EXPERIMENTAL RESULTS +5.1 +Implementation details +We conduct training and evaluation on the upgraded AVS- +Bench dataset, with both convolution-based and transformer- +based backbones, ResNet-50 [69] and Pyramid Vision Trans- +former (PVT-v2) [33]. Both of the backbones are pretrained on +the ImageNet [70] dataset. All the video frames are resized to +a shape of 224 × 224. The channel sizes of the four stages are +C1:4 = [256, 512, 1024, 2048] and C1:4 = [64, 128, 320, 512] +for ResNet-50 and PVT-v2, respectively. The channel size of +the ASPP module is set to C = 256. We use the VGGish +model to extract audio features, a VGG-like network [62] +pretrained on the AudioSet [60] dataset. The audio signals are +converted to one-second splits as the network inputs. We use +the Adam optimizer with a learning rate of 1e-4 for training. +The batch size is set to 4 and the number of training epochs +are 15, 30, and 60 respectively for the semi-supervised S4, +the fully-supervised MS3, and the AVSS settings. The λ in +Eq. (2) is empirically set to 0.5. +5.2 +Comparison with methods from related tasks +Predictions under the S4 and MS3 settings are binary +segmentation maps while they are semantic maps under the +AVSS setting. Methods from different related tasks need to be +compared under these settings. We introduce the comparison +results of the former two settings in Sec. 5.2.1 and the AVSS +setting in Sec. 5.2.2. +5.2.1 +Comparison under the S4 and MS3 settings +For the audio-visual segmentation under S4 and MS3 settings, +we compare our baseline framework with the methods +from three related tasks, including sound source localization +(SSL), video object segmentation (VOS), and salient object +detection (SOD). For each task, we report the results of +two SOTA methods on our AVSBench-object dataset, i.e., +LVS [25] and MSSL [27] for SSL, 3DC [65] and SST [66] +for VOS, iGAN [67] and LGVT [68] for SOD. We select +these methods as they are state-of-the-art in their fields: +1) LVS uses the background and the most confident regions +of sounding objects to design a contrastive loss for audio- +visual representation learning and the localization map is +obtained by computing the audio-visual similarity. 2) MSSL +is a two-stage method for multiple sound source localization +and the localization map is obtained by Grad-CAM [40]. +3) 3DC adopts an architecture that is fully constructed +by powerful 3D convolutions to encode video frames and +predict segmentation masks. 4) SST introduces a transformer + +↑zi +TX hiXWiXC ++ +1×1×1 +^ T×hiXWi×C +reshape + ThiWiXC +ThiWiX ThiWi +ThiW;XC +ThiWiXC +CxThiWi +reshape +reshape +reshape +T×hiXWiXC +T×hiXWiXC +T×hiXW;XC +g:1×1×1 +0:1×1×1 +Φ:1×1×1 +linear +epeat +T×hiXWiXC +; XW;XC +A +AJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +TABLE 3 +Comparison with methods from related tasks on audio-visual segmentation under the S4 and MS3 settings. The compared methods come +from the tasks of sound source localization (SSL), video object segmentation (VOS), and salient object detection (SOD). Results of mIoU (%) and +F-score are reported. +Metric +Setting +SSL +VOS +SOD +AVS +LVS [25] +MSSL [27] +3DC [65] +SST [66] +iGAN [67] +LGVT [68] +ResNet50 +PVT-v2 +mIoU +S4 +37.94 +44.89 +57.10 +66.29 +61.59 +74.94 +72.79 +78.74 +MS3 +29.45 +26.13 +36.92 +42.57 +42.89 +40.71 +47.88 +54.00 +F-score +S4 +.510 +.663 +.759 +.801 +.778 +.873 +.848 +.879 +MS3 +.330 +.363 +.503 +.572 +.544 +.593 +.578 +.645 +architecture to achieve sparse attention of the features in +the spatiotemporal domain. 5) iGAN is a ResNet-based +generative model for saliency detection, considering about +the inherent uncertainty of saliency detection. 6) LGVT is a +saliency detection method based on Swin transformer [71], +whose long-range dependency modeling ability leads to +better global context modeling. We adopt the architecture +of these methods and fit them into our semi-supervised S4 +and fully-supervised MS3 settings. For a fair comparison, +the backbones of these methods are all pretrained on the +ImageNet [70]. +Quantitative +comparison +between +AVS +and +SSL/SOD/VOS. The quantitative results are shown in +Table 3, with Mean Intersection over Union (mIoU) and +F-score1.There is a substantial gap between the results of +SSL methods and those of our baseline, mainly because the +SSL methods cannot provide a pixel-level prediction. Also, +our baseline framework shows a consistent superiority to +the VOS and SOD methods in both semi-supervised S4 and +fully-supervised MS3 settings. It is worth noting that the +state-of-the-art SOD method LGVT [68] slightly outperforms +our ResNet50-based baseline under the Single-source set +(74.94% mIoU vs. 72.79% mIoU), mainly because LGVT uses +the strong Swin Transformer backbone [71]. However, when +it comes to the Multi-sources setting, the performance of +LGVT is obviously worse than that of our ResNet50-based +baseline (40.71% mIoU vs. 47.88% mIoU). This is because +the SOD method relies on the dataset prior, and cannot +handle situations where sounding objects change but +visual contents remain the same. Instead, the audio signals +guide our method to identify which object to segment, +leading to better performance. Moreover, if also using a +transformer-based backbone, our method is stronger than +LGVT in both settings. Besides, we notice that although SSL +methods utilize both audio and visual signals, they cannot +match the performance of VOS or SOD methods that only +use visual frames. It indicates the significance of pixel-wise +scene understanding. The proposed AVS baselines achieve +satisfactory performance under the semi-supervised S4 +setting (around 70% mIoU), which verifies that one-shot +annotation is sufficient for single-source cases. +Qualitative comparison between AVS and SSL/VOS/SOD. +We provide some qualitative examples to compare our AVS +framework with the SSL methods, LVS [25] and MSSL [27]. +As shown in the left sample of Fig. 6, LVS over-locates the +1. F-score +considers +both +the +precision +and +recall: +Fβ += +(1+β2)×precision×recall +β2×precision+recall +, where β2 is set to 0.3 in our experiments. +sounding object violin. At the same time, MSSL fails to locate +the piano of the right sample. Both the results of these two +methods are blurry and they cannot accurately locate the +sounding objects. Instead, the proposed AVS framework can +not only accurately segment all the sounding objects, but +also nicely outline the object shapes. +Besides, we also compare the proposed AVS framework +with the state-of-the-art methods from VOS and SOD, i.e., +SST [66] and LGVT [68], respectively. As shown in Fig. 7, +SST and LGVT can predict their objects of interest in a +pixel-wise manner. However, their predictions rely on the +visual saliency and the dataset prior, which cannot satisfy +our problem setting. For example, in the left sample of Fig. 7, +the dog keeps quiet in the first two frames and should not +be viewed as an object of interest in our problem setting. +Our AVS method correctly follows the guidance of the audio +signal, i.e., accurately segmenting the baby at the first two +frames and both the sounding objects at the last three frames, +with their shapes complete. Instead, the VOS method SST +misses the barking dog at the last three frames. The SOD +method LGVT masks out both the baby and dog over all the +frames mainly because these two objects usually tend to be +‘salient’, which is not desired in this sample. When it comes +to the right sample of Fig. 7, we can observe that LGVT +almost fails to capture the violin, since the violin is relatively +small. The VOS method SST can find the rough location of +the violin, with the help of the information from temporal +movement. In contrast, our AVS framework can accurately +depict the shapes and locations of the violin and piano. +5.2.2 +Comparison under the AVSS setting +For the audio-visual semantic segmentation (AVSS) setting, +the experiments are conducted on the Semantic-labels subset. +We compare the proposed baseline to two methods from +the VOS task since they can also generate semantic maps +from videos. Specifically, we include the aforementioned +3DC [65] and a newly proposed SOTA method AOT [72] +in our comparison. We select AOT as a referenced method +because it proposes a new long-short term transformer layer +and can effectively handle multi-object scenarios, whereas +our AVS model also focuses on multiple sounding objects. +Quantitative comparison between AVS and VOS. As +shown in Table 4, the strong AOT model surpasses our +ResNet50-based AVS model but the PVT-based AVS model +keeps the top performance (29.77% mIoU, 0.352 F-score). +Besides, we found the performance under the AVSS setting is +much lower than the S4 and MS3 settings. For example, the +mIoU is 54.00% under the MS3 setting while it is 29.77% + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +Raw +image +Ground +truth +LVS +Audio +9xp46AwF9BY_3 +violin, piano +MSSL +AVS +violin, piano +violin, piano +violin, piano +violin, piano +SWLG_3suH7w_0 +guitar +guitar +violin +violin +violin +Fig. 6. Qualitative examples of the SSL methods and our AVS framework under the fully-supervised MS3 setting. The SSL methods (LVS +[25] and MSSL [27]) can only generate rough location maps, while the AVS framework can accurately segment the pixels of sounding objects and +nicely outline their shapes. +TABLE 4 +Comparison with methods from VOS task on audio-visual +segmentation under the AVSS setting. Results of mIoU (%) and +F-score are reported. +Metric +VOS +AVS +3DC [65] +AOT [72] +ResNet50 +PVT-v2 +mIoU +17.27 +25.40 +20.18 +29.77 +F-score +.216 +.310 +.252 +.352 +under the AVSS setting, using the same PVT-v2 based +backbone. One of the main reason should be that the AVSS +setting needs to further predict the category semantic of each +pixel. Notably, there are more multi-source videos covering +70 classes in the dataset and some objects are hard to identify +in appearance or sound. +Qualitative comparison between AVS and VOS. We also +display some qualitative examples to compare the AOT +method with our AVS model under the AVSS setting. As +shown in Fig. 8(a), the VOS method AOT segments the +cello at the lower right corner over the video frames which +is actually not making sound and predict the guitar with +incorrect category. In contrast, our AVS model accurately +segments the sounding guitar with correct semantic. In +Fig. 8(b), when the sounding objects changes (green boxes), +the AOT still segments both the man and the gun while our +AVS model enables to merely segment the sounding one, +i.e., the speaking man in the third and fourth frames and +the gun in the last two frames. These results again verify +that audio information is helpful under the more challenging +audio-visual semantic segmentation. +TABLE 5 +Impact of audio signal and TPAVI. Results (mIoU) of AVS model both +with and without the TPAVI module. The middle row indicates directly +adding the audio and visual features, which already improves +performance under the MS3 and the AVSS settings. The TPAVI module +further enhances the results over all settings and backbones. +Method +S4 +MS3 +AVSS +Res50 PVT- +v2 +Res50 PVT- +v2 +Res50 PVT- +v2 +without TPAVI 70.12 77.76 +43.56 48.21 +17.34 27.71 +with A⊕V +70.54 77.65 +45.69 51.55 +19.85 28.94 +with TPAVI +72.79 78.74 +46.64 53.06 +20.18 29.77 +5.3 +Model Analysis +Impact of audio signal and TPAVI. As illustrated in Fig. 5, +the TPAVI module is used to formulate the audio-visual in- +teractions from a temporal and pixel-wise level, introducing +the audio information to explore the visual segmentation. We +conduct an ablation study to explore its impact as shown in +Table 5. Two rows show the proposed AVS method with or +without the TPAVI module, while “A⊕V” indicates directly +adding the audio to visual features. It will be noticed that +adding the audio features to the visual ones does not result +in a clear difference under the S4 setting, but lead to a distinct +gain under the MS3 and AVSS settings. This is consistent with +our hypothesis that audio is especially beneficial to samples +with multiple sound sources, because the audio signals can +guide which object(s) to segment. Furthermore, with the +power of our TPAVI module, we can achieve a temporal and +pixel-wise mapping. With TPAVI, each visual pixel hears +the current sound and the sounds at other times, while + +20000 +20000 +0 +1 +2 +20000 +0 +一 +-20000 +0 +2 +3 +4 +5 +time (seconds)5000 +2500 +-2500 +5000 +0 +5000 +0 +-5000 +0 +1 +2 +3 +4 +5 +time (seconds)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +9xp46AwF9BY_0 +Raw +image +Ground +truth +LGVT +AVS +Audio +Q1pZUmvQbWk_4 +SST +violin, piano +violin, piano +violin, piano +violin, piano +violin, piano +baby +baby +baby, dog +baby, dog +baby, dog +Fig. 7. Qualitative examples of the VOS, SOD, and our AVS methods under the fully-supervised MS3 setting. We pick the state-of-the-art +VOS method SST [66] and SOD method LGVT [68]. As can be verified in the left sample, SST or LGVT cannot capture the change of sounding +objects (from ‘baby’ to ‘baby and dog’), while the AVSS accurately conducts prediction under the guidance of the audio signal. +simultaneously interacting with other pixels. The physical +interpretation is that the pixels with high similarity to the +same sound are more likely to belong to one object. TPAVI +helps further enhance the performance over various settings +and backbones, e.g., 72.79% vs. 70.54% and 20.18% vs. 19.85% +when using ResNet50 as the backbone under the S4 and the +AVSS settings, and 53.06% vs. 51.55% if using PVT-v2 under +the MS3 setting. +We also visualize some qualitative examples to reflect the +impact of TPAVI on AVS task under different settings. For +the S4 setting, as shown in Fig. 9, the baseline method with +TPAVI depicts the shape of sounding object better, e.g., the +guitar in the left video, while it can only segment several +parts of the guitar without TPAVI. Such benefit can also be +observed in the MS3 setting, as shown in Fig. 10, the model +enables to ignore those pixels of human hands with TPAVI. +More importantly, with TPAVI, the model is able to segment +the correct sounding object and ignore the potential sound +sources which actually do not make sounds, e.g., the man on +the right of Fig. 9. Also, the “AVS w. TPAVI” has a stronger +ability to capture multiple sound sources. As shown on the +right of Fig. 10, the person who is singing is mainly segmented +with TPAVI but is almost lost without TPAVI. The impact +of audio and TPAVI can also be verified under the AVSS +setting. As shown in Fig. 11a, “AVS wo. TPAVI” tends to +segment the audio-unrelated part, i.e., the woman. Besides, the +sounding object suona is not recognized in most of the video +frames or recognized with incorrect semantics using “AVS +wo. TPAVI”. While the AVS model with TPAVI enables to +focus on segmenting the truly sounding objects. In Fig. 11b, +both “AVS w. TPAVI” and “AVS wo. TPAVI” incorrectly +segments the silent man at the initial frame. The reason may +be that the background noise misleads the model to give +unnecessary predictions. But “AVS w. TPAVI” successfully +recognizes the speaking man at the fifth frame and generates +a more complete shape with more accurate semantics of the +sounding dogs in the subsequent frames (green boxes in the +figure). We argue that it is hard for a model without audio +guidance to predict for AVS task because the model only +learns to fit the provided ground truth and will not perceive +the audio-visual correspondence. These results show the +advantages of utilizing the audio signals, which helps to +segment more accurate audio-visual semantic-corresponding +pixels. +Besides, we also visualize the audio-visual attention +matrices to explore what happens in the cross-modal fusion +process of TPAVI. In detail, the attention matrix is obtained +from αi in Eq. (1) of the fourth stage TPAVI. We upsample it +to have the same shape as the video frame. This is visually +similar to the localization heatmap of these SSL methods, +but only the intermediate result in our AVS method. As +shown in Fig. 12, the high response area basically overlaps +the region of sounding objects. It suggests that TPAVI builds +a mapping from the visual pixels to the audio signals, which +is semantically consistent. +Effectiveness of LAVM. We expect that constructing the +mapping between audio and visual features will enhance the +network’s ability to identify the correct objects. Therefore, +we propose a LAVM loss to introduce a soft constraint for +training. We only apply LAVM in the fully-supervised MS3 +setting and AVSS setting because the change of sounding +objects only happens there. +As shown in Table 6, we explore two variants of the LAVM +loss. LAVM-AV is the one introduced in Eq. (3). It encourages +the visual features masked by the segmentation result to +be consistent with the corresponding audio features in a + +10000 +0 +-10000 +-20000 +0 +20000 +10000 +0 +-10000 +0 +3 +4 +5 +time (seconds)20000 +10000 +0 +-10000 +-20000 +0 +1 +3 +5 +20000 +10000 +0 +-10000 +20000 +0 +1 +2 +3 +4 +5 +time (seconds)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +Raw +image +Ground +truth +AOT +Audio +sitar +(a) +Raw +image +Ground +truth +Audio +man +(b) +sitar +sitar +sitar +sitar +sitar +sitar +sitar +sitar +sitar +AVS +man +man +gun +gun +background +background +background +background +background +AOT +AVS +Fig. 8. Qualitative examples of the VOS method AOT [72] and our AVS method under the fully-supervised AVSS setting. AVS model with +audio guidance performs better to segment the correct audio-related objects and give more accurate semantic prediction. +TABLE 6 +Effectiveness of LAVM. The two variants of LAVM both bring a clear +performance gain compared with only using a standard BCE loss. +Objective function +MS3 (mIoU) +AVSS (mIoU) +ResNet50 +PVT-v2 +ResNet50 +PVT-v2 +LBCE +46.64 +53.06 +18.88 +29.17 +LBCE + LAVM-VV +46.71 +53.77 +19.65 +29.62 +LBCE + LAVM-AV +47.88 +54.00 +20.18 +29.77 +statistical way, i.e., both depicting the sounding objects. +Alternatively, LAVM-VV first finds the closest audio partner +for each candidate audio, and then computes the KL distance +of the corresponding visual features (also masked by the +segmentation results). This is based on the idea that if +two clips share similar audio signals, the visual features +of their sounding objects should also be similar. As shown in +Table 6, both variants achieve a clear performance gain. For +example, LAVM-AV improves the mIoU by around 1% under +the MS3 and AVSS settings. This demonstrates the benefits of +introducing such an audio-visual constraint. We use LAVM-AV, +since LAVM-VV inconveniently requires a ranking operation. +Cross-modal fusion at various stages. The TPAVI module +is a plug-in architecture that can be applied in any stage +for cross-modal fusion. As shown in Table 7, when the +TPAVI module is used in different single stages, the segmen- +TABLE 7 +Cross-modal fusion at various stages, measured by mIoU (%). In +all the settings, the model achieves the best performance when the +TPAVI module is used in all four stages. +Setting +Backbone +i-th stage of Encoder, i ∈ {1, 2, 3, 4} +1 +2 +3 +4 +3,4 +2,3,4 +1,2,3,4 +S4 +ResNet50 +68.55 69.56 71.30 69.99 +71.29 71.98 72.79 +PVT-v2 +78.30 78.58 78.02 77.70 +78.19 78.47 78.74 +MS3 +ResNet50 +41.62 42.37 43.02 42.29 +44.84 45.98 47.88 +PVT-v2 +46.16 48.79 47.35 49.01 +49.79 50.53 54.00 +AVSS +ResNet50 +19.29 18.39 18.89 17.96 +18.16 18.44 20.18 +PVT-v2 +28.62 29.19 29.07 28.59 +28.78 28.73 29.77 +tation performance fluctuates. For the variant based on the +ResNet50 backbone, the model achieves the best performance +when employing the TPAVI module at the third stage under +both S4 and MS3 settings and at the first stage under AVSS +setting. As for the PVT-v2 based model, it is better to use +the TPAVI module at the second stage in the S4 and AVSS +settings and at the fourth stage under MS3 setting. The AVSS +setting needs to further predict the semantic label for each +pixel and thus may benefit more from the early stage having +a large receptive field. Since our decoder architecture adopts +a skip-connection, it would be beneficial to apply the TPAVI +modules in multiple stages, as verified in the right part +of Table 7. For example, under the MS3 setting, applying + +FJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +12 +playing_acoustic_guitar/-FIGpCo9VoM +Raw +image +Ground +truth +without +TPAVI +with +TPAVI +Audio +cap_gun_shooting/2eEPiCh9bZo +gun shooting +gun shooting +gun shooting +gun shooting +gun shooting +playing guitar +playing guitar +playing guitar +playing guitar +playing guitar +Fig. 9. Qualitative results under the semi-supervised S4 setting. Predictions are generated by the ResNet50-based AVS model. Two benefits +are noticed by introducing the audio signal (TPAVI): 1) learning the shape of the sounding object, e.g., guitar in the video (LEFT); 2) segmenting +according to the correct sound source, e.g., the gun rather than the man (RIGHT). +s6dj_CqT0Mk_1 +Raw +image +Ground +truth +without +TPAVI +with +TPAVI +Audio +o1bZB4fKv2U_1 +violin, +people singing +violin, +people singing +violin, +people singing +people singing +playing piano, +people singing +guitar, ukulele +guitar, ukulele +guitar, ukulele +guitar, ukulele +guitar, ukulele +Fig. 10. Qualitative results under the fully-supervised MS3 setting. The predictions are obtained by the PVT-v2 based AVS model. Note that +AVS with TPAVI uses audio information to perform better in terms of 1) filtering out the distracting visual pixels that do not correspond to the audio, +i.e., the human hands (LEFT); 2) segmenting the correct sound source in the visual frames that matches the audio more accurately, i.e., the singing +person (RIGHT). +TPAVI at all four stages would increase the metric mIoU +from 49.01% to 54.00%, with a gain of 4.99%. It indicates +the model has the ability to fuse and balance the features +from multiple stages. +Pre-training on the Single-source subset. As introduced +in Sec. 3 of the paper, the videos in the Multi-sources +subset share similar categories to those in the Single-source +subset. A natural idea is whether we can pre-train the model +on the Single-source subset to help deal with the MS3 +problem. As shown in Table 8, we test two initialization +strategies, i.e., from scratch or pretrained on the Single- +source subset. It is verified that the pre-training strategy +is beneficial in all the settings, whether we use the audio +information (“w. TPAVI”) or not (“wo. TPAVI”). Taking the +PVT-v2 based AVS model for example, the mIoU is improved +from 48.21% to 50.59% (by 2.38%) and from 54.00% to +57.34% (by 3.34%), respectively without or with TPAVI. +The phenomenon is more obvious if using ResNet50 as +the backbone and adopting the TPAVI module, where the +mIoU increases from 47.88% to 54.33% (by 6.45%). With +pre-training on the Single-source subset, the model can learn +prior knowledge about the audio-visual correspondence, i.e., +the matching relationship between the visual objects and +sounds. This kind of knowledge is naturally beneficial. +T-SNE visualization analysis. We also visualize the visual +features with or without TPAVI module to analyze whether + +20000 +20000 +10000 +0 +-10000 +0 +2 +.. +4 +5 +time (seconds)20000 +0 +-20000 +20000 +0 +-20000 +2 +3 +5 +time (seconds)5000 +0 +-5000 +0 +5000 +0 +-5000 +0 +3 +4 +5 +time (seconds)20000 +20000 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +20000 +0 +-20000 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +time (seconds)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +13 +(b) +background +dog +dog +dog +dog; man +dog +dog +dog +dog +dog +-iPvFoDVJLM_23000_33000 +Raw +image +Ground +truth +AVS wo. +TPAVI +AVS w. +TPAVI +Audio +(a) +suona +suona +suona +suona +suona +suona +suona +suona +suona +suona +Raw +image +Ground +truth +AVS wo. +TPAVI +AVS w. +TPAVI +Audio +Fig. 11. Qualitative results under the fully-supervised AVSS setting. The predictions are obtained by the PVT-v2 based AVS model. With the +TPAVI module, the AVS model focuses on segmenting the objects which are making sounds, and with more complete shape and correct semantics. +xGeqjlPz4kw_4 +Ground +truth +Audio- +visual +attention +Audio +Vwdib3HWRBI_0_2 +tabla +pxa8kn8h5ew_0.mp4_4.png +FhF2q_P-vAA_0.mp4_2.png +GsfG9ZC8rUU_0.mp4_2.png +I8K17mzV-QU_0.mp4_2.png +QYvhdbgEJPM_0.mp4_2 +xLXcCv45AYg_22.mp4_4.png +q6Vwbg3SOSc_0.mp4_5.png +0bzkGQLy7b4_2.mp4_5 +people +gun +tabla +piano +computer +keyboard +people +violin +dog +people +Fig. 12. Audio-visual attention maps that come from the fourth stage TPAVI. Darker brown color indicates a higher response. Such heatmaps +are usually adopted as the final results for the SSL task, while they are just the intermediate output of the TPAVI module in our AVSS framework. +These results reveal that the TPAVI helps the model focus more on the visual regions that are semantic-corresponding to the audio. +the network has built a connection between the audio and +the visual features. Specifically, on the test split of the Multi- +sources set, we use the PVT-v2 based AVS model to obtain +the visual features. Since the Multi-source set do not have +category labels (its videos may contain several categories), +we use the principal component analysis (PCA) to divide +the audio features into K = 20 clusters. Then we assign the +audio cluster labels to the corresponding visual features. In +this case, if the audio and the visual features are correlated, +the visual features should be clustered as well. We use the +t-SNE visualization to verify this assumption. As shown in +Fig. 13a, without audio signals, the learned visual features +distribute chaotically; whereas in Fig. 13b, the visual features +sharing the same audio labels tend to gather together. This +indicates that the distribution of the visual features and audio +features are highly correlated. +Segmenting unseen objects. We restrict the study under the +MS3 setting as it does not need the model to predict the actual +category labels for unseen objects but still requires the model +to predict the sounding objects. We display some qualitative +visualizations on real-world videos whereas the category of +sounding objects are barely not appeared in the training set +of AVS model. As shown in Fig. 14, the pretrained AVS model +has a certain ability to segment the correct sounding objects in + +K-920000 +10000 +-10000 +-20000 +-30000 +0 +2 +3 +5 +事 +20000 +10000 +0 +-10000 +-20000 +-30000 +0 +2 +3 +4 +5 +time (seconds)Aber ofeach chord +Maw.PstOUITAR.ov. +IG|D)Em|CAber of each chord +wa.P.s/OUIAR.covk +IG|DJEm|C20000 +-20000 +0 +5 +20000 +0 +-20000 +0 +3 +4 +5 +time (seconds)20000 +0 +-20000 +2 +20000 +10000 +0 +-10000 +20000 +2 +3 +5 +time (seconds)20000 +0 +-20000 +5 +20000 +0 +-20000 +1 +2 +3 +4 +5 +time (seconds)20000 +10000 +0 +-10000 +-20000 +0 +2 +3 +4 +5 +10000 +0 +-10000 +0 +1 +2 +3 +4 +5 +time (seconds)20000 +0 +-20000 +0 +20000 +0 +-20000 +o +1 +2 +3 +4 +5 +time (seconds)10000 +-10000 +0 +1 +2 +3 +10000 +0 +-10000 +0 +1 +2 +3 +4 +5 +time (seconds)5000 +0 +5000 +0 +2 +5000 +0 +-5000 +0 +2 +3 +4 +5 +time (seconds)20000 +-20000 +20000 +0 +-20000 +2 +3 +5 +time (seconds)20000 +10000 +0 +-10000 +-20000 +0 +1 +2 +20000 +10000 +-10000 +-20000 +0 +1 +2 +4 +5 +time (seconds)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +14 +(a) without audio +(b) with audio +Fig. 13. T-SNE [73] visualization of the visual features, trained with or without audio. These results are from the test split of the Multi-sources +subset. We first use principal component analysis (PCA) to divide the audio features into K = 20 clusters. Then we assign the audio cluster labels to +the corresponding visual features and conduct t-SNE visualization. The points with the same color share the same audio cluster labels. It can be +seen that when training is accompanied by audio signals (right), the visual features illustrate a closer trend with the audio feature distribution, i.e., +points with the same colors gather together, which indicates an audio-visual correlation has been learned. (Best viewed in color.) +Video +frames +Predic- +tion +(a) gorilla +Video +frames +Predic- +tion +(d) clarinet and piano +(b) accordion +(c) sweeping robot +Fig. 14. Qualitative examples of applying the pretrained AVS model under the MS3 setting to unseen videos. The caption in each sub-figure +indicates the sounding object(s) accordingly. There are almost no videos having the same category as these sounding objects during AVS model +training. The pretrained AVS model gains the ability to segment the correct sounding object(s) in both single and multi sources. +TABLE 8 +Performance with different initialization strategies under the MS3 +setting. Compared to training from scratch under the MS3 setting, we +observe a significant performance improvement if pre-training the model +on the Single-source subset. Note the proposed LAVM loss is used in all +the experiments of the Table. The metric is mIoU. +Method +From scratch +Pretrained on Single-source +ResNet50 +PVT-v2 +ResNet50 +PVT-v2 +wo. TPAVI +43.56 +48.21 +45.50 +50.59 +w. TPAVI +47.88 +54.00 +54.33 +57.34 +the case of a single sound source (a), multiple visible objects +(b, c), and multiple sound sources (d). We speculate that the +pretrained AVS model learned some prior knowledge about +audio-visual correspondence from the AVSBench dataset that +helps it generalize to even unseen videos and give possibly +accurate pixel-level segmentation. +6 +CONCLUSION +We explore the task of audio-visual segmentation (AVS), +which aims to generate pixel-level segmentation masks for +sounding objects in audible videos. To facilitate research +on AVS, we build and enrich the audio-visual segmentation +benchmark (AVSBench) that contains the single-source, multi- +sources and semantic-labels subsets. Accordingly, three task +settings are explored: the semi-supervised single-source +AVS (S4), fully-supervised multi-source AVS (MS3) and +the fully-supervised audio-visual semantic segmentation +(AVSS). 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Yang, “Associating objects with transformers +for video object segmentation,” in Advances in Neural Information +Processing Systems (NeurIPS), 2021, pp. 1–20. +[73] L. Van der Maaten and G. Hinton, “Visualizing data using t-sne.” +Journal of Machine Learning Research (JMLR), 2008. + diff --git a/ltFPT4oBgHgl3EQf3TXn/content/tmp_files/load_file.txt b/ltFPT4oBgHgl3EQf3TXn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e71437985bda4ba73f7e57aa49b9717291377a3a --- /dev/null +++ b/ltFPT4oBgHgl3EQf3TXn/content/tmp_files/load_file.txt @@ -0,0 +1,1640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf,len=1639 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 1 Audio-Visual Segmentation with Semantics Jinxing Zhou∗, Xuyang Shen∗, Jianyuan Wang∗, Jiayi Zhang, Weixuan Sun, Jing Zhang, Stan Birchfield, Dan Guo, Lingpeng Kong, Meng Wang†, Fellow, IEEE, and Yiran Zhong† Abstract—We propose a new problem called audio-visual segmentation (AVS), in which the goal is to output a pixel-level map of the object(s) that produce sound at the time of the image frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To facilitate this research, we construct the first audio-visual segmentation benchmark, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', AVSBench, providing pixel-wise annotations for sounding objects in audible videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It contains three subsets: AVSBench-object (Single-source subset, Multi-sources subset) and AVSBench-semantic (Semantic-labels subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Accordingly, three settings are studied: 1) semi-supervised audio-visual segmentation with a single sound source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2) fully-supervised audio-visual segmentation with multiple sound sources, and 3) fully-supervised audio-visual semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The first two settings need to generate binary masks of sounding objects indicating pixels corresponding to the audio, while the third setting further requires generating semantic maps indicating the object category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To deal with these problems, we propose a new baseline method that uses a temporal pixel-wise audio-visual interaction module to inject audio semantics as guidance for the visual segmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also design a regularization loss to encourage audio-visual mapping during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Quantitative and qualitative experiments on AVSBench compare our approach to several existing methods for related tasks, demonstrating that the proposed method is promising for building a bridge between the audio and pixel-wise visual semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='com/OpenNLPLab/AVSBench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Online benchmark is available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='avlbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='opennlplab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Index Terms—Audio-visual segmentation, Multi-modal segmentation, Audio-visual learning, AVSBench, Semantic segmentation, Video segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 1 INTRODUCTION H UMANS largely rely on visual and auditory cues to understand their environmental surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, a dog barking can be distinguished from a bird calling based on both their sound and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Such audio-visual information is integrated with the brain in a synthesis process [1], crucial for comprehensively perceiving the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Inspired by this cognitive ability of humans, we explore audio-visual learning with deep models via the integration of multi-modal signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Over the years, researchers have studied various prob- lems within audio-visual artificial perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For instance, some researchers investigate the audio-visual correspon- dence (AVC) problem [2], [3], [4], which aims to determine whether an audio signal and a visual image describe the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVC is based on the phenomenon that these two signals usually occur simultaneously, such as a barking dog, a singing person, and a humming car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Others study the audio- Jinxing Zhou, Dan Guo and Meng Wang are with Key Laboratory of Knowledge Engineering with Big Data (HFUT), Ministry of Education and School of Computer Science and Information Engineering, Hefei University of Technology, Hefei, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Xuyang Shen is with the Sensetime Research, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Jianyuan Wang is with Visual Geometry Group, University of Oxford, Oxford, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Jiayi Zhang is with School of Computer Science and Engineering, the Beihang University, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Weixuan Sun and Jing Zhang are with School of Computing, the Australian National University, Canberra, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Stan Birchfield is with Nvidia, Redmond, WA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Lingpeng Kong is with the University of Hong Kong, Hong Kong, China and Shanghai AI Lab, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Yiran Zhong is with Shanghai AI Lab, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' ∗: These authors have equal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' †: Meng Wang and Yiran Zhong are corresponding authors (e-mail: eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='mengwang@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='com, zhongyiran@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' visual event localization (AVEL) [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], which classifies the segments of a video using a set of pre-defined event labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Similarly, some research explores audio-visual video parsing (AVVP) [15], [16], [17], [18], [19], [20], [21], whose goal is to divide a video into several events and classify them as audible, visible, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Due to a lack of pixel-level annotations, all these scenarios are restricted to the frame/temporal level, thus reducing the problem to audible image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' A related problem, known as sound source localization (SSL), aims to locate the visual regions within the image frames that correspond to the sound [2], [3], [22], [23], [24], [25], [26], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Compared to AVC/AVEL/AVVP, SSL seeks patch-level scene understanding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the results are usually presented by a heat map that is obtained either by visualizing the similarity matrix of the audio feature and the visual feature map, or by class activation mapping (CAM) [29]—without considering the actual shape of the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Building on this research, in this work we propose the pixel-level audio-visual segmentation (AVS) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This problem requires the network to densely predict whether each pixel corresponds to the given audio, so that a mask of the sounding object(s) is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 1 illustrates the differences between SSL and AVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As can be seen, the AVS task is more challenging as it requires the network to not only locate the audible frames but also delineate the shape of the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Moreover, the AVS finally needs to classify the category semantics of different sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 1, each type of sounding object is assigned a specific color indicating its unique semantic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To facilitate this research, we release the AVSBench dataset, which is the first pixel-level audio-visual segmen- tation benchmark that provides ground truth labels for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='13190v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='CV] 30 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 2 Video SSL AVS- object Audio 28XJmUz906g_0 dog barking lawn mower UsTf7brftGg_0 man talking man talking man talking, playing piano playing piano playing piano (a) AVS-S4 dog barking lawn mower dog barking lawn mower dog barking lawn mower dog barking lawn mower AVS- semantic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Comparison of the proposed AVS task with the Sound source localization (SSL) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' SSL aims to estimate an approximate location of the sounding objects in the visual frame, at a patch level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In contrast, AVS estimates pixel-wise masks for all the sounding objects, regardless of the number of visible sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The segmentation masks can be binary or semantic under different task settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The binary masks indicate objects making sounds while the semantic masks further distinguish the object category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In the last row, the ground truths are displayed with the semantic masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The dataset is divided into three subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In the first subset, there is a single sound source in the video, leading to the task we call semi-supervised Single Sound Source Segmentation (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In the second subset, there are multiple sound sources, leading to the task of fully-supervised Multiple Sound Source Segmentation (MS3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For these two subsets, the ground truths are binary masks indicating pixels emitting the sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We study these two settings to have a basic perception of audio-visual segmentation from pixel- level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' While, the third subset is a Semantic-labels subset that introduces semantic labels of the sounding objects, exploring the task of fully-supervised Audio-Visual Semantic Segmentation (AVSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Compared to the S4 and MS3 settings, AVSS requires generating semantic maps that further tell the category information of the masked sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown on the left example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 1, pixels of the dog and lawn mower are assigned with different colors indicating the unique semantic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For convenience, we denote that the first two subsets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', Single-source and Multi-sources subsets, constitute the AVSBench-object dataset, and the third Semantic-labels subset is also called the AVSBench-semantic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For all the settings, the goal is to segment the object(s) from the visual frames that are producing sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Compared with traditional semantic segmentation [30], [31], [32], [33], [34] task or video object segmentation [35], [36], [37] task, AVS is a multi-modal segmentation problem that necessitates the alignment of visual and audio semantics rather than classifying each pixel solely based on visual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To deal with the aforementioned three settings, we test several methods from related tasks on AVSBench dataset and provide a new AVS method as a strong baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It utilizes a standard encoder- decoder architecture but with a novel temporal pixel-wise audio-visual interaction (TPAVI) module to better introduce the audio semantics for guiding visual segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also propose a loss function to utilize the correlation of audio-visual signals, which further enhances segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' At last, we remind that the audio-visual segmentation problem is first introduced in our previous work [38] that has been published in ECCV 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Compared with the conference version, we add the following new extensions in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Firstly, we expand upon our previous work by incorporating a new and challenging setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the fully-supervised AVSS, which can be viewed as an independent task by the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also conduct extensive ablation studies in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These explorations help us gain a deeper understanding of audio-visual scenarios and design a more realistic model that enables us to perceive pixel-wise semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Secondly, we propose the AVSBench-semantic dataset containing a Semantic-labels subset that newly pro- vides pixel-wise semantic labels, as a significant complement of the original AVSBench dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVSBench-semantic dataset includes significantly more event categories (70 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 23) and frames (80k vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 10k) compared to the original AVSBench dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' More extending details of the video statistic and annotation are introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Thirdly, we update the previous AVS model [38] to predict the semantic maps and add extensive experiments on the new AVSBench dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the convenience of the community, we build an online benchmark suite at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='avlbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='opennlplab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2 RELATED WORK Sound Source Localization (SSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Perhaps the most closely related problem to ours is SSL, which aims to locate the regions in the visual frames responsible for the sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The prediction of SSL is usually computed from the similarity matrix of the learned audio feature and the visual feature map [2], [3], [22], [23], [24], [25], displayed as a heat map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' SSL can also be divided into two settings according to the complexity of sound sources, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', single and multiple sound 20000 10000 0 10000 20000 0 1 2 3 5 20000 10000 0 10000 20000 0 1 4 5 time (seconds)20000 +++++ 10000 0 10000 20000 0 2 3 5 20000 10000 0 10000 20000 0 Y 2 3 4 5 time (seconds)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 3 source(s) localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Here we focus on the challenging setting of multiple sources, which requires accurately lo- calizing the true sound source among multiple potential candidates [26], [27], [28], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In pioneering work, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' [26] divide the audio and visual features into multiple cluster centers and take the center distance as a supervision signal to rank the paired audio-visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' [27] first train an audio-visual correspondence model to extract coarse feature representations of audio and visual signals, and then use Grad-CAM [40] to visualize the class- specific features for localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Furthermore, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' [28] adopt a two-stage method, which first learns audio-visual semantics in the single sound source condition, using such learned knowledge to help with multiple sound sources localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Rouditchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' [41] tackles this problem by disentangling category concepts in the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This method is actually more related to the task of sound source separation [42], [43], [44], [45] and shows sub-optimal performance regarding visual localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Although these existing SSL methods indicate which regions in the image are making sound, the results do not clearly delineate the shape of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Rather, the location map is computed by up-sampling the audio-visual similarity matrix from a low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Moreover, the methods above all rely on unsupervised learning when capturing the shape of sounding objects, which partly suffers from the lack of an annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To overcome these limitations, this paper provides an audio-visual segmentation dataset with pixel- level ground truth labels, which enables to achieve more accurate segmentation predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Video Object Segmentation (VOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The VOS task aims to segment the object of interest throughout the entire video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It is divided into two settings: the semi-supervised and unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the semi-supervised VOS, the target object is decided given a one-shot mask of the first sampled video frame [35], [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As for unsupervised VOS, it needs to automatically segment all the primary objects [46], [47], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Many excellent works are proposed and proven to achieve impressive segmentation performance [49], [50], [51], [52], [53], [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' However, these fancy designs are limited to a single visual modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Recently, referring video object segmentation (R-VOS) attracts more attention [55], [56], [57], [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The target object in R-VOS task is referred by a short language expression, whereas the proposed AVS task focuses on the audio-aligned visual objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the object of interest is determined by the audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Unlike the language used in R- VOS has clear semantics, the proposed AVSS requires a joint semantic classification for both audio and visual information, which makes it more challenging than the R-VOS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Audio-Visual Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To the best of our knowledge, there are no publicly available datasets that provide segmentation masks for the sounding visual objects with audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Here we briefly introduce the popular datasets in the audio- visual community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, the AVE [7] and LLP [15] datasets are respectively collected for audio-visual event localization and video parsing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' They only have category annotations for video frames, and hence cannot be used for pixel-level segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the sound source localization problem, researchers usually use the Flickr-SoundNet [22] and VGG-SS [25] datasets, where the videos are sampled from the large-scale Flickr [4] and VGGSound [59] datasets, TABLE 1 AVSBench statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The videos are split into train/valid/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The asterisk (∗) indicates one annotation per video whereas others are one annotation per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' ⋄ in the last row indicates that 1,000 videos are withheld for online benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' subsets classes videos train/valid/test labeled frames single-source 23 4,932 3,452∗/740/740 10,852 multi-source 23 424 296/64/64 2,120 semantic-labels 70 12,356⋄ 8,498/1,304/1,554 82,972 TABLE 2 Existing audio-visual dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Each benchmark is shown with the number of videos and the annotated frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The final column indicates whether the frames are labeled by category, bounding boxes, or pixel-level masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench extension provides pixel-level semantic labels with object category information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' benchmark videos frames classes types annotations AVE [7] 4,143 41,430 28 video category LLP [15] 11,849 11,849 25 video category Flickr-SoundNet [22] 5,000 5,000 50 image bbox VGG-SS [25] 5,158 5,158 220 image bbox AVSBench-object [38] 5,356 12,972 23 video pixel AVSBench-semantic 12,356 82,972 70 video pixel & category respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The authors provide bounding boxes to outline the location of the target sound source, which could serve as patch-level supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' However, this still inevitably suffers from incorrect evaluation results since the sounding objects are usually irregular in shape and some regions within the bounding box actually do not correspond to the real sound source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The proposed AVSBench dataset provides pixel-wise semantic masks that accurately outline the shape of sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This is beneficial for the research of pixel-level audio- visual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3 THE AVSBENCH DATASET The AVSBench dataset is first proposed in our previous work [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It contains a Single-source and a Multi-sources subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Ground truths of these two subsets are binary segmentation maps indicating pixels of the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Recently, we collected a new Semantic-labels subset that provides semantic segmentation maps as labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We add it to the original AVSBench dataset as the third subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For convenience, we denote the original AVSBench dataset as AVSBench-object, and the newly added Semantic-labels subset as AVSBench-semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In this section, we first introduce the video statistics and annotations of the AVSBench-object and then provide the extending details of AVSBench-semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Lastly, we introduce three benchmark settings on the updated AVSBench dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='1 Dataset Statistics AVSBench-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We collected the videos using the tech- niques introduced in VGGSound [59] to ensure the audio and visual clips correspond to the intended semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench- object [38] contains two subsets—Single-source and Multi- sources—depending on the number of sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' All JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AUGUST 2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='296312 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='406 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5310743 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='225 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Statistics of the AVSBench dataset extension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the AVSBench-semantic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' There are 70 categories in the extension and the video number of each category is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' videos were downloaded from YouTube under the Creative Commons license, and each video was trimmed to 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The Single-source subset contains 4, 932 videos over 23 categories, covering sounds from humans, animals, vehicles, and musical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To collect the Multi-sources subset, we selected the videos that contain multiple sounding objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', a video of baby laughing, man speaking, and then woman singing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To be specific, we randomly chose two or three category names from the Single-source subset as keywords to search for online videos, then manually filtered out videos to ensure 1) each video has multiple sound sources, 2) the sounding objects are visible in the frames, and 3) there is no deceptive sound, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', canned laughter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In total, this process yielded 424 videos for the Multi-sources subset out of more than six thousand candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The ratio of train/validation/test split percentages is set as 70/15/15 for both subsets, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Several video examples are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3, where the red text indicates the name of sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench-semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Since the original version of AVSBench dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', AVSBench-object, we have extended the dataset by adding a third Semantic-labels subset that provides semantic segmentation maps as labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench-semantic is enriched in video amount and audio-visual scene categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In total, it contains 12,356 videos covering 70 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2, we show the category names and the video number for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This extension reserves all 5,356 videos from the original and upgrades them to 720p resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In addition, we further collect another 7,000 multi-source videos following the principle of collecting multi-sources subset of the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We reserve 1,000 videos for online evaluation and it will only be available for contestants in the future AVS Benchmark competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These newly collected videos are trimmed to 10 seconds which helps to train a segmentation model with the ability to encode long-range audio-visual sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Except for the 1,000 withheld videos, the rest of the videos are split into 8,498 for training, 1,304 for validation, 1,554 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also display some video examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVSBench-object and the AVSBench-semantic to- gether form the updated AVSBench dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We make a com- parison between AVSBench with other popular audio-visual benchmarks in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVE [7] dataset contains 4,143 videos covering 28 event categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The LLP [15] dataset consists of 11,849 YouTube video clips spanning 25 cate- gories, collected from AudioSet [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Both the AVE and LLP datasets are labeled at a frame level, through audio-visual event boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Meanwhile, the Flickr-SoundNet [22] dataset and VGG-SS [25] dataset are proposed for sound source localization (SSL), labeled at a patch level through bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVSBench-object (original AVSBench dataset [38]) contains 5,356 videos with 12,972 pixel-wise annotated frames which is designed to facilitate research on fine-grained audio-visual segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench-semantic further extends it from three aspects: 1) the video quantity is expanded to 12,356 and focuses more on the multi-source case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2) the number of object categories is enlarged from 23 to 70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3) annotations are updated from pixel-wise binary mask to semantic masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The recent AVSBench dataset provides accurate semantic maps as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This makes it beneficial not only for the proposed audio-visual segmentation but also for sound source localization, which could help the training of SSL methods and serve as an evaluation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2 Annotation AVSBench-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Videos in AVSBench-object are trimmed to 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We divide each 5-second video into five equal 1- second clips, and we provide manual pixel-level annotations for the 1-second clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The ground truth label is a binary mask indicating the pixels of sounding objects, according to the audio at the corresponding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, in the Multi-sources subset, even though a dancing person shows drastic movement spatially, it would not be labeled as long as no sound was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In clips where objects do not make sound, the object should not be masked, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the piano in the first two clips of the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Similarly, when more than one object emits sound, all the emitting objects are annotated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the guitar and ukulele in the first row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Also, when the sounding objects in the video are changing dynamically, the difficulty is further increased, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the second, third, and fourth rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We use two types of labeling strategies, based on the different difficulties between the Single-source and the Multi- sources subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the videos in the training split of Single- source, we only annotate the first sampled frame (with the assumption that the information from one-shot annotation is sufficient, as the Single-source subset has a single and consistent sounding object over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This assumption is verified by the quantitative experimental results shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the more challenging Multi-sources subset, all JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 5 mynah bird singing ambulance siren horse clip-clop man with violin and piano guitar with ukulele man with piano (a) Video examples in Single-source subset of AVSBench (b) Video examples in Multi-sources subset of AVSBench man with vacuum-cleaner (c) Video examples in Semantic-labels subset of AVSBench SPr6nTlav0_181000_191000 u5tbtyhBMA_35000_45000 024k2nxvSRI_144000_154000 donkey with woman girl with guitar 68Mm2Nc-W4w_50000_60000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVSBench dataset contains the Single-source subset (a), Multi-sources subset (b), and Semantic-labels subset which mainly contains the multi-source videos (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Each video is divided into 5 clips for the first two, while 10 clips for the latter, as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Annotated clips are indicated by brown framing rectangles while the green rectangles represent there are no sounding objects in those frames;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' the name of sounding objects is indicated by red text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Binary masks of the sounding objects are annotated in the first two, reflected by the orange masks in (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The third subset provides colorful semantic masks indicating different object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Note that for the Single-source training set of AVSBench, only the first frame of each video is annotated, whereas all of the extracted frames are annotated for all other sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' clips are annotated for training, since the sounding objects may change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Note that for validation and test splits, all clips are annotated, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVSBench-semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVSBench-semantic subset uses the videos from AVSBench-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For these videos, we update the annotated binary masks to semantic masks by adding category information of the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As for the newly collected 10-second videos, we sample ten video frames and provide their semantic annotations, similar to the annotation process of AVSBench-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We show some annotation examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown, the sounding object is highlighted with unique color indicating its category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Also, when there is no sound or the sounding object is out of the screen (green boxes in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3(c)), that video frame will not be annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='3 Benchmark Setting We provide three benchmark settings: the semi-supervised Single Sound Source Segmentation (S4), the fully supervised Multiple Sound Source Segmentation (MS3), and the fully supervised audio-visual semantic segmentation (AVSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The former two settings are based on the AVSBench-object dataset while the AVSS is conducted on the AVSBench-semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For ease of expression, we denote the video sequence as S, which consists of T non-overlapping yet continuous clips {Sv t , Sa t }T t=1, where Sv and Sa are the visual and audio components, T is equal to 5 for AVSBench-object while 10 for AVSBench-semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In practice, we extract the video frame at the end of each second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Semi-supervised S4 corresponds to the Single-source subset of AVSBench-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It is termed as semi-supervised because only part of the ground truth is given during training (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the first sampled frame of the videos) but all the video frames require a prediction during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We denote the pixel-wise label as Y s t=1 ∈ RH×W , where H and W are the frame height and width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Y s t=1 is a binary matrix where 1 indicates sounding objects while 0 corresponds to background or silent objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Fully-supervised MS3 deals with the Multi-sources subset of AVSBench-object, where the labels of all five frames of each video are available for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The ground truth is denoted as {Y m t }T t=1, where Y m t ∈ RH×W is the binary label for the t-th video clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2bestathire NatonalTool&EquipmentHire 0F-0723bestathire Naonal Tool&EquipmentHire 6723bestathire National Tool&EquipmentHire 0323bestathire 632ebestataire Natonalbestathire Na Carpet Cieaner Hirebestathirebestathire Nason Tool8FiinJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 6 stage 1 stage 2 stage 3 stage 4 𝑇×𝐻×𝑊×3 𝑭!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' : 𝑇× 𝐻 4 × 𝑊 4 ×𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 𝑭": 𝑇× 𝐻 8 × 𝑊 8 ×𝐶" 𝑭#: 𝑇× 𝐻 16 × 𝑊 16 ×𝐶# 𝑭$: 𝑇× 𝐻 32 × 𝑊 32 ×𝐶$ ASPP TPAVI visual encoder visual encoder visual encoder visual encoder 𝑨: 𝑇×𝑑 𝑇 seconds audio encoder ASPP ASPP ASPP TPAVI TPAVI TPAVI 𝑴: 𝑇×𝐻×𝑊×𝐾 𝑷$: 𝑇× 𝐻 2 × 𝑊 2 ×𝐶 𝑷#: 𝑇× 𝐻 4 × 𝑊 4 ×𝐶 𝑷": 𝑇× 𝐻 8 × 𝑊 8 ×𝐶 𝑷!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' : 𝑇× 𝐻 16 × 𝑊 16 ×𝐶 stage 4 stage 3 stage 2 stage 1 Encoder Decoder 𝑽!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 𝑽" 𝑽# ASPP 𝑽$ 𝒁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 𝒁" 𝒁# 𝒁$ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Overview of the Baseline, which follows a hierarchical Encoder-Decoder pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The encoder takes the video frames and the entire audio clip as inputs, and outputs visual and audio features, respectively denoted as Fi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The visual feature map Fi at each stage is further sent to the ASPP [61] module and then our TPAVI module (introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' ASPP provides different receptive fields for recognizing visual objects, while TPAVI focuses on the temporal pixel-wise audio-visual interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The decoder progressively enlarges the fused feature maps by four stages and finally generates the output mask M for sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Fully-supervised AVSS deals with the Semantic-labels subset of AVSBench-semantic, where the semantic masks of all ten frames of each video are known during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The ground truth can be denoted as {Yt}T t=1, where Yt ∈ RH×W ×K is the semantic label for the t-th video clip, K is the total category number of the sounding objects in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The goal for all the settings is to correctly segment the sounding object(s) for each video clip by utilizing the audio and visual cues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', Sa and Sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Different from S4 and MS3 settings, AVSS setting needs to further output the category of the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Generally, it is expected Sa to indicate the target object, while Sv provides information for fine-grained segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The predictions are denoted as {Mt}T t=1, Mt ∈ RH×W ×K, where K = 1 under S4 and MS3 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4 A BASELINE We propose a new baseline method for the pixel-level audio- visual segmentation problem as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Following the convention of semantic segmentation methods [30], [31], [33], [34], our method adopts an encoder–decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The Encoder: We extract audio and visual features inde- pendently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Given an audio clip Sa, we first process it to a spectrogram via the short-time Fourier transform and then send it to a convolutional neural network, VGGish [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We use the weights that are pretrained on AudioSet [60] to extract audio features A ∈ RT ×d, where d = 128 is the feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For a video frame Sv, we extract visual features with popular convolution-based or vision transformer-based backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We try both two options in the experiments and they show similar performance trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These backbones produce hierarchical visual feature maps during the encoding process, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We denote the features as Fi ∈ RT ×hi×wi×Ci, where (hi, wi) = (H, W)/2i+1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The number of levels is set to n = 4 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Cross-Modal Fusion: We use Atrous Spatial Pyramid Pool- ing (ASPP) modules [61] to further post-process the visual features Fi to Vi ∈ RT ×hi×wi×C, where C = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These modules employ multiple parallel filters with different rates and hence help to recognize visual objects with different receptive fields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', different-sized moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Then, we consider introducing the audio information to build the audio-visual mapping to assist with identifying the sounding object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This is particularly essential for the MS3 and AVSS settings where there are multiple dynamic sound sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Our intuition is that, although the auditory and visual signals of the sound sources may not appear simultaneously, they usually exist in more than one video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Therefore, integrating the audio and visual signals of the whole video should be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Motivated by [63] that uses the non-local block to encode space-time relation, we adopt a similar module to encode the temporal pixel-wise audio-visual interaction (TPAVI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5, the current visual feature map Vi and the audio feature A of the entire video are sent into the TPAVI module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Specifically, the audio feature A is first transformed to a feature space with the same dimension as the visual feature Vi, by a linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Then it is spatially duplicated hiwi times and reshaped to the same size as Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We denote such processed audio feature as ˆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Next, it is expected to find those pixels of visual feature map Vi that have a high response to the audio counterpart ˆ A through the entire video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2000 1000 0 1000 2000 0 3 4 1000 0 1000 0 2 3 4 5 time (seconds)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The TPAVI module takes the i-th stage visual feature Vi and the audio feature A as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The colored boxes represent 1 × 1 × 1 convolutions, while the yellow boxes indicate reshaping operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The symbols “⊗” and “⊕” denote matrix multiplication and element-wise addition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Such an audio-visual interaction can be measured by dot- product, then the updated feature maps Zi at the i-th stage can be computed as, Zi = Vi + µ(αi g(Vi)), where αi = θ(Vi) φ( ˆ A) ⊤ N (1) where θ, φ, g and µ are 1×1×1 convolutions, N = T ×hi×wi is a normalization factor, αi denotes the audio-visual sim- ilarity, and Zi ∈ RT ×hi×wi×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Each visual pixel interacts with all the audio through the TPAVI module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We provide a visualization of the audio-visual attention in TPAVI later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 12, which shows a similar “appearance” to the prediction of SSL methods because it constructs a pixel-to- audio mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The Decoder: We adopt the decoder of Panoptic-FPN [64] in this work for its flexibility and effectiveness, though any valid decoder architecture could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In short, at the j-th stage, where j = 2, 3, 4, both the outputs from stage Z5−j and the last stage Z6−j of the encoder are utilized for the decoding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The decoded features are then upsampled to the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The final output of the decoder is M ∈ RT ×H×W ×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For S4 and MS3 settings, K = 1, and the output is then activated by sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' During inference of the AVSS setting, the output is further processed by a softmax operation along the K channel, and the index with the highest probability represents the category of the sounding object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Objective function: Given the prediction M and the pixel- wise label Y , we adapt the binary cross entropy (BCE) loss as the main supervision function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Besides, we use an additional regularization term LAVM to force the audio-visual mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Specifically, we use the Kullback–Leibler (KL) divergence to ensure the masked visual features have similar distributions with the corresponding audio features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In other words, if the audio features of some frames are close in the feature space, the corresponding sounding objects are expected to be close in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The total objective function L can be computed as follows: L = BCE(M, Y ) + λLAVM(M, Z, A), (2) LAVM = n � i=1 (KL(avg (Mi ⊙ Zi), Ai), (3) where λ is a balance weight, ⊙ denotes element-wise mul- tiplication, and avg denotes the average pooling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' At each stage, we down-sample the prediction M to Mi via average pooling to have the same shape as Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The vector Ai is a linear transformation of A that has the same feature dimension with Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the semi-supervised S4 setting, we found that the audio-visual regularization loss does not help, so we set λ = 0 in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5 EXPERIMENTAL RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='1 Implementation details We conduct training and evaluation on the upgraded AVS- Bench dataset, with both convolution-based and transformer- based backbones, ResNet-50 [69] and Pyramid Vision Trans- former (PVT-v2) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Both of the backbones are pretrained on the ImageNet [70] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' All the video frames are resized to a shape of 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The channel sizes of the four stages are C1:4 = [256, 512, 1024, 2048] and C1:4 = [64, 128, 320, 512] for ResNet-50 and PVT-v2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The channel size of the ASPP module is set to C = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We use the VGGish model to extract audio features, a VGG-like network [62] pretrained on the AudioSet [60] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The audio signals are converted to one-second splits as the network inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We use the Adam optimizer with a learning rate of 1e-4 for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The batch size is set to 4 and the number of training epochs are 15, 30, and 60 respectively for the semi-supervised S4, the fully-supervised MS3, and the AVSS settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' (2) is empirically set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2 Comparison with methods from related tasks Predictions under the S4 and MS3 settings are binary segmentation maps while they are semantic maps under the AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Methods from different related tasks need to be compared under these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We introduce the comparison results of the former two settings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='1 and the AVSS setting in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='1 Comparison under the S4 and MS3 settings For the audio-visual segmentation under S4 and MS3 settings, we compare our baseline framework with the methods from three related tasks, including sound source localization (SSL), video object segmentation (VOS), and salient object detection (SOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For each task, we report the results of two SOTA methods on our AVSBench-object dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', LVS [25] and MSSL [27] for SSL, 3DC [65] and SST [66] for VOS, iGAN [67] and LGVT [68] for SOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We select these methods as they are state-of-the-art in their fields: 1) LVS uses the background and the most confident regions of sounding objects to design a contrastive loss for audio- visual representation learning and the localization map is obtained by computing the audio-visual similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2) MSSL is a two-stage method for multiple sound source localization and the localization map is obtained by Grad-CAM [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3) 3DC adopts an architecture that is fully constructed by powerful 3D convolutions to encode video frames and predict segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 4) SST introduces a transformer ↑zi TX hiXWiXC + 1×1×1 ^ T×hiXWi×C reshape ThiWiXC ThiWiX ThiWi ThiW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='XC ThiWiXC CxThiWi reshape reshape reshape T×hiXWiXC T×hiXWiXC T×hiXW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='XC g:1×1×1 0:1×1×1 Φ:1×1×1 linear epeat T×hiXWiXC ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' XW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='XC A AJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 8 TABLE 3 Comparison with methods from related tasks on audio-visual segmentation under the S4 and MS3 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The compared methods come from the tasks of sound source localization (SSL), video object segmentation (VOS), and salient object detection (SOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Results of mIoU (%) and F-score are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Metric Setting SSL VOS SOD AVS LVS [25] MSSL [27] 3DC [65] SST [66] iGAN [67] LGVT [68] ResNet50 PVT-v2 mIoU S4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='94 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='89 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='10 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='29 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='59 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='94 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='74 MS3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='45 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='13 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='92 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='57 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='89 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='71 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00 F-score S4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='510 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='663 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='759 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='801 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='778 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='873 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='848 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='879 MS3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='330 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='363 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='503 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='572 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='544 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='593 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='578 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='645 architecture to achieve sparse attention of the features in the spatiotemporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5) iGAN is a ResNet-based generative model for saliency detection, considering about the inherent uncertainty of saliency detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 6) LGVT is a saliency detection method based on Swin transformer [71], whose long-range dependency modeling ability leads to better global context modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We adopt the architecture of these methods and fit them into our semi-supervised S4 and fully-supervised MS3 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For a fair comparison, the backbones of these methods are all pretrained on the ImageNet [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Quantitative comparison between AVS and SSL/SOD/VOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The quantitative results are shown in Table 3, with Mean Intersection over Union (mIoU) and F-score1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='There is a substantial gap between the results of SSL methods and those of our baseline, mainly because the SSL methods cannot provide a pixel-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Also, our baseline framework shows a consistent superiority to the VOS and SOD methods in both semi-supervised S4 and fully-supervised MS3 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It is worth noting that the state-of-the-art SOD method LGVT [68] slightly outperforms our ResNet50-based baseline under the Single-source set (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='94% mIoU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79% mIoU), mainly because LGVT uses the strong Swin Transformer backbone [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' However, when it comes to the Multi-sources setting, the performance of LGVT is obviously worse than that of our ResNet50-based baseline (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='71% mIoU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88% mIoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This is because the SOD method relies on the dataset prior, and cannot handle situations where sounding objects change but visual contents remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Instead, the audio signals guide our method to identify which object to segment, leading to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Moreover, if also using a transformer-based backbone, our method is stronger than LGVT in both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Besides, we notice that although SSL methods utilize both audio and visual signals, they cannot match the performance of VOS or SOD methods that only use visual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It indicates the significance of pixel-wise scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The proposed AVS baselines achieve satisfactory performance under the semi-supervised S4 setting (around 70% mIoU), which verifies that one-shot annotation is sufficient for single-source cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative comparison between AVS and SSL/VOS/SOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We provide some qualitative examples to compare our AVS framework with the SSL methods, LVS [25] and MSSL [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in the left sample of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 6, LVS over-locates the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' F-score considers both the precision and recall: Fβ = (1+β2)×precision×recall β2×precision+recall , where β2 is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='3 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' sounding object violin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' At the same time, MSSL fails to locate the piano of the right sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Both the results of these two methods are blurry and they cannot accurately locate the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Instead, the proposed AVS framework can not only accurately segment all the sounding objects, but also nicely outline the object shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Besides, we also compare the proposed AVS framework with the state-of-the-art methods from VOS and SOD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', SST [66] and LGVT [68], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 7, SST and LGVT can predict their objects of interest in a pixel-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' However, their predictions rely on the visual saliency and the dataset prior, which cannot satisfy our problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, in the left sample of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 7, the dog keeps quiet in the first two frames and should not be viewed as an object of interest in our problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Our AVS method correctly follows the guidance of the audio signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', accurately segmenting the baby at the first two frames and both the sounding objects at the last three frames, with their shapes complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Instead, the VOS method SST misses the barking dog at the last three frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The SOD method LGVT masks out both the baby and dog over all the frames mainly because these two objects usually tend to be ‘salient’, which is not desired in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' When it comes to the right sample of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 7, we can observe that LGVT almost fails to capture the violin, since the violin is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The VOS method SST can find the rough location of the violin, with the help of the information from temporal movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In contrast, our AVS framework can accurately depict the shapes and locations of the violin and piano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2 Comparison under the AVSS setting For the audio-visual semantic segmentation (AVSS) setting, the experiments are conducted on the Semantic-labels subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We compare the proposed baseline to two methods from the VOS task since they can also generate semantic maps from videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Specifically, we include the aforementioned 3DC [65] and a newly proposed SOTA method AOT [72] in our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We select AOT as a referenced method because it proposes a new long-short term transformer layer and can effectively handle multi-object scenarios, whereas our AVS model also focuses on multiple sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Quantitative comparison between AVS and VOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Table 4, the strong AOT model surpasses our ResNet50-based AVS model but the PVT-based AVS model keeps the top performance (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77% mIoU, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='352 F-score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Besides, we found the performance under the AVSS setting is much lower than the S4 and MS3 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, the mIoU is 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00% under the MS3 setting while it is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77% JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 9 Raw image Ground truth LVS Audio 9xp46AwF9BY_3 violin, piano MSSL AVS violin, piano violin, piano violin, piano violin, piano SWLG_3suH7w_0 guitar guitar violin violin violin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative examples of the SSL methods and our AVS framework under the fully-supervised MS3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The SSL methods (LVS [25] and MSSL [27]) can only generate rough location maps, while the AVS framework can accurately segment the pixels of sounding objects and nicely outline their shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TABLE 4 Comparison with methods from VOS task on audio-visual segmentation under the AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Results of mIoU (%) and F-score are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Metric VOS AVS 3DC [65] AOT [72] ResNet50 PVT-v2 mIoU 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='27 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='40 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='18 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77 F-score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='310 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='252 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='352 under the AVSS setting, using the same PVT-v2 based backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' One of the main reason should be that the AVSS setting needs to further predict the category semantic of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Notably, there are more multi-source videos covering 70 classes in the dataset and some objects are hard to identify in appearance or sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative comparison between AVS and VOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also display some qualitative examples to compare the AOT method with our AVS model under the AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8(a), the VOS method AOT segments the cello at the lower right corner over the video frames which is actually not making sound and predict the guitar with incorrect category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In contrast, our AVS model accurately segments the sounding guitar with correct semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8(b), when the sounding objects changes (green boxes), the AOT still segments both the man and the gun while our AVS model enables to merely segment the sounding one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the speaking man in the third and fourth frames and the gun in the last two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These results again verify that audio information is helpful under the more challenging audio-visual semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TABLE 5 Impact of audio signal and TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Results (mIoU) of AVS model both with and without the TPAVI module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The middle row indicates directly adding the audio and visual features, which already improves performance under the MS3 and the AVSS settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The TPAVI module further enhances the results over all settings and backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Method S4 MS3 AVSS Res50 PVT- v2 Res50 PVT- v2 Res50 PVT- v2 without TPAVI 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='12 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='76 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='56 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='21 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='34 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='71 with A⊕V 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='54 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='65 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='69 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='55 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='85 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='94 with TPAVI 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='74 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='64 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='06 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='18 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='3 Model Analysis Impact of audio signal and TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 5, the TPAVI module is used to formulate the audio-visual in- teractions from a temporal and pixel-wise level, introducing the audio information to explore the visual segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We conduct an ablation study to explore its impact as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Two rows show the proposed AVS method with or without the TPAVI module, while “A⊕V” indicates directly adding the audio to visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It will be noticed that adding the audio features to the visual ones does not result in a clear difference under the S4 setting, but lead to a distinct gain under the MS3 and AVSS settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This is consistent with our hypothesis that audio is especially beneficial to samples with multiple sound sources, because the audio signals can guide which object(s) to segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Furthermore, with the power of our TPAVI module, we can achieve a temporal and pixel-wise mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' With TPAVI, each visual pixel hears the current sound and the sounds at other times, while 20000 20000 0 1 2 20000 0 一 20000 0 2 3 4 5 time (seconds)5000 2500 2500 5000 0 5000 0 5000 0 1 2 3 4 5 time (seconds)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 10 9xp46AwF9BY_0 Raw image Ground truth LGVT AVS Audio Q1pZUmvQbWk_4 SST violin, piano violin, piano violin, piano violin, piano violin, piano baby baby baby, dog baby, dog baby, dog Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative examples of the VOS, SOD, and our AVS methods under the fully-supervised MS3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We pick the state-of-the-art VOS method SST [66] and SOD method LGVT [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As can be verified in the left sample, SST or LGVT cannot capture the change of sounding objects (from ‘baby’ to ‘baby and dog’), while the AVSS accurately conducts prediction under the guidance of the audio signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' simultaneously interacting with other pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The physical interpretation is that the pixels with high similarity to the same sound are more likely to belong to one object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI helps further enhance the performance over various settings and backbones, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='54% and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='18% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='85% when using ResNet50 as the backbone under the S4 and the AVSS settings, and 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='06% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='55% if using PVT-v2 under the MS3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also visualize some qualitative examples to reflect the impact of TPAVI on AVS task under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the S4 setting, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 9, the baseline method with TPAVI depicts the shape of sounding object better, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the guitar in the left video, while it can only segment several parts of the guitar without TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Such benefit can also be observed in the MS3 setting, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 10, the model enables to ignore those pixels of human hands with TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' More importantly, with TPAVI, the model is able to segment the correct sounding object and ignore the potential sound sources which actually do not make sounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the man on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Also, the “AVS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI” has a stronger ability to capture multiple sound sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 10, the person who is singing is mainly segmented with TPAVI but is almost lost without TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The impact of audio and TPAVI can also be verified under the AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 11a, “AVS wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI” tends to segment the audio-unrelated part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Besides, the sounding object suona is not recognized in most of the video frames or recognized with incorrect semantics using “AVS wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' While the AVS model with TPAVI enables to focus on segmenting the truly sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 11b, both “AVS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI” and “AVS wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI” incorrectly segments the silent man at the initial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The reason may be that the background noise misleads the model to give unnecessary predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' But “AVS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI” successfully recognizes the speaking man at the fifth frame and generates a more complete shape with more accurate semantics of the sounding dogs in the subsequent frames (green boxes in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We argue that it is hard for a model without audio guidance to predict for AVS task because the model only learns to fit the provided ground truth and will not perceive the audio-visual correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These results show the advantages of utilizing the audio signals, which helps to segment more accurate audio-visual semantic-corresponding pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Besides, we also visualize the audio-visual attention matrices to explore what happens in the cross-modal fusion process of TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In detail, the attention matrix is obtained from αi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' (1) of the fourth stage TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We upsample it to have the same shape as the video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This is visually similar to the localization heatmap of these SSL methods, but only the intermediate result in our AVS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 12, the high response area basically overlaps the region of sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It suggests that TPAVI builds a mapping from the visual pixels to the audio signals, which is semantically consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Effectiveness of LAVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We expect that constructing the mapping between audio and visual features will enhance the network’s ability to identify the correct objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Therefore, we propose a LAVM loss to introduce a soft constraint for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We only apply LAVM in the fully-supervised MS3 setting and AVSS setting because the change of sounding objects only happens there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Table 6, we explore two variants of the LAVM loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' LAVM-AV is the one introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It encourages the visual features masked by the segmentation result to be consistent with the corresponding audio features in a 10000 0 10000 20000 0 20000 10000 0 10000 0 3 4 5 time (seconds)20000 10000 0 10000 20000 0 1 3 5 20000 10000 0 10000 20000 0 1 2 3 4 5 time (seconds)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 11 Raw image Ground truth AOT Audio sitar (a) Raw image Ground truth Audio man (b) sitar sitar sitar sitar sitar sitar sitar sitar sitar AVS man man gun gun background background background background background AOT AVS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative examples of the VOS method AOT [72] and our AVS method under the fully-supervised AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' AVS model with audio guidance performs better to segment the correct audio-related objects and give more accurate semantic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TABLE 6 Effectiveness of LAVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The two variants of LAVM both bring a clear performance gain compared with only using a standard BCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Objective function MS3 (mIoU) AVSS (mIoU) ResNet50 PVT-v2 ResNet50 PVT-v2 LBCE 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='64 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='06 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='17 LBCE + LAVM-VV 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='71 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='65 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='62 LBCE + LAVM-AV 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='18 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77 statistical way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', both depicting the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Alternatively, LAVM-VV first finds the closest audio partner for each candidate audio, and then computes the KL distance of the corresponding visual features (also masked by the segmentation results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This is based on the idea that if two clips share similar audio signals, the visual features of their sounding objects should also be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Table 6, both variants achieve a clear performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, LAVM-AV improves the mIoU by around 1% under the MS3 and AVSS settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This demonstrates the benefits of introducing such an audio-visual constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We use LAVM-AV, since LAVM-VV inconveniently requires a ranking operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Cross-modal fusion at various stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The TPAVI module is a plug-in architecture that can be applied in any stage for cross-modal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Table 7, when the TPAVI module is used in different single stages, the segmen- TABLE 7 Cross-modal fusion at various stages, measured by mIoU (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In all the settings, the model achieves the best performance when the TPAVI module is used in all four stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Setting Backbone i-th stage of Encoder, i ∈ {1, 2, 3, 4} 1 2 3 4 3,4 2,3,4 1,2,3,4 S4 ResNet50 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='55 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='56 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='30 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='99 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='29 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='98 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79 PVT-v2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='30 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='58 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='02 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='70 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='47 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='74 MS3 ResNet50 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='62 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='37 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='02 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='29 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='84 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='98 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88 PVT-v2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='16 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='35 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='01 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='79 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='53 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00 AVSS ResNet50 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='29 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='39 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='89 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='96 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='16 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='44 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='18 PVT-v2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='62 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='19 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='07 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='59 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='78 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='73 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='77 tation performance fluctuates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For the variant based on the ResNet50 backbone, the model achieves the best performance when employing the TPAVI module at the third stage under both S4 and MS3 settings and at the first stage under AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As for the PVT-v2 based model, it is better to use the TPAVI module at the second stage in the S4 and AVSS settings and at the fourth stage under MS3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The AVSS setting needs to further predict the semantic label for each pixel and thus may benefit more from the early stage having a large receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Since our decoder architecture adopts a skip-connection, it would be beneficial to apply the TPAVI modules in multiple stages, as verified in the right part of Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For example, under the MS3 setting, applying FJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 12 playing_acoustic_guitar/-FIGpCo9VoM Raw image Ground truth without TPAVI with TPAVI Audio cap_gun_shooting/2eEPiCh9bZo gun shooting gun shooting gun shooting gun shooting gun shooting playing guitar playing guitar playing guitar playing guitar playing guitar Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative results under the semi-supervised S4 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Predictions are generated by the ResNet50-based AVS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Two benefits are noticed by introducing the audio signal (TPAVI): 1) learning the shape of the sounding object, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', guitar in the video (LEFT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2) segmenting according to the correct sound source, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the gun rather than the man (RIGHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' s6dj_CqT0Mk_1 Raw image Ground truth without TPAVI with TPAVI Audio o1bZB4fKv2U_1 violin, people singing violin, people singing violin, people singing people singing playing piano, people singing guitar, ukulele guitar, ukulele guitar, ukulele guitar, ukulele guitar, ukulele Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative results under the fully-supervised MS3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The predictions are obtained by the PVT-v2 based AVS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Note that AVS with TPAVI uses audio information to perform better in terms of 1) filtering out the distracting visual pixels that do not correspond to the audio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the human hands (LEFT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 2) segmenting the correct sound source in the visual frames that matches the audio more accurately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the singing person (RIGHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI at all four stages would increase the metric mIoU from 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='01% to 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00%, with a gain of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It indicates the model has the ability to fuse and balance the features from multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Pre-training on the Single-source subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 3 of the paper, the videos in the Multi-sources subset share similar categories to those in the Single-source subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' A natural idea is whether we can pre-train the model on the Single-source subset to help deal with the MS3 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Table 8, we test two initialization strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', from scratch or pretrained on the Single- source subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It is verified that the pre-training strategy is beneficial in all the settings, whether we use the audio information (“w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI”) or not (“wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Taking the PVT-v2 based AVS model for example, the mIoU is improved from 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='21% to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='59% (by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='38%) and from 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00% to 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='34% (by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='34%), respectively without or with TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The phenomenon is more obvious if using ResNet50 as the backbone and adopting the TPAVI module, where the mIoU increases from 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88% to 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='33% (by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='45%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' With pre-training on the Single-source subset, the model can learn prior knowledge about the audio-visual correspondence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', the matching relationship between the visual objects and sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This kind of knowledge is naturally beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' T-SNE visualization analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We also visualize the visual features with or without TPAVI module to analyze whether 20000 20000 10000 0 10000 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='. 4 5 time (seconds)20000 0 20000 20000 0 20000 2 3 5 time (seconds)5000 0 5000 0 5000 0 5000 0 3 4 5 time (seconds)20000 20000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 20000 0 20000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='0 time (seconds)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 13 (b) background dog dog dog dog;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' man dog dog dog dog dog iPvFoDVJLM_23000_33000 Raw image Ground truth AVS wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI AVS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI Audio (a) suona suona suona suona suona suona suona suona suona suona Raw image Ground truth AVS wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI AVS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI Audio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative results under the fully-supervised AVSS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The predictions are obtained by the PVT-v2 based AVS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' With the TPAVI module, the AVS model focuses on segmenting the objects which are making sounds, and with more complete shape and correct semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' xGeqjlPz4kw_4 Ground truth Audio- visual attention Audio Vwdib3HWRBI_0_2 tabla pxa8kn8h5ew_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='mp4_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='png FhF2q_P-vAA_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='mp4_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='png GsfG9ZC8rUU_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='mp4_2.' metadata={'source': 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+page_content='mp4_5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='png 0bzkGQLy7b4_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='mp4_5 people gun tabla piano computer keyboard people violin dog people Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Audio-visual attention maps that come from the fourth stage TPAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Darker brown color indicates a higher response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Such heatmaps are usually adopted as the final results for the SSL task, while they are just the intermediate output of the TPAVI module in our AVSS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These results reveal that the TPAVI helps the model focus more on the visual regions that are semantic-corresponding to the audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' the network has built a connection between the audio and the visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Specifically, on the test split of the Multi- sources set, we use the PVT-v2 based AVS model to obtain the visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Since the Multi-source set do not have category labels (its videos may contain several categories), we use the principal component analysis (PCA) to divide the audio features into K = 20 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Then we assign the audio cluster labels to the corresponding visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' In this case, if the audio and the visual features are correlated, the visual features should be clustered as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We use the t-SNE visualization to verify this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 13a, without audio signals, the learned visual features distribute chaotically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' whereas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 13b, the visual features sharing the same audio labels tend to gather together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' This indicates that the distribution of the visual features and audio features are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Segmenting unseen objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We restrict the study under the MS3 setting as it does not need the model to predict the actual category labels for unseen objects but still requires the model to predict the sounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We display some qualitative visualizations on real-world videos whereas the category of sounding objects are barely not appeared in the training set of AVS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='time (seconds)JOURNAL OF LATEX CLASS FILES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 14 (a) without audio (b) with audio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' T-SNE [73] visualization of the visual features, trained with or without audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' These results are from the test split of the Multi-sources subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We first use principal component analysis (PCA) to divide the audio features into K = 20 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Then we assign the audio cluster labels to the corresponding visual features and conduct t-SNE visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The points with the same color share the same audio cluster labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' It can be seen that when training is accompanied by audio signals (right), the visual features illustrate a closer trend with the audio feature distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=', points with the same colors gather together, which indicates an audio-visual correlation has been learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' (Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=') Video frames Predic- tion (a) gorilla Video frames Predic- tion (d) clarinet and piano (b) accordion (c) sweeping robot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Qualitative examples of applying the pretrained AVS model under the MS3 setting to unseen videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The caption in each sub-figure indicates the sounding object(s) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' There are almost no videos having the same category as these sounding objects during AVS model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The pretrained AVS model gains the ability to segment the correct sounding object(s) in both single and multi sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TABLE 8 Performance with different initialization strategies under the MS3 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Compared to training from scratch under the MS3 setting, we observe a significant performance improvement if pre-training the model on the Single-source subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Note the proposed LAVM loss is used in all the experiments of the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' The metric is mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Method From scratch Pretrained on Single-source ResNet50 PVT-v2 ResNet50 PVT-v2 wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='56 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='21 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='50 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='59 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' TPAVI 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='88 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='00 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='33 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content='34 the case of a single sound source (a), multiple visible objects (b, c), and multiple sound sources (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We speculate that the pretrained AVS model learned some prior knowledge about audio-visual correspondence from the AVSBench dataset that helps it generalize to even unseen videos and give possibly accurate pixel-level segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 6 CONCLUSION We explore the task of audio-visual segmentation (AVS), which aims to generate pixel-level segmentation masks for sounding objects in audible videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' To facilitate research on AVS, we build and enrich the audio-visual segmentation benchmark (AVSBench) that contains the single-source, multi- sources and semantic-labels subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' Accordingly, three task settings are explored: the semi-supervised single-source AVS (S4), fully-supervised multi-source AVS (MS3) and the fully-supervised audio-visual semantic segmentation (AVSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We presented a new pixel-level method to serve as a strong baseline and work for those three settings, which JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' 8, AUGUST 2015 15 includes a TPAVI module to encode the pixel-wise audio- visual interactions within temporal video sequences and a regularization loss that is designed to help the model learn audio-visual correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' We compared our method with several existing state-of-the-art methods from related tasks on AVSBench, and further demonstrated that our method can build a connection between the sound and the appearance of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' For future work, we will create a large-scale synthetic dataset for model pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': 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+page_content=' Hinton, “Visualizing data using t-sne.” Journal of Machine Learning Research (JMLR), 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQf3TXn/content/2301.13190v1.pdf'} diff --git a/ntE0T4oBgHgl3EQfqAHb/vector_store/index.faiss b/ntE0T4oBgHgl3EQfqAHb/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e2b8bfc6c6a3643a30a41137351d6fd4a65427dc --- /dev/null +++ b/ntE0T4oBgHgl3EQfqAHb/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1e511a58702ae24b6fa5fa84ed0130bab73d1cb44288113851bb1b4d58553f74 +size 4194349 diff --git a/ntE0T4oBgHgl3EQfqAHb/vector_store/index.pkl b/ntE0T4oBgHgl3EQfqAHb/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ebd3592d6c6189ecb62c569a1d250968c8553fcb --- /dev/null +++ b/ntE0T4oBgHgl3EQfqAHb/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3d9b37eb582d628ce5e1b2025d0d863a963ceb62488ca3a418c18672b912e3dc +size 131941 diff --git a/o9E0T4oBgHgl3EQf9AJc/content/tmp_files/2301.02795v1.pdf.txt b/o9E0T4oBgHgl3EQf9AJc/content/tmp_files/2301.02795v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58800bfd1c72ca372c3f763c0613e3dcbd82ebf8 --- /dev/null +++ b/o9E0T4oBgHgl3EQf9AJc/content/tmp_files/2301.02795v1.pdf.txt @@ -0,0 +1,968 @@ +1 +A Lite Fireworks Algorithm for Optimization +Haimiao Mo*, Min Zeng +Abstract—The fireworks algorithm is an optimization algorithm for simulating the explosion phenomenon of fireworks. Because of its +fast convergence and high precision, it is widely used in pattern recognition, optimal scheduling, and other fields. However, most of +the existing research work on the fireworks algorithm is improved based on its defects, and little consideration is given to reducing the +number of parameters of the fireworks algorithm. The original fireworks algorithm has too many parameters, which increases the cost of +algorithm adjustment and is not conducive to engineering applications. In addition, in the fireworks population, the unselected individuals +are discarded, thus causing a waste of their location information. To reduce the number of parameters of the original Fireworks Algorithm +and make full use of the location information of discarded individuals, we propose a simplified version of the Fireworks Algorithm. +It reduces the number of algorithm parameters by redesigning the explosion operator of the fireworks algorithm and constructs an +adaptive explosion radius by using the historical optimal information to balance the local mining and global exploration capabilities. The +comparative experimental results of function optimization show that the overall performance of our proposed LFWA is better than that of +comparative algorithms, such as the fireworks algorithm, particle swarm algorithm, and bat algorithm. +Index Terms—Fireworks Algorithm, Lite Fireworks Algorithm (LFWA), Function Optimization, Algorithm’s parameter reduction. +! +1 +INTRODUCTION +I +NSPIRED by the natural phenomenon that fireworks ex- +plode in the night sky to generate sparks and illuminate +the surrounding area, Tan et al. proposed the Fireworks +Algorithm (FWA) in 2010 [1]. In this algorithm, fireworks are +regarded as a feasible solution in the solution space of the +optimization problem, so the process of fireworks explosion +to produce a certain number of sparks is the process of +searching the neighborhood. The fireworks algorithm is +widely used in pattern recognition [2], optimal scheduling +[3], function optimization [4], and other domains because it +has fast convergence and high precision and can balance +local mining and global exploration capabilities through +exploding sparks. +However, according to previous studies, there are some +flaws in the design of the original fireworks algorithm [5]. +First, when the fireworks of the original fireworks algorithm +explode, each dimension produces the same displacement. It +may lead to insufficient diversity of the population, which in +turn limits the exploration ability of the algorithm. Second, +the original fireworks select the next-generation fireworks +through a strategy based on distance, which leads to inef- +ficiency in selecting offspring fireworks and increased com- +putational cost. Third, the original firework algorithm has +different subpopulations, such as fireworks subpopulation, +explosion spark subpopulation, explosion spark subpopula- +tion, and Gaussian spark subpopulation. The local mining +and global exploration capacities of the fireworks algorithm +are constrained by the absence of information-sharing and +coordination mechanisms among these subpopulations. +Improvement efforts for this algorithm are mostly clas- +• +Haimiao Mo* is with the School of Management, Hefei University of +Technology, Anhui Hefei 23009, China, also with the Key Laboratory +of Process Optimization and Intelligent Decision-Making, Ministry of +Education, China.(Email: mhm hfut@163.com) +• +Min Zeng is with College of Computer Science, Guangdong Uni- +versity of Science and Technology, Dongguan, Guangdong Province, +China.(Email: zm6102@163.com) +sified into two categories based on the shortcomings of the +original fireworks method [6]. The first type of improvement +work is to improve explosion operators (such as explosion +strength, explosion radius, and displacement operations), +mutation operators, mapping rules, and selection strate- +gies. Another type of improvement work is to improve the +fireworks algorithm by combining the advantages of other +heuristic algorithms [5]. +In the first type of improvement work, there are three +main improvement strategies. First, the explosion operator +(mutation operator) is improved by using the effective in- +formation of fireworks (mapping rules, or pros and cons +of explosion spark difference vector) to enhance the local +mining and global exploration capabilities (increase popula- +tion diversity) of the fireworks algorithm. This improvement +strategy mainly enhances the optimization ability of the al- +gorithm by improving the explosion radius of the fireworks +algorithm [7], [8], [9], [10] random mapping rules or bound- +ary mapping rules [11], [12], and population diversity [13]. +Second, due to the low efficiency and high cost of selecting +offspring, this type of improvement strategy is mainly im- +proved for the offspring selection strategy of the fireworks +algorithm [14], [15], [16], such as ”random-elite” strategy, +and tournament strategy. Third, the improved firework al- +gorithm establishes an effective collaborative optimization +mechanism between sub-populations through information +exchange between sub-populations [9], [17], [18], [19], [20]. +It can not only improve the information utilization effi- +ciency among the subpopulations of the original fireworks +algorithm, and select excellent subpopulations to guide the +population to find the optimal solution, but also effectively +enhance the optimization ability of the algorithm. +The second type of improvement work mainly enhances +the performance of the hybrid algorithm by introducing +the advantages of other evolutionary algorithms, such as +Particle Swarm Optimization (PSO) [21], Moth Search Al- +gorithm (MSA) [22], Gray Wolf Optimizer (GWO) [23], +Biogeography-based Optimization Algorithm (BOA) [24], +arXiv:2301.02795v1 [cs.NE] 7 Jan 2023 + +2 +Genetic Algorithm (GA) [25], [26]. The original fireworks +algorithm’s local mining and global exploration capabilities +are significantly enhanced by integrating the benefits of +these evolutionary algorithms. +However, most of the existing research work is to im- +prove the defects of the fireworks algorithm and rarely +considers reducing the number of parameters of the fire- +works algorithm. When calculating the explosion intensity, +the original firework algorithm [1] needs to use a constant +to control the number of explosions sparks. The original +fireworks algorithm requires predefining the threshold of +the maximum explosion radius before computing the ex- +plosion radius. The original fireworks algorithm has too +many parameters, which increases the cost of parameter +adjustment and is not conducive to the wide application +of the algorithm. +In addition, when the fireworks algorithm selects off- +spring fireworks, individuals that are not selected as the +next generation will be discarded, which greatly wastes the +historical information of these individuals. We simplify the +parameters of the fireworks algorithm and make full use of +the historical information of the unselected individuals to +construct an adaptive explosion radius, to balance the local +mining and global exploration capabilities of the algorithm. +Therefore, we propose a Lite Fireworks Algorithm +(LFWA) to reduce the number of parameters of the original +fireworks algorithm and adaptively adjust the explosion +radius. The contributions of our LFWA are as follows. +1) By redesigning the explosion operator of the fireworks +algorithm, the algorithm is simplified to reduce the number +of parameters of the algorithm, which is more conducive +to the parameter adjustment and application practice of the +algorithm. +2) By using the historical optimal information of the +fireworks algorithm to construct a new explosion radius, +the fireworks population can better balance the local search +and global search capabilities to improve the performance +of the algorithm. +2 +THE LITE FIREWORKS ALGORITHM +2.1 +Explosion Intensity +Since the original fireworks algorithm needs to predefine the +maximum number of explosion sparks, too many parame- +ters may harm the application of the algorithm. Equation +(1) illustrates the new strategy for calculating explosion +intensity, which reduces the number of parameters of the +fireworks algorithm. It limits the maximum number of +explosion sparks by the fitness value of individual fireworks +and the population size of fireworks. +Si = +� +M +fmax−f(xi) +fmax−fmin+ξ +� +(1) +where Si is the explosion intensity of the fireworks xi, M is +the population of fireworks. ⌈⌉ is a ceil function to limit the +number of Si in the integer range of [1, M]. +2.2 +Explosion Radius +When selecting the next generation of fireworks, the original +fireworks algorithm will discard the unselected individuals, +which greatly wastes the historical information of these +discarded individuals. Therefore, the historical information +of these individual locations is used to construct a new +explosion radius to balance the local mining and global +exploration capabilities of the fireworks algorithm. The new +explosion radius is calculated as shown in Equations (2) and +(3). +Ri = +� pbesti − xi, Si < SAvg +xCF − xi, Si ≥ SAvg +(2) +SAvg = 1 +M +M +� +i=1 +Si +(3) +where pbesti is the historical best location information of +the i-th fireworks xi, and xCF is the best fireworks among +all pbest = {pbest1, pbest2, ..., pesbtM}, also called Core +Fireworks (CF). SAvg is the average explosion intensity, +which is also the average number of all explosion sparks. +2.3 +Displacement Operation +When exploding, the fireworks xi complete the displace- +ment and then generates explosion sparks with the number +of Si. The displacement operation is given by Equation (4). +ESj = xi + β × Ri +(4) +where j = 1, 2, ..., Si. And xi is the position of i-th fire- +works, and ESj is the j-th Explosion Sparks (ES) in the i-th +subpopulation generated by the i-th fireworks xi after an +explosion. β is a random number with a uniform distribu- +tion in the range [0,1], Ri is the explosion radius of the i-th +fireworks xi. +From Equations (2) to (4), LFWA adopts xCF and pbest +to construct a new explosion radius that can adaptively +adjust the explosion radius according to the explosion in- +tensity. If the explosion intensity of the i-th fireworks xi +is less than the average explosion intensity SAvg, then the +fireworks learn from pbest. Otherwise, the fireworks learn +from the core fireworks xCF . In this way, LFWA can adap- +tively adjust the step size by introducing xCF and pbest to +construct a new explosion radius. +2.4 +Mutation Factor +The LFWA maintains the diversity of the fireworks pop- +ulation through the mutation strategy, so as to avoid the +fireworks population falling into the local optimal solu- +tion too quickly. The mutation strategy of LFWA is shown +in Equation (5), which randomly selects n dimensions of +fireworks for Gaussian mutation. Gaussian Sparks (GS) is +generated after the fireworks undergo Gaussian mutation. +GSij = xij × (N(0, 1) + 1) +(5) +where j = 1, 2, ..., n. N(0, 1) is a Gaussian distribution +function with a mean of 0 and a standard deviation of 1. +d is the total dimensions of xi. + +3 +TABLE 1 +Benchmark functions +Function Name +Function +Domain +dim +optimum +Sphere +f1(x) = +n +� +i=1 +xi +[−100, 100]n +30 +0 +Rosenbrock +f2(x) = +n +� +i=1 +[100(xi+1 − xi)2 + (xi − 1)2] +[−10, 10]n +30 +0 +Rosenbrock +f3(x) = +n +� +i=1 +[x2 +i − 10 cos(2πxi) + 10] +[−5.12, 5.12]n +30 +0 +Griewank +f4(x) = +1 +4000 +n +� +i=1 +x2 +i − +n� +i=1 +cos( xi +√ +i ) + 1 +[−600, 600]n +30 +0 +Ackley +f5(x) = −20 exp(−0.2 +� +n +� +i=1 +x2 +i ) − exp( 1 +n ) +n +� +i=1 +cos(2πxi) + 20 + e +[−32, 32]n +30 +0 +Schwefel +f6(x) = +n +� +i=1 +−xi sin( +� +|xi|) +[−100, 100]n +30 +0 +Six-Hump Camel-Back +f7(x) = 4x2 +1 − 2.1x4 +1 + 3.1x6 +1 + x1x2 − 4x2 +2 + 4x4 +2 +[−5, 5]n +2 +-1.0316285 +Goldstein Price +f8(x) = [1 + (x1 + x2 + 1)2(19 − 14x1 + 3x2 +1 − 14x2 + 6x1x2 + 3x4 +2)] × +[30+(2x1 − 3x2)2(18 − 32x1 + 12x2 +1 + 48x2 − 36x1x2 + 27x2 +2)] +[−2, 2]n +2 +3 +Schaffer’s F6 +f9(x) +sin2� +x2 +1+x2 +2−0.5 +[1+0.001(x2 +1+x2 +2)]2 + 0.5 +[−100, 100]n +2 +0 +2.5 +Mapping Rules +Each dimension of the positions of fireworks, explosion +sparks, and Gaussian sparks needs to be processed using +the mapping rules of Equation (6) to prevent them from +going out of bounds. +xnew +ij += LB + β × (UB − LB) +(6) +where UB and LB are the upper and lower bounds of the +algorithm’s search range, respectively. +2.6 +Selection Strategy +The LFWA adopts the ”Elite-Random” strategy [14] to select +the next generation of fireworks from the candidate set, +which consists of fireworks x = {x1, x2, ..., xM}, pbest, +xCF , explosion sparks ES, and Gaussian fireworks GF. +That is to say, the core firework with the best fitness is +selected as the next generation of fireworks individuals, and +the other fireworks individuals of the next generation are +randomly selected. +3 +COMPARATIVE EXPERIMENTS +3.1 +Benchmark Functions +Benchmark functions are utilized in comparison tests to +examine how well the suggested LFWA performs. Table 1 +displays the function expression. +3.2 +Parameters Setting +The performance of LFWA is compared with other algo- +rithms (e.g. BA, SPSO, FWA) by testing the benchmark +functions shown in Table 1. A fixed tolerance λ ≤ 10−5 is +used in experiments to measure the error between the search +value and the optimal value of the objective function. If the +error is less than or equal to λ, it means that the algorithm +successfully finds the optimal solution. +TABLE 2 +The comparative experimental results of the four algorithms +f +Alg +worst +best +mean +SD +f1 +LFWA +0 +0 +0 +0 +BA +1.698E-03 +7.972E-04 +1.367E-03 +1.547E-04 +SPSO +7.170E+01 +5.100E+00 +2.207E+01 +1.164E+01 +FWA +1.625E-173 +4.067E-260 +1.625E-175 +0 +f2 +LFWA +2.883E+01 +2.553E+01 +2.749E+01 +1.018E+00 +BA +8.094E+01 +2.419E+01 +2.810E+01 +5.502E+00 +SPSO +2.563E+05 +2.344E+03 +4.160E+04 +4.982E+04 +FWA +2.890E+01 +2.706E+01 +2.852E+01 +4.811E-01 +f3 +LFWA +0 +0 +0 +0 +BA +9.397E-05 +4.027E-05 +7.027E-05 +1.063E-05 +SPSO +8.792E-01 +2.158E-01 +5.811E-01 +1.355E-01 +FWA +0 +0 +0 +0 +f4 +LFWA +0 +0 +0 +0 +BA +5.496E+01 +1.410E+01 +2.879E+01 +8.334E+00 +SPSO +6.407E+02 +2.054E+02 +3.339E+02 +8.129E+01 +FWA +0 +0 +0 +0 +f5 +LFWA +8.882E-16 +8.882E-16 +8.882E-16 +0 +BA +3.225E+00 +1.902E+00 +2.733E+00 +2.978E-01 +SPSO +2.048E+01 +5.323E+00 +1.903E+01 +3.471E+00 +FWA +8.882E-16 +8.882E-16 +8.882E-16 +0 +f6 +LFWA +1.197E-08 +6.414E-16 +6.693E-10 +1.448E-09 +BA +1.197E-08 +6.414E-16 +6.693E-10 +1.448E-09 +SPSO +0 +0 +0 +0 +FWA +1.916E-10 +6.322E-37 +2.513E-12 +1.974E-11 +f7 +LFWA +-1.0316285 +-1.0316285 +-1.0316285 +1.508E-12 +BA +-1.0316282 +-1.0316285 +-1.0316284 +6.612E-08 +SPSO +-1.0316285 +-1.0316285 +-1.0316285 +1.558E-15 +FWA +-1.0316089 +-1.0316285 +-1.0316273 +2.492E-06 +f8 +LFWA +3.000000 +3.000000 +3.000000 +7.177E-11 +BA +84.000027 +3.000000 +10.560005 +1.539E+01 +SPSO +3.000000 +3.000000 +3.000000 +1.085E-15 +FWA +3.000236 +3.000000 +3.000010 +2.824E-05 +f9 +LFWA +0 +0 +0 +0 +BA +1.2699E-01 +1.4234E-10 +6.8488E-03 +1.788E-02 +SPSO +9.7159E-03 +0 +1.7489E-03 +3.752E-03 +FWA +0 +0 +0 +0 + +4 +3.3 +Analysis of Convergence +The experiment results of worst, best, mean, and Standard +Deviation (SD), with 1000 iterations and 100 runs, are shown +as Table 2. From Table 2, the mean of LFWA is better than +the other three algorithms for f1 and f2 at the end of the +iteration, so the optimization accuracy of LFWA is better +than the other three algorithms. For f3, f4, f5 and f9, the +mean of LFWA is the same as FWA, but better than the other +two algorithms, so the optimization accuracy of LFWA and +FWA is better than the other two algorithms at the end of the +iteration. For f6, f7, and f8, the mean of LFWA is the same +as SPSO, but better than the other two algorithms, so the +optimization accuracy of LFWA and SPSO is better than the +other two algorithms. In a word, the overall optimization +accuracy of LFWA is better than the other three algorithms +at the end of the iteration. +We only give partial evolution curves for four algorithms +shown in Figure 1(a) to Figure 1(g) that the convergence +speed of LFWA is better than the other three algorithms. +The evolution curves except Figure 1(b) and Figure 1(g) are +partly overlapping, and their ordinates are log10(Fitness). +From the Figure 1(a) to Figure 1(g) except Figure 1(f), it +can be seen that during the early phase of optimizing +function, the evolution of LFWA is steeper than these of +the other three algorithms, and the convergence speed of +LFWA is better than other three algorithms. Meanwhile, the +optimization accuracy of LFWA is better than the other three +algorithms when the iteration is the same. +In Figure 1(f), it can be seen that the convergence speed +of LFWA is better than that of BA but worse than the other +two algorithms during the early phase of optimization. +However, from the mean of f8 shown in Table 2, it can +be seen that at the end of the iteration, the optimization +accuracy of LFWA is the same as SPSO and better than +the other two algorithms. In summary, LFWA outperforms +the other three algorithms in terms of overall optimization +accuracy and convergence speed. +3.4 +Analysis of Robustness +The standard deviation reflects the robustness of an algo- +rithm. In other words, a smaller standard deviation presents +better robustness for an algorithm, and reversely, presents +worse robustness for an algorithm. From the standard de- +viation of Table 2, the robustness of LFWA is the same as +FWA for f1, f3, f4, f5, and f9, but better than the other +two algorithms. For f2, the robustness of LFWA is worse +than FWA but better than the other two algorithms. For f6, +the robustness of LFWA is the same as SPSO, but better +than the other two algorithms. For f2, the robustness of +LFWA is worse than FWA but better than the other two +algorithms. For f6, the robustness of LFWA is the same as +SPSO, but better than the other two algorithms. In a word, +the overall robustness of LFWA is better than the other three +algorithms. +3.5 +Analysis of success rate +If the accuracy satisfies the fixed tolerance λ , it is recorded +successfully once. The Success Rate (SR) for four algorithms +are shown as Table 3. From Table 3, for f2, f7, the success +Fig. 1. Evolution curves for partial functions of Table 1. +TABLE 3 +The success rate (%) of four algorithms +alg +f1 +f3 +f4 +f5 +f6 +f8 +f9 +f2, f7 +LFWA +100 +100 +100 +100 +100 +100 +100 +0 +BA +0 +0 +0 +0 +100 +73 +67 +0 +SPSO +0 +0 +0 +0 +100 +100 +82 +0 +FWA +100 +100 +100 +100 +100 +81 +100 +0 + +Sphere dim = 30 +Rastrigin dim = 30 +50 +5举 +0*0 +A +0 . +-50 上 +θ—BA +-5 +*一SPSO +△-FWA +θ—BA +*SPsO +-200 +FWA +-250 +-15 +-300 +-350 +-20 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +0 +Number of Function Iteration +Number of Function Iteration +(a) Evolution curve of fi +(b) Evolution curve of f3 +Griewank dim = 30 +Ackley dim = 30 +0 +—O—BA +*一SPSO +△—FWA +PA +(sse +—BA +-6 +米一SPSO +A—FWA +-PA +-10 +-12 +-15+ +-14 +-20 +-16 +200 +400 +600 +800 +100C +0 +200 +400 +600 +800 +1000 +0 +Number of Function Iteration +Number of Function Iteration +(d) Evolution curve of fs +(c) Evolution curve of f4 +Schwefel dim = 30 +Goldstein Price dim = 2 +50 +2.4 - +O—BA +2.2 +0袋 +*一SPSO +△-FWA +米(米 +2 4 +-BA +-50 上 +PA +米一SPSO +1.8 * +FWA +-100 +1.6d +10(Fitne +1.4 + 1.2 +-250 +0.8 +-300 +0.6类 +A +-350 +0.4 +200 +400 +600 +800 +1001 +200 +400 +600 +800 +100C +0 +Number of Function Iteration +Number of Function Iteration +(e) Evolution curve of f. +(f) Evolution curve of f. +Schaffers F6 dim = 2 +O—BA +*一 SPSO +△—FWA +PA +-15 +-20 L +200 +400 +600 +800 +1000 +0 +Number of Function Iteration +(g) Evolution curve of f,5 +rate of the four algorithms is 0%. For f1 , f3, f4 and f5, the +success rate of LFWA and FWA is 100%, they are better than +other two algorithms. For f6, the SR of the four algorithms +is 100%. For f8, the SR of LFWA and SPSO is 100%, and +they are better than other two algorithms. For f9, the SR of +LFWA and FWA is 100%, and they are better than other two +algorithms. Therefore, the overall success rate of LFWA is +better than other three algorithms for optimizing f1 to f9. +3.6 +Discussion +As shown in the experiments, the LFWA has a faster conver- +gence speed, a better optimization accuracy, a better robust- +ness and a higher optimization success rate, compared to the +BA, SPSO and FWA. The reason why LFWA outperforms +other comparative algorithms lies in the following three +aspects. +First, LFWA reduces the number of parameters of the +original fireworks algorithm, and presents a more concise +expression, which reduces the cost of parameter adjustment +during the optimization process of the algorithm. +Second, LFWA draws on the memory mechanism of the +particle swarm algorithm, and introduces the locally opti- +mal fireworks individual pbest and core fireworks xCF into +the original fireworks algorithm during the optimization +process. Then an adaptive explosion radius is constructed +by utilizing the historical optimal information. In this way, +LFWA can share and inherit the historical optimal informa- +tion during the optimization process, so LFWA has a faster +convergence speed. +Third, the ”Elite-Rand” strategy and variation factors +maintain the diversity of LFWA. 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Cpałka, “On the application of a hybrid genetic- +firework algorithm for controllers structure and parameters selec- +tion,” in Information Systems Architecture and Technology: Proceedings +of 36th International Conference on Information Systems Architecture +and Technology–ISAT 2015–Part I. +Springer, 2016, pp. 111–123. +[26] T. S. Babu, J. P. Ram, K. Sangeetha, A. Laudani, and N. Rajasekar, +“Parameter extraction of two diode solar pv model using fire- +works algorithm,” Solar energy, vol. 140, pp. 265–276, 2016. + diff --git a/o9E0T4oBgHgl3EQf9AJc/content/tmp_files/load_file.txt b/o9E0T4oBgHgl3EQf9AJc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e86ca094109aa66f3ebe88c65f7ac1fc0809ea4 --- /dev/null +++ b/o9E0T4oBgHgl3EQf9AJc/content/tmp_files/load_file.txt @@ -0,0 +1,524 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf,len=523 +page_content='1 A Lite Fireworks Algorithm for Optimization Haimiao Mo*, Min Zeng Abstract—The fireworks algorithm is an optimization algorithm for simulating the explosion phenomenon of fireworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Because of its fast convergence and high precision, it is widely used in pattern recognition, optimal scheduling, and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' However, most of the existing research work on the fireworks algorithm is improved based on its defects, and little consideration is given to reducing the number of parameters of the fireworks algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The original fireworks algorithm has too many parameters, which increases the cost of algorithm adjustment and is not conducive to engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In addition, in the fireworks population, the unselected individuals are discarded, thus causing a waste of their location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' To reduce the number of parameters of the original Fireworks Algorithm and make full use of the location information of discarded individuals, we propose a simplified version of the Fireworks Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' It reduces the number of algorithm parameters by redesigning the explosion operator of the fireworks algorithm and constructs an adaptive explosion radius by using the historical optimal information to balance the local mining and global exploration capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The comparative experimental results of function optimization show that the overall performance of our proposed LFWA is better than that of comparative algorithms, such as the fireworks algorithm, particle swarm algorithm, and bat algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Index Terms—Fireworks Algorithm, Lite Fireworks Algorithm (LFWA), Function Optimization, Algorithm’s parameter reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 1 INTRODUCTION I NSPIRED by the natural phenomenon that fireworks ex- plode in the night sky to generate sparks and illuminate the surrounding area, Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' proposed the Fireworks Algorithm (FWA) in 2010 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In this algorithm, fireworks are regarded as a feasible solution in the solution space of the optimization problem, so the process of fireworks explosion to produce a certain number of sparks is the process of searching the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The fireworks algorithm is widely used in pattern recognition [2], optimal scheduling [3], function optimization [4], and other domains because it has fast convergence and high precision and can balance local mining and global exploration capabilities through exploding sparks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' However, according to previous studies, there are some flaws in the design of the original fireworks algorithm [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' First, when the fireworks of the original fireworks algorithm explode, each dimension produces the same displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' It may lead to insufficient diversity of the population, which in turn limits the exploration ability of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Second, the original fireworks select the next-generation fireworks through a strategy based on distance, which leads to inef- ficiency in selecting offspring fireworks and increased com- putational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Third, the original firework algorithm has different subpopulations, such as fireworks subpopulation, explosion spark subpopulation, explosion spark subpopula- tion, and Gaussian spark subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The local mining and global exploration capacities of the fireworks algorithm are constrained by the absence of information-sharing and coordination mechanisms among these subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Improvement efforts for this algorithm are mostly clas- Haimiao Mo* is with the School of Management, Hefei University of Technology, Anhui Hefei 23009, China, also with the Key Laboratory of Process Optimization and Intelligent Decision-Making, Ministry of Education, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' (Email: mhm hfut@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='com) Min Zeng is with College of Computer Science, Guangdong Uni- versity of Science and Technology, Dongguan, Guangdong Province, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' (Email: zm6102@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='com) sified into two categories based on the shortcomings of the original fireworks method [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The first type of improvement work is to improve explosion operators (such as explosion strength, explosion radius, and displacement operations), mutation operators, mapping rules, and selection strate- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Another type of improvement work is to improve the fireworks algorithm by combining the advantages of other heuristic algorithms [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In the first type of improvement work, there are three main improvement strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' First, the explosion operator (mutation operator) is improved by using the effective in- formation of fireworks (mapping rules, or pros and cons of explosion spark difference vector) to enhance the local mining and global exploration capabilities (increase popula- tion diversity) of the fireworks algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' This improvement strategy mainly enhances the optimization ability of the al- gorithm by improving the explosion radius of the fireworks algorithm [7], [8], [9], [10] random mapping rules or bound- ary mapping rules [11], [12], and population diversity [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Second, due to the low efficiency and high cost of selecting offspring, this type of improvement strategy is mainly im- proved for the offspring selection strategy of the fireworks algorithm [14], [15], [16], such as ”random-elite” strategy, and tournament strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Third, the improved firework al- gorithm establishes an effective collaborative optimization mechanism between sub-populations through information exchange between sub-populations [9], [17], [18], [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' It can not only improve the information utilization effi- ciency among the subpopulations of the original fireworks algorithm, and select excellent subpopulations to guide the population to find the optimal solution, but also effectively enhance the optimization ability of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The second type of improvement work mainly enhances the performance of the hybrid algorithm by introducing the advantages of other evolutionary algorithms, such as Particle Swarm Optimization (PSO) [21], Moth Search Al- gorithm (MSA) [22], Gray Wolf Optimizer (GWO) [23], Biogeography-based Optimization Algorithm (BOA) [24], arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='02795v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='NE] 7 Jan 2023 2 Genetic Algorithm (GA) [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The original fireworks algorithm’s local mining and global exploration capabilities are significantly enhanced by integrating the benefits of these evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' However, most of the existing research work is to im- prove the defects of the fireworks algorithm and rarely considers reducing the number of parameters of the fire- works algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' When calculating the explosion intensity, the original firework algorithm [1] needs to use a constant to control the number of explosions sparks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The original fireworks algorithm requires predefining the threshold of the maximum explosion radius before computing the ex- plosion radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The original fireworks algorithm has too many parameters, which increases the cost of parameter adjustment and is not conducive to the wide application of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In addition, when the fireworks algorithm selects off- spring fireworks, individuals that are not selected as the next generation will be discarded, which greatly wastes the historical information of these individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' We simplify the parameters of the fireworks algorithm and make full use of the historical information of the unselected individuals to construct an adaptive explosion radius, to balance the local mining and global exploration capabilities of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Therefore, we propose a Lite Fireworks Algorithm (LFWA) to reduce the number of parameters of the original fireworks algorithm and adaptively adjust the explosion radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The contributions of our LFWA are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 1) By redesigning the explosion operator of the fireworks algorithm, the algorithm is simplified to reduce the number of parameters of the algorithm, which is more conducive to the parameter adjustment and application practice of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 2) By using the historical optimal information of the fireworks algorithm to construct a new explosion radius, the fireworks population can better balance the local search and global search capabilities to improve the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 2 THE LITE FIREWORKS ALGORITHM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1 Explosion Intensity Since the original fireworks algorithm needs to predefine the maximum number of explosion sparks, too many parame- ters may harm the application of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Equation (1) illustrates the new strategy for calculating explosion intensity, which reduces the number of parameters of the fireworks algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' It limits the maximum number of explosion sparks by the fitness value of individual fireworks and the population size of fireworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Si = � M fmax−f(xi) fmax−fmin+ξ � (1) where Si is the explosion intensity of the fireworks xi, M is the population of fireworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' ⌈⌉ is a ceil function to limit the number of Si in the integer range of [1, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2 Explosion Radius When selecting the next generation of fireworks, the original fireworks algorithm will discard the unselected individuals, which greatly wastes the historical information of these discarded individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Therefore, the historical information of these individual locations is used to construct a new explosion radius to balance the local mining and global exploration capabilities of the fireworks algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The new explosion radius is calculated as shown in Equations (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Ri = � pbesti − xi, Si < SAvg xCF − xi, Si ≥ SAvg (2) SAvg = 1 M M � i=1 Si (3) where pbesti is the historical best location information of the i-th fireworks xi, and xCF is the best fireworks among all pbest = {pbest1, pbest2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=', pesbtM}, also called Core Fireworks (CF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' SAvg is the average explosion intensity, which is also the average number of all explosion sparks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='3 Displacement Operation When exploding, the fireworks xi complete the displace- ment and then generates explosion sparks with the number of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The displacement operation is given by Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' ESj = xi + β × Ri (4) where j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=', Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' And xi is the position of i-th fire- works, and ESj is the j-th Explosion Sparks (ES) in the i-th subpopulation generated by the i-th fireworks xi after an explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' β is a random number with a uniform distribu- tion in the range [0,1], Ri is the explosion radius of the i-th fireworks xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' From Equations (2) to (4), LFWA adopts xCF and pbest to construct a new explosion radius that can adaptively adjust the explosion radius according to the explosion in- tensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' If the explosion intensity of the i-th fireworks xi is less than the average explosion intensity SAvg, then the fireworks learn from pbest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Otherwise, the fireworks learn from the core fireworks xCF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In this way, LFWA can adap- tively adjust the step size by introducing xCF and pbest to construct a new explosion radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='4 Mutation Factor The LFWA maintains the diversity of the fireworks pop- ulation through the mutation strategy, so as to avoid the fireworks population falling into the local optimal solu- tion too quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The mutation strategy of LFWA is shown in Equation (5), which randomly selects n dimensions of fireworks for Gaussian mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Gaussian Sparks (GS) is generated after the fireworks undergo Gaussian mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' GSij = xij × (N(0, 1) + 1) (5) where j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' N(0, 1) is a Gaussian distribution function with a mean of 0 and a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' d is the total dimensions of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 3 TABLE 1 Benchmark functions Function Name Function Domain dim optimum Sphere f1(x) = n � i=1 xi [−100, 100]n 30 0 Rosenbrock f2(x) = n � i=1 [100(xi+1 − xi)2 + (xi − 1)2] [−10, 10]n 30 0 Rosenbrock f3(x) = n � i=1 [x2 i − 10 cos(2πxi) + 10] [−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='12, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='12]n 30 0 Griewank f4(x) = 1 4000 n � i=1 x2 i − n� i=1 cos( xi √ i ) + 1 [−600, 600]n 30 0 Ackley f5(x) = −20 exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2 � n � i=1 x2 i ) − exp( 1 n ) n � i=1 cos(2πxi) + 20 + e [−32, 32]n 30 0 Schwefel f6(x) = n � i=1 −xi sin( � |xi|) [−100, 100]n 30 0 Six-Hump Camel-Back f7(x) = 4x2 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1x4 1 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1x6 1 + x1x2 − 4x2 2 + 4x4 2 [−5, 5]n 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 Goldstein Price f8(x) = [1 + (x1 + x2 + 1)2(19 − 14x1 + 3x2 1 − 14x2 + 6x1x2 + 3x4 2)] × [30+(2x1 − 3x2)2(18 − 32x1 + 12x2 1 + 48x2 − 36x1x2 + 27x2 2)] [−2, 2]n 2 3 Schaffer’s F6 f9(x) sin2� x2 1+x2 2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='5 [1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='001(x2 1+x2 2)]2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='5 [−100, 100]n 2 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='5 Mapping Rules Each dimension of the positions of fireworks, explosion sparks, and Gaussian sparks needs to be processed using the mapping rules of Equation (6) to prevent them from going out of bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' xnew ij = LB + β × (UB − LB) (6) where UB and LB are the upper and lower bounds of the algorithm’s search range, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='6 Selection Strategy The LFWA adopts the ”Elite-Random” strategy [14] to select the next generation of fireworks from the candidate set, which consists of fireworks x = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=', xM}, pbest, xCF , explosion sparks ES, and Gaussian fireworks GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' That is to say, the core firework with the best fitness is selected as the next generation of fireworks individuals, and the other fireworks individuals of the next generation are randomly selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 3 COMPARATIVE EXPERIMENTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1 Benchmark Functions Benchmark functions are utilized in comparison tests to examine how well the suggested LFWA performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Table 1 displays the function expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2 Parameters Setting The performance of LFWA is compared with other algo- rithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' BA, SPSO, FWA) by testing the benchmark functions shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' A fixed tolerance λ ≤ 10−5 is used in experiments to measure the error between the search value and the optimal value of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' If the error is less than or equal to λ, it means that the algorithm successfully finds the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' TABLE 2 The comparative experimental results of the four algorithms f Alg worst best mean SD f1 LFWA 0 0 0 0 BA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='698E-03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='972E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='367E-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='547E-04 SPSO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='170E+01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='100E+00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='207E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='164E+01 FWA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='625E-173 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='067E-260 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='625E-175 0 f2 LFWA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='883E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='553E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='749E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='018E+00 BA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='094E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='419E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='810E+01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='502E+00 SPSO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='563E+05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='344E+03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='160E+04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='982E+04 FWA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='890E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='706E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='852E+01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='811E-01 f3 LFWA 0 0 0 0 BA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='397E-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='027E-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='027E-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='063E-05 SPSO 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='792E-01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='158E-01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='811E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='355E-01 FWA 0 0 0 0 f4 LFWA 0 0 0 0 BA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='496E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='410E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='879E+01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='334E+00 SPSO 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='407E+02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='054E+02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='339E+02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='129E+01 FWA 0 0 0 0 f5 LFWA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='882E-16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='882E-16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='882E-16 0 BA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='225E+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='902E+00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='733E+00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='978E-01 SPSO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='048E+01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='323E+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='903E+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='471E+00 FWA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='882E-16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='882E-16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='882E-16 0 f6 LFWA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='197E-08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='414E-16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='693E-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='448E-09 BA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='197E-08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='414E-16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='693E-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='448E-09 SPSO 0 0 0 0 FWA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='916E-10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='322E-37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='513E-12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='974E-11 f7 LFWA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='508E-12 BA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316282 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316284 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='612E-08 SPSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='558E-15 FWA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316089 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316285 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0316273 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='492E-06 f8 LFWA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='177E-11 BA 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000027 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='560005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='539E+01 SPSO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='085E-15 FWA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000236 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='000010 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='824E-05 f9 LFWA 0 0 0 0 BA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2699E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='4234E-10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='8488E-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='788E-02 SPSO 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='7159E-03 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='7489E-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='752E-03 FWA 0 0 0 0 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='3 Analysis of Convergence The experiment results of worst, best, mean, and Standard Deviation (SD), with 1000 iterations and 100 runs, are shown as Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' From Table 2, the mean of LFWA is better than the other three algorithms for f1 and f2 at the end of the iteration, so the optimization accuracy of LFWA is better than the other three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f3, f4, f5 and f9, the mean of LFWA is the same as FWA, but better than the other two algorithms, so the optimization accuracy of LFWA and FWA is better than the other two algorithms at the end of the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f6, f7, and f8, the mean of LFWA is the same as SPSO, but better than the other two algorithms, so the optimization accuracy of LFWA and SPSO is better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In a word, the overall optimization accuracy of LFWA is better than the other three algorithms at the end of the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' We only give partial evolution curves for four algorithms shown in Figure 1(a) to Figure 1(g) that the convergence speed of LFWA is better than the other three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The evolution curves except Figure 1(b) and Figure 1(g) are partly overlapping, and their ordinates are log10(Fitness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' From the Figure 1(a) to Figure 1(g) except Figure 1(f), it can be seen that during the early phase of optimizing function, the evolution of LFWA is steeper than these of the other three algorithms, and the convergence speed of LFWA is better than other three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Meanwhile, the optimization accuracy of LFWA is better than the other three algorithms when the iteration is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In Figure 1(f), it can be seen that the convergence speed of LFWA is better than that of BA but worse than the other two algorithms during the early phase of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' However, from the mean of f8 shown in Table 2, it can be seen that at the end of the iteration, the optimization accuracy of LFWA is the same as SPSO and better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In summary, LFWA outperforms the other three algorithms in terms of overall optimization accuracy and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='4 Analysis of Robustness The standard deviation reflects the robustness of an algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In other words, a smaller standard deviation presents better robustness for an algorithm, and reversely, presents worse robustness for an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' From the standard de- viation of Table 2, the robustness of LFWA is the same as FWA for f1, f3, f4, f5, and f9, but better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f2, the robustness of LFWA is worse than FWA but better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f6, the robustness of LFWA is the same as SPSO, but better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f2, the robustness of LFWA is worse than FWA but better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f6, the robustness of LFWA is the same as SPSO, but better than the other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In a word, the overall robustness of LFWA is better than the other three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='5 Analysis of success rate If the accuracy satisfies the fixed tolerance λ , it is recorded successfully once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The Success Rate (SR) for four algorithms are shown as Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' From Table 3, for f2, f7, the success Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Evolution curves for partial functions of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' TABLE 3 The success rate (%) of four algorithms alg f1 f3 f4 f5 f6 f8 f9 f2, f7 LFWA 100 100 100 100 100 100 100 0 BA 0 0 0 0 100 73 67 0 SPSO 0 0 0 0 100 100 82 0 FWA 100 100 100 100 100 81 100 0 Sphere dim = 30 Rastrigin dim = 30 50 5举 0*0 A 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='50 上 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='θ—BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='一SPSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='△-FWA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='θ—BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='SPsO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='FWA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Number of Function Iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Number of Function Iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='(a) Evolution curve of fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='(b) Evolution curve of f3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Griewank dim = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Ackley dim = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='—O—BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='一SPSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='△—FWA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='(sse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='—BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='米一SPSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='A—FWA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='15+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='100C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Number of Function Iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Number of Function Iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='(d) Evolution curve of fs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='(c) Evolution curve of f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Schwefel dim = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='Goldstein Price dim = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='4 - O—BA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2 0袋 一SPSO △-FWA 米(米 2 4 BA 50 上 PA 米一SPSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='8 * FWA 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='6d 10(Fitne 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='2 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='8 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='6类 A 350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='4 200 400 600 800 1001 200 400 600 800 100C 0 Number of Function Iteration Number of Function Iteration (e) Evolution curve of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' (f) Evolution curve of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Schaffers F6 dim = 2 O—BA 一 SPSO △—FWA PA 15 20 L 200 400 600 800 1000 0 Number of Function Iteration (g) Evolution curve of f,5 rate of the four algorithms is 0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f1 , f3, f4 and f5, the success rate of LFWA and FWA is 100%, they are better than other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f6, the SR of the four algorithms is 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f8, the SR of LFWA and SPSO is 100%, and they are better than other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' For f9, the SR of LFWA and FWA is 100%, and they are better than other two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Therefore, the overall success rate of LFWA is better than other three algorithms for optimizing f1 to f9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content='6 Discussion As shown in the experiments, the LFWA has a faster conver- gence speed, a better optimization accuracy, a better robust- ness and a higher optimization success rate, compared to the BA, SPSO and FWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The reason why LFWA outperforms other comparative algorithms lies in the following three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' First, LFWA reduces the number of parameters of the original fireworks algorithm, and presents a more concise expression, which reduces the cost of parameter adjustment during the optimization process of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Second, LFWA draws on the memory mechanism of the particle swarm algorithm, and introduces the locally opti- mal fireworks individual pbest and core fireworks xCF into the original fireworks algorithm during the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Then an adaptive explosion radius is constructed by utilizing the historical optimal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In this way, LFWA can share and inherit the historical optimal informa- tion during the optimization process, so LFWA has a faster convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Third, the ”Elite-Rand” strategy and variation factors maintain the diversity of LFWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Therefore, LFWA has a strong ability to avoid premature convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' 4 CONCLUSIONS Compared with the bat algorithm, standard particle swarm algorithm, and fireworks algorithm, we can conclude that the LFWA has a promising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' The LFWA out- performs the other three algorithms on benchmark func- tions in terms of optimization accuracy, convergence speed, robustness, and optimization success rate, which endues the LFWA with a promising prospect of application and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' In future work, we will seek a deep theoretical analysis of the LFWA and try to apply the LFWA to some special engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQf9AJc/content/2301.02795v1.pdf'} +page_content=' Tan and Y.' metadata={'source': 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b/u9E4T4oBgHgl3EQfWwwm/content/tmp_files/2301.05035v1.pdf.txt @@ -0,0 +1,725 @@ +arXiv:2301.05035v1 [physics.optics] 12 Jan 2023 +Tunable in situ Near-UV Pulses by Transient +Plasmonic Resonance in Nanocomposites +ANTON HUSAKOU,1,* IHAR BABUSHKIN,1,2,3 OLGA FEDOTOVA,4 +RYHOR RYSETSKY,4 TATSIANA SMIRNOVA,5 OLEG KHASANOV,4 +ALEXANDER FEDOTOV,5 USMAN SAPAEV,6 TZVETA APOSTOLOVA7,8 +1Max Born Institute, Max Born Str. 2a, 12489 Berlin, Germany +2Institute of Quantum Optics, Leibnitz Hannover University, Welfengarten 1, 30167 Hannover, Germany +3Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering – Innovation Across Disciplines), +Welfengarten 1, 30167 Hannover, Germany +4 Scientific and Practical Materials Research Center, Belarus NAS, Brovky 17, 220072 Minsk, Belarus +5Belarus State University, Niezalie˘znasci avenue 4, 220030 Minsk, Belarus +6Tashkent State Technical University, 2 uy 2 Qatartol ko’chasi, 100097 Tashkent, Uzbekistan +7Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko +Chausse 72, 1784 Sofia, Bulgaria +8Institute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, Bulgaria +*gusakov@mbi-berlin.de +Abstract: +We propose a new concept for generation of ultrashort pulses based on transient +plasmonic resonance in nanoparticle composites. +Photoionization and free-carriers plasma +change the susceptibility of nanoparticles on a few-femtosecond scale. This results in a narrow +time window during the pump pulse duration when the system is in plasmonic resonance, +accompanied by a short burst of the local field. During this process, frequency-tunable few-fs +pulses are generated. We elucidate the details of the above mechanism, and investigate the +influences of different contributing processes. +© 2023 Optica Publishing Group +1. +Introduction +Numerous fields of modern ultrafast optics, such as tracing of atomic motion in molecules [1], +chemistry on electronic timescale [2], steering of ultrafast electron dynamics in the valence shell +of solids, nanoparticles, and clusters [3,4],two-dimensional electronic spectroscopy experiments +[5], generation and diagnosis of warm-dense matter [6], material modification [7] and so on, +require short, sub-10-fs intense pulses at UV or near-UV frequencies. The ultraviolet (UV) +wavelength range is of great interest for ultrafast spectroscopic investigations because of the +possible resonance with electronic transitions of many small molecules with fairly simple excited +state energy level structure, whose photo-induced dynamics can be accurately modeled using ab +initio computational approaches. Transient absorption spectroscopy in the UV can thus be used +for the study of the optical response of biomolecules and can benchmark the accuracy of such +methods. +In the visible and near infrared ranges, ultrashort pulses are routinely generated using different +approaches: directly from a laser oscillator [8], by a non-collinear optical parametric amplifier +[9], or by spectral broadening in a nonlinear medium (e.g. solid or hollow-core optical fiber) [10] +based on self-phase modulation due to the optical Kerr-effect [11] or time-dependent plasma +density [12, 13]. The extension of these techniques to the UV range is challenging due to the +lack of broadband laser gain media in the UV (except for excimers of noble gas halides [14]) and +strong two-photon absorption for high-energy photons [15] required for UV optical parametric +amplifiers. +For these reasons, broadband UV pulses are typically generated in a two-step +approach: first, few-optical-cycle pulses in the visible or near-infrared ranges are generated, and + +then nonlinear frequency up-conversion is used to reach the UV range. +For frequency up-conversion in gases, the usually used nonlinear processes are third- or +higher- order harmonic generation or four-wave-mixing between the fundamental wavelength +and the second harmonic of the driving pulse, which is typically obtained from a Ti:sapphire +laser. +To name a few examples, 16-fs pulses at 266 nm were produced by third-harmonic +generation in air from the 20-fs pulses [16], much shorter sub-4-fs pulses at 270 nm were +generated in Ne from the 6-fs laser output spectrally broadened in a hollow-core fiber [17], 8-fs +pulses at 266 nm were generated by four-wave-mixing between the second harmonic and the +fundamental wavelength from a 20-fs pulse [18], and 5-nJ pulses at 133 nm were generated +by cascaded four-wave-mixing between third, fourth and fifth harmonic in a filament [19]. UV +pulses were also generated by second-order nonlinear frequency conversion in crystals, typically +훽-barium borate. For example, sub-10 fs 400 nm pulses were reported [20], with sufficient +spectral bandwidth obtained via frequency-doubling using broadband phase-matching enabled +by grating recollimation. Furthermore, UV pulses with 9.7 fs pulse duration were generated +by second harmonic generation using a spectrally shaped 1.9 mJ, 8 fs few-cycle near-infrared +pulses [21]. +Recently, new trends appeared in the generation of UV pulses. Resonant dispersive-wave +generation based on Kerr nonlinearlity during the optical soliton propagation in waveguides +combines intrinsically short pulse duration and easy tunability over the full ultraviolet spectral +range as well as the entire visible spectrum when using infrared pump pulses, as was recently +demonstrated [22,23]. These advances also raise the prospect of compact on-chip integrated +nonlinear devices [24]. +Due to a persistently high demand for short pulses in near-UV range and adjacent visible +frequencies, it is promising and timely to search and investigate alternative approaches to their +generation, preferably directly from relatively long infrared pulses without compression step. +A possibility to generate short UV pulses by a very small device in situ would be an important +and highly desirable feature, particularly in view of biological applications. Indeed, during +propagation in any transparent condensed matter, a ∼10-fs pulse will very quickly, on the +sub-mm scale, become much longer due to group-velocity dispersion. Pre-compensation of +group-velocity dispersion is in principle possible but challenging and requires precise a priori +knowledge of the material properties, which is rarely available. Therefore it would be highly +valuable to suggest a technique which allows generation of short pulse directly inside of a +transparent material at the desired position. On top of that, spectral tunability of the pulses +would strongly enhance their application potential, e.g. by allowing to address different optical +transitions. +Here we propose and investigate such a method, based on a nanoparticle (NP) composite. +The key idea of the paper is illustrated in Fig. 1. The incident long near-ir pulse, shown by +green curve, leads to transition of electrons from valence zone to first and higher conduction +zones inside the dielectric nanostructure. The motion of these electrons is almost free (possibly +with modified effective mass). Therefore they provide a negative Drude-type contribution to the +dielectric function of the NPs 휖푖 (푡), which decreases in time due to growing density of carriers +휌 (illustrated by blue curve in Fig. 1). The plasmonic resonance is determined by the condition +2휖ℎ + 휖푖 (푡) = 0, where 휖ℎ is the dielectric function of host material. For 2휖ℎ + 휖푖(−∞) > 0, the +plasmonic resonance can be reached only for a short time range when the relative free-carried +density is close to a certain value 휌res. In this time range, the local field inside of the NPs, +proportional to 1/(2휖ℎ + 휖ℎ(푡)), shows a burst as illustrated by the red curve in Fig. 1. As +will be discussed later, this burst and associated nonlinear processes lead to generation of short +pulses at new frequencies (above the frequency of the pump field). All of the above aims: short +near-UV pulse generation directly from long IR pulses, generation in situ at the position on NPs, +as well as spectral tunability, are met by this design. + + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 +-20 -15 -10 -5 + 0 + 5 10 15 20 + 0 + 0.25 + 0.5 + 0.75 + 1 +I/I0 +ρ +t (fs) +* +ρres +Fig. 1. +The schematic representation of the key idea of the investigation. +In a +nanocomposite, the intensity inside the spherical nanoparticles (red curve) can be high +compared to incident intensity (green curve) during the short time range when the +plasmonic resonance (indicated by the yellow asterisk) is reached. The resonance +takes place when the relative plasma density (blue curve) crosses the resonant value +휌res indicated by the dashed black curve. +The paper is organized as follows. In section 2, we present the physical model used for +simulation of nonlinear pulse propagation. In section 3, we show the results regarding the +nonlinear dynamics and short pulse generation, both in time and spectral domains. In section 4, +we discuss the tunability of short pulses and the responsible mechanisms. The summary of the +results is provided in the conclusion. +2. +Theoretical model +The model used in this paper is based on the formalism described in our recent work (for +details, see Ref. [25] and discussion therein), with important modifications pertinent to the +time-dependent contribution of free-carrier plasma. +We consider a composite consisting of a homogeneous host material and spherical NPs, +randomly distributed in space, with diameter well below the light wavelength so that effective- +medium theory can be applied. The following effects are included into account: linear dispersion +including intrinsic and scattering losses, second- and third-order optical nonlinearities, as well +as photoionization accompanied by ionization losses and plasma dynamics. A unidirectional +(1+1)D propagation equation [26,27] is the most suitable for this kind of situations: +휕퐸(푧, 휔) +휕푧 += +−푖 [푛eff(휔) − 푛푔(휔0)]휔 +푐 +퐸(푧, 휔) +− +푖휔 +2푐푛eff(휔0) 푃NL(푧, 휔), +(1) +where 퐸(푧, 휔) is the Fourier transform of the electric field 퐸(푧,푡), 푧 is the propagation coordinate, +푛eff(휔) is the refractive index, 푛푔(휔) is the group refractive index, 휔0 is a characteristic +frequency of the pulse spectrum, and 푃NL(푧, 휔) is the Fourier transform of the nonlinear part of +the polarization. No slowly-varying envelope approximation is used, and 퐸(푧, 푡) represents the +real-valued field including the carrier oscillations. This approach provides a unified treatment +for a pulse with an arbitrary spectral content. +The effective-mediumtheory allows to describe the nanocompositematerial as a homogenised +medium with appropriately defined effective material parameters. For low volume filling frac- +tions of the spherical NPs 푓 and moderate scattering loss, the effective refractive index of a + +-20 +-10 + 0 + 10 + 20 +-40 +-20 + 0 + 20 + 40 + 0 + 0.25 + 0.5 + 0.75 + 1 +E (GV/m) +ρ +t (fs) +(a) +-20 +-10 + 0 + 10 + 20 +-40 +-20 + 0 + 20 + 40 + 0 + 0.25 + 0.5 + 0.75 + 1 +E (GV/m) +ρ +t (fs) +(d) +-20 +-10 + 0 + 10 + 20 +-40 +-20 + 0 + 20 + 40 + 0 + 0.25 + 0.5 + 0.75 + 1 +E (GV/m) +ρ +t (fs) +(g) +-3 +-2 +-1 + 0 + 1 + 2 + 3 +-10 -8 -6 -4 -2 0 2 4 6 8 10 +E (GV/m) +t (fs) +(c) +-3 +-2 +-1 + 0 + 1 + 2 + 3 +-10 -8 -6 -4 -2 0 2 4 6 8 10 +E (GV/m) +t (fs) +(f) +-3 +-2 +-1 + 0 + 1 + 2 + 3 +-10 -8 -6 -4 -2 0 2 4 6 8 10 +E (GV/m) +t (fs) +(i) +10-6 +10-4 +10-2 +1 + 0 + 1 + 2 + 3 + 4 +E (GV/m) +ω/ω0 +(b) +10-6 +10-4 +10-2 +1 + 0 + 1 + 2 + 3 + 4 +E (GV/m) +ω/ω0 +(e) +10-6 +10-4 +10-2 +1 + 0 + 1 + 2 + 3 + 4 +E (GV/m) +ω/ω0 +(h) +Fig. 2. The temporal properties [left column, (a),(d),(g)], spectra of the average field +Eav [middle column, (b),(e),(h)], and temporal profiles generated short pulses [right +column, (c),(f),(i)] for propagation distances of 15 nm [top row, (a),(b),(c)], 45 nm +[middle row, (d),(e),(f)], 105 nm [bottom row, (g),(h),(i)]. Input pulse centered at 800 +nm has a duration of 25 fs and intensity of 25 TW/cm2. A composite of AlN NPs with +identical radius of 2.5 nm and filling factor of 푓 = 0.003 in SiO2 host is considered. +In left column, composite-averaged electric field (green curve), local field inside the +NPs (red curve), and relative plasma density (blue curve) are shown. +composite is given by [27] +푛eff = √휖 ℎ + 3 +2 +푓 (휖푖 − 휖ℎ) +2휖ℎ + 휖푖 ++ 푖√휖ℎ푐 +� 휖ℎ − 휖푖 +2휖ℎ + 휖푖 +�2 �푟NP휔 +푐 +�3 +, +(2) +where 푟NP is the NPs radius, and 휖ℎ,푖 are the dielectric functions of the host and of the NPs, +correspondingly. +We consider the situation when the presence of plasma leads to significant Drude-type modi- +fication of the dielectric susceptibility of the NPs: +휖푖 → 휖푖 − +푁휌(푡)푒2 +휖0푚푒휔(휔 + 푖휈) , +(3) +where 푁 is the density of the neutral atoms or molecules before the ionization, 휌(푡) is the time- +dependent relative density of the plasma, 푚푒 is the effective electron mass in the conduction +zone, and 휈 is the collision rate of the conduction electrons. Note that we consider composites +with ionization rate in host material lower than that in NPs and neglect the plasma contribution +to 휖ℎ. +One can see that the above Eq. (3) combines quantities defined both in frequency (휔) and +time (푡) domain. This issue can be resolved by formally substituting 휔 → 푖휕푡. Let us consider +the ratio 푥 = 퐸loc/퐸av between the local field inside the NPs 퐸loc, and the average field in the +composite 퐸av. For stationary materials (i.e., those without time-dependent parameters), it is +given by: +푥(휔) = +3휖ℎ(휔) +2휖ℎ(휔) + 휖푖(휔) . +(4) + +In a general time-dependent case, knowing 푥 allows to directly calculate the factor 1/(2휖ℎ(휔) + +휖푖(휔)) = 푥/(3휖ℎ) which appears in Eq. (2). Combining Eq. (3) and (4), we obtain +� 휕2 +휕푡2 + 휈 휕 +휕푡 +� +퐸loc(푡) = 휅휌(푡)퐸loc(푡) + +3휖ℎ +2휖ℎ + 휖푖 +� 휕2 +휕푡2 + 휈 휕 +휕푡 +� +퐸av(푡), +(5) +휕휌(푡) +휕푡 += Γ(퐸loc(푡)), +(6) +where Γ is the ionization rate (see more details below). +We have neglected the frequency +dependence of 휖ℎ and 휖푖 in this term (but not in linear polarization) for pump frequency well +below the bandgap. For each position in 푧, 퐸av(푡) is a known function determined from the +propagation equation (1). This system of equations is the key novel contribution of the presented +model. It allows to calculate the time-dependent ratio 푥 = 퐸loc/퐸av including the contribution +of plasma. +The second equation in the above system describes the photoionization inside of NPs, which is +induced by the local field inside of the NPs 퐸loc(푡). Due to high relevance of the photoionization +for the considered process, it is critically important to develop an accurate formalism for the +ionization rate in agreement with the experiment for both of the NP materials considered in +this paper: AlN and ZnO. Two models for the photoionization were considered: one based on +Ivanov-Yudin formalism [28] and one based on ADK formula [29]. In femtosecond regime the +damage threshold (DT) is associated [30] with the intensity at which the plasma contribution +would lead to strong backreflection in bulk (corresponding to bulk dielectric function near zero), +which allows us to benchmark the models by the experimental data. For AlN, data regarding +the DT is available for several values of the pulse duration [31, 32], suggesting not the typical +휏1/2 [33] but a 휏1/4 [34] law for increase of DT with pulse duration. Our calculations show that +while Ivanov-Yudin model provides poor agreement with the experiment, the ADK formula is +surprisingly accurate in predicting the DT for a range of the pulse durations, therefore ADK +photoionization rate was used, with an insignificant phenomenological pre-factor of 1.35. For +ZnO, less data regarding the DT is available [35], therefore for ZnO we augmented the ADK +rate by a first-principle calculation of the ionization rate based on the numerical solution of +the time-dependent 3D Schrodinger equation in single active electron approximation [36]. In +this approach the empirical pseudopotential method was used for the electron band structure of +ZnO [37]. These calculations provided outstanding agreement with the available experimental +data, with numericalthreshold differing fromthe experimentalone by 5%. Thereforea pre-factor +obtained from the time-dependent 3D Schrodinger equation was utilized for the photionization +rate Γ(퐸loc(푡)). +The second- and third-order nonlinear processes can also be described in the framework of +the effective-medium theory. The expressions for the effective second- and third-order order +susceptibility in a stationary medium look like [38] +휒(2) +eff (휔1 = 휔2 + 휔3; 휔2, 휔3) = (1 − 푓 )휒(2) +ℎ ++ 푓 푥(휔1)푥(휔2)푥(휔3)휒(2) +푖 +, +(7) +휒(3) +eff (휔1 = 휔2 + 휔3 + 휔4; 휔2, 휔3, 휔4) = (1 − 푓 )휒(3) +ℎ ++ +푓 푥(휔1)푥(휔2)푥(휔3)푥(휔4)휒(3) +푖 +, +(8) +where 휒(2) +ℎ +and 휒(2) +푖 +are the susceptibilities of host and NP materials, correspondingly. Note +that we neglected the frequency dependence of the bulk susceptibilities of host and NPs, which + +is a good assumption far from bulk resonances, as well as thermal effects which happen on +picosecond time scale. For the considered dynamic case, we derive the following expressions +for the second- and third-order polarizations: +푃(2) (푡) = (1 − 푓 )휒(2) +ℎ 퐸av(푡)2 + 푓 휒(2) +푖 +푥퐸loc(푡)2, +(9) +푃(3) (푡) = (1 − 푓 )휒(2) +ℎ 퐸av(푡)3 + 푓 휒(3) +푖 +푥퐸loc(푡)3. +(10) +We solve the propagation equation by an extended split-step method, whereby each of the +contributionsto the polarization is treated subsequently,which allows to reducethe accumulation +of numerical error, using the Runge-Kutta method of the order 4. Fixed step of the grid both in +time and in the propagation coordinate is used. The appearance of numerical artifacts during the +propagation is monitored by tracing the total pulse energy as well as the total energy absorbed +at the boundaries of the numerical time window. +3. +Numerical results +In Fig. +2, the numerical results for 25-fs, 25 TW/cm2 pulses at 800 nm propagating in a +composite of AlN particles (volume filling fraction 푓 = 0.003) in SiO2 host are presented. We +have used availableSellmeyer-typeexpressions to model the dispersion of both materials [39,40], +phenomenological values of the nonlinear susceptibilites [41–43], and the bandgap of 6.01 eV +for the AlN [44]. Note that the fluence of above pulses is below the DT for fused silica of 1 +J/cm2, suggesting that ionization will be predominantly happening in the AlN NPs. We note +parenthetically that absence or presence of backreflection is determined by effective refractive +index, therefore for AlN NP composite (as opposed to bulk AlN) due to low filling factor 푓 even +significant levels of relative ionization in NPs will not lead to backreflection, and damage can +be avoided even for 휌 ∼ 1. +From blue curves in left column in Fig. 2, one can see that indeed high levels of relative +ionization are reached during the pulse. Slightly before the maximum of the pulse, the system +passes through the plasmonic resonance, which manifests itself as a sharp peak of the field +inside the NPs Eloc (red curve) as compared to the average field Eav (green curve). At later +stages of propagation, the input pulse is modified and depleted in the center of the pulse due to +photoionization, as can be seen in Fig. 2(g). In the spectral domain, at the initial stage of the +propagation a pronounced peak is formed at roughly (but not exactly) double the input frequency +휔0, which later broadens and extends to higher frequencies. In the right column of Fig. 2, +we show the temporal profile corresponding to this higher-frequency spectral components, by +leaving only the spectral range from 1.2휔0 to 3.5휔0. It is important to note that we do not +calculate the Fourier-limited pulse, rather, all the spectral phases which result from propagation +are preserved. One can see that after only 35 nm of propagation, a well-isolated short pulse +with a FWHM duration of 1.9 fs and weak pedestal is formed, with energy efficiency of +1.2% (corresponding to the efficiency determined from the peak field ratio of roughly 14%). +Subsequent propagation, as illustrated in Fig. 2(i), shows further efficiency increase, which is +however accompanied by longer and less regular pulse shape. +4. +Pulse tunability and generation mechanism +We explored the possibility to influence the position of the spectral peak visible in Fig. 2(b) by +varying the pump pulse intensity and, correspondingly, the relative density of plasma after the +pulse, 휌(+∞). In Fig. 3(a) we show that for different pump intensities it is possible to shift the +peak (and the corresponding short pulse) in a significant spectral range, from 410 to 545 nm. In +Fig. 3(b), the maximum wavelength of the peak is presented as a function of the relative density +of plasma after the pulse, 휌(+∞). + + 400 + 420 + 440 + 460 + 480 + 500 + 520 + 540 + 560 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 +λpeak (nm) +ρ +(b) + 0 + 0.5 + 1 + 350 + 400 + 450 + 500 + 550 + 600 +I(λ) (arb. units) +λ (nm) +(a) +Fig. 3. Spectra of generated short pulse for different intensities (a) and the dependence +of the central wavelength on the after-pulse plasma density (b). In (a), the intensities +of 15.045 TW/cm2, 15.05 TW/cm2, 15.075 TW/cm2, 15.2 TW/cm2, 15.5 TW/cm2, +17 TW/cm2, 25 TW/cm2 (from right to left) are considered. In (b), by solid curve the +analytical dependence given by Eq. (11) is shown. +-20 +-10 + 0 + 10 + 20 +-40 +-20 + 0 + 20 + 40 + 0 + 0.1 + 0.2 + 0.3 + 0.4 +E (GV/m) +ρ +t (fs) +(a) +-2 +-1 + 0 + 1 + 2 + 5 + 10 15 20 25 30 35 40 45 +E (GV/m) +t (fs) +(b) +10-6 +10-4 +10-2 +1 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +E (GV/m) +ω/ω0 +(c) +Fig. 4. The temporal properties (a), temporal profile of the generated short pulse (b), +and spectrum (c) for propagation distance of 10 nm. Input pulse centered at 800 nm +has a duration of 55 fs and intensity of 10 TW/cm2. A composite of ZnO NPs with +identical radius of 2.5 nm and filling factor of 0.0003 in SiO2 host is considered. In (a), +composite-averaged electric field (green curve), local field inside the NPs (red curve), +and relative plasma density (blue curve) are shown. +The mechanism responsible for the generation of this peak is highly relevant to understand its +tunability and further features. We note that it cannot be explained by the well-known plasma- +induced blue shift of the spectrum, since such shift is proportional to the nonlinear phase +accumulated during propagation, and therefore the position of the peak would be 푧-dependent, +in contradiction to the numerical findings. Also, the energy of the peak grow quadratically with +propagation length, which excludes amplification-like processes. Rather, we speculate that a +short burst at the plasmonic resonance contains many spectral components. After the plasmonic +resonance, the relative plasma density 휌(+∞) > 휌res corresponds to the plasmonic resonance +at frequency 휔∗ > 휔0. Spectral components at or around this frequency could be preserved +and grow. To confirm this conjecture, in Fig. 3(b) we plot the wavelength corresponding to the +resonant frequency after the pulse, the latter being given by +휔∗ = 0.91 +� +푁푒2 +휖0푚푒[2휖ℎ + 휖푖(+∞)] . +(11) +In order to fit the numerical data, we have introduced a prefactor of 1.21, which is justified by +the fact that the peak is generated under highly dynamical conditions with plasma density quickly +changing in time. In Fig. 3(b), the prediction given by Eq. (11) is shown by the red curve. An +almost perfect agreement with numerical results is obtained. Even with a fit parameter, such +agreement is highly indicative that proposed mechanism indeed describes the peak generation +in our system. +With the aim to investigate the influence of the material choice of the transient plasmonic +resonance, in Fig. 4 we show the numerical results for the 55-fs, 10 TW/cm2 pulses at 800 + +nm propagating in a composite of ZnO NPs (volume filling fraction 푓 = 0.0003) in SiO2 host. +Similar to the case of AlN NPs, phenomenological bulk material parameters were used [45–48]. +ZnO has a much lower bandgap of 3.37 eV, which has significant influence on the dynamics. +The dependence of the ionization rate on the intensity is smoother and does not have a strongly +pronounced threshold-like character. Therefore the growth of the relative ionization, as shown +in Fig. 4(a), occurs slower, and the systems spends a longer time in the plasmonic resonance, +as can be seen from comparison of local field (red curve) and average field (green curve) in +Fig. 4(a). Correspondingly, the generated pulse is longer with FWHM of 9.5 fs, whereas the +efficiency of 1.3% is comparable to AlN case. Despite the quantitative differences to AlN case, +the generation of the short pulse is based on the same mechanism, as can be seen from the +spectrum in Fig. 4(c) showing clear isolated feature around 1.4휔0. We would like to stress that, +despite the different ionization dynamics, for ZnO NPs the Eq. (11) provides accurate estimation +of the peak spectral position using the same fitting factor of 0.91. We conclude that short pulse +generation is possible for composites with host bandgap larger than inclusion bandgap, however, +the latter should not be below roughly 4 eV for photoionization to be threshold-like. In addition, +pedestal-free and sufficiently strong input pulses (intensity above 10 TW/cm2) are required. +5. +Conclusion +In conclusion, we have developed a model for simulation of nonlinear pulse propagation under +the condition of rapid free carrier generation. We showed that a transient plasmonic resonance +in a nanoparticle composite can lead to a very short burst of the local field inside the NPs. +We predict a direct generation of tunable few-fs near-UV pulses from much longer near-IR +pulses, with efficiencies in the range of 1%. The generation mechanism is connected to growth +of spectral components which are in plasmonic resonance after the pulse peak. The above +nonlinear dynamics is explored for two NP materials, AlN and ZnO. +Funding. +Authors acknowledge financial support from European Union project H2020-MSCA-RISE- +2018-823897 "Atlantic". I.B. thanks Cluster of Excellence PhoenixD (EXC 2122, project ID 390833453) +for financial support. Support from the BNSF under Contract No. KP-06-COST/7 is acknowledged (T.A.) +Disclosures. +The authors declare no conflicts of interest. +Data availability. +Data underlying the results presented in this paper are not publicly available at this +time but may be obtained from the authors upon reasonable request. +References +1. +A. Zewail, “Femtochemistry: Atomic-scale dynamics of the chemical bond,” J. Phys. Chem. A 104, 5660–5694 +(2000). +2. +F. Remacle and R. D. Levine, “An electronic timescale in chemistry,” PNAS 103, 6793–6798 (2005). +3. +F. Remacle, M. Nest, and R. D. Levine, "Laser Steered Ultrafast Quantum Dynamics of Electrons in LiH", Phys. +Rev. Lett. 99, 183902 (2007). +4. +S. Lunnemann, A. I. Kuleff, and L. S. Cederbaum, "Ultrafast charge migration in 2-phenylethyl-N,N-dimethylamine", +Chem. Phys. Lett. 450, 232-235 (2008). +5. +A. Nenov, I. Rivalta, G. Cerullo, S. Mukamel, and M. 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Ya Gayvoronsky, and R. +Le Dantec, "Averaged third-order susceptibility of ZnO nanocrystals from Third Harmonic Generation and Third +Harmonic Scattering", Optical Materials 84, 579-585 (2018). + diff --git a/u9E4T4oBgHgl3EQfWwwm/content/tmp_files/load_file.txt b/u9E4T4oBgHgl3EQfWwwm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20eae8b1b65555f6b410c33e8a9de5323a1f7875 --- /dev/null +++ b/u9E4T4oBgHgl3EQfWwwm/content/tmp_files/load_file.txt @@ -0,0 +1,777 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf,len=776 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='05035v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='optics] 12 Jan 2023 Tunable in situ Near-UV Pulses by Transient Plasmonic Resonance in Nanocomposites ANTON HUSAKOU,1,* IHAR BABUSHKIN,1,2,3 OLGA FEDOTOVA,4 RYHOR RYSETSKY,4 TATSIANA SMIRNOVA,5 OLEG KHASANOV,4 ALEXANDER FEDOTOV,5 USMAN SAPAEV,6 TZVETA APOSTOLOVA7,8 1Max Born Institute, Max Born Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 12489 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Germany 2Institute of Quantum Optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Leibnitz Hannover University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Welfengarten 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 30167 Hannover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Germany 3Cluster of Excellence PhoenixD (Photonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' and Engineering – Innovation Across Disciplines),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Welfengarten 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 30167 Hannover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Germany 4 Scientific and Practical Materials Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Belarus NAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Brovky 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 220072 Minsk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Belarus 5Belarus State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Niezalie˘znasci avenue 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 220030 Minsk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Belarus 6Tashkent State Technical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2 uy 2 Qatartol ko’chasi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 100097 Tashkent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Uzbekistan 7Institute for Nuclear Research and Nuclear Energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Bulgarian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Tsarigradsko Chausse 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 1784 Sofia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Bulgaria 8Institute for Advanced Physical Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' New Bulgarian University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 1618 Sofia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Bulgaria gusakov@mbi-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='de Abstract: We propose a new concept for generation of ultrashort pulses based on transient plasmonic resonance in nanoparticle composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Photoionization and free-carriers plasma change the susceptibility of nanoparticles on a few-femtosecond scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' This results in a narrow time window during the pump pulse duration when the system is in plasmonic resonance, accompanied by a short burst of the local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' During this process, frequency-tunable few-fs pulses are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We elucidate the details of the above mechanism, and investigate the influences of different contributing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' © 2023 Optica Publishing Group 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Introduction Numerous fields of modern ultrafast optics, such as tracing of atomic motion in molecules [1], chemistry on electronic timescale [2], steering of ultrafast electron dynamics in the valence shell of solids, nanoparticles, and clusters [3,4],two-dimensional electronic spectroscopy experiments [5], generation and diagnosis of warm-dense matter [6], material modification [7] and so on, require short, sub-10-fs intense pulses at UV or near-UV frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The ultraviolet (UV) wavelength range is of great interest for ultrafast spectroscopic investigations because of the possible resonance with electronic transitions of many small molecules with fairly simple excited state energy level structure, whose photo-induced dynamics can be accurately modeled using ab initio computational approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Transient absorption spectroscopy in the UV can thus be used for the study of the optical response of biomolecules and can benchmark the accuracy of such methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In the visible and near infrared ranges, ultrashort pulses are routinely generated using different approaches: directly from a laser oscillator [8], by a non-collinear optical parametric amplifier [9], or by spectral broadening in a nonlinear medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' solid or hollow-core optical fiber) [10] based on self-phase modulation due to the optical Kerr-effect [11] or time-dependent plasma density [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The extension of these techniques to the UV range is challenging due to the lack of broadband laser gain media in the UV (except for excimers of noble gas halides [14]) and strong two-photon absorption for high-energy photons [15] required for UV optical parametric amplifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For these reasons, broadband UV pulses are typically generated in a two-step approach: first, few-optical-cycle pulses in the visible or near-infrared ranges are generated, and then nonlinear frequency up-conversion is used to reach the UV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For frequency up-conversion in gases, the usually used nonlinear processes are third- or higher- order harmonic generation or four-wave-mixing between the fundamental wavelength and the second harmonic of the driving pulse, which is typically obtained from a Ti:sapphire laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' To name a few examples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 16-fs pulses at 266 nm were produced by third-harmonic generation in air from the 20-fs pulses [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' much shorter sub-4-fs pulses at 270 nm were generated in Ne from the 6-fs laser output spectrally broadened in a hollow-core fiber [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 8-fs pulses at 266 nm were generated by four-wave-mixing between the second harmonic and the fundamental wavelength from a 20-fs pulse [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' and 5-nJ pulses at 133 nm were generated by cascaded four-wave-mixing between third,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' fourth and fifth harmonic in a filament [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' UV pulses were also generated by second-order nonlinear frequency conversion in crystals, typically 훽-barium borate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For example, sub-10 fs 400 nm pulses were reported [20], with sufficient spectral bandwidth obtained via frequency-doubling using broadband phase-matching enabled by grating recollimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Furthermore, UV pulses with 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='7 fs pulse duration were generated by second harmonic generation using a spectrally shaped 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='9 mJ, 8 fs few-cycle near-infrared pulses [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Recently, new trends appeared in the generation of UV pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Resonant dispersive-wave generation based on Kerr nonlinearlity during the optical soliton propagation in waveguides combines intrinsically short pulse duration and easy tunability over the full ultraviolet spectral range as well as the entire visible spectrum when using infrared pump pulses, as was recently demonstrated [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' These advances also raise the prospect of compact on-chip integrated nonlinear devices [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Due to a persistently high demand for short pulses in near-UV range and adjacent visible frequencies, it is promising and timely to search and investigate alternative approaches to their generation, preferably directly from relatively long infrared pulses without compression step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' A possibility to generate short UV pulses by a very small device in situ would be an important and highly desirable feature, particularly in view of biological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Indeed, during propagation in any transparent condensed matter, a ∼10-fs pulse will very quickly, on the sub-mm scale, become much longer due to group-velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Pre-compensation of group-velocity dispersion is in principle possible but challenging and requires precise a priori knowledge of the material properties, which is rarely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Therefore it would be highly valuable to suggest a technique which allows generation of short pulse directly inside of a transparent material at the desired position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' On top of that, spectral tunability of the pulses would strongly enhance their application potential, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' by allowing to address different optical transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Here we propose and investigate such a method, based on a nanoparticle (NP) composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The key idea of the paper is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The incident long near-ir pulse, shown by green curve, leads to transition of electrons from valence zone to first and higher conduction zones inside the dielectric nanostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The motion of these electrons is almost free (possibly with modified effective mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Therefore they provide a negative Drude-type contribution to the dielectric function of the NPs 휖푖 (푡), which decreases in time due to growing density of carriers 휌 (illustrated by blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The plasmonic resonance is determined by the condition 2휖ℎ + 휖푖 (푡) = 0, where 휖ℎ is the dielectric function of host material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For 2휖ℎ + 휖푖(−∞) > 0, the plasmonic resonance can be reached only for a short time range when the relative free-carried density is close to a certain value 휌res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In this time range, the local field inside of the NPs, proportional to 1/(2휖ℎ + 휖ℎ(푡)), shows a burst as illustrated by the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' As will be discussed later, this burst and associated nonlinear processes lead to generation of short pulses at new frequencies (above the frequency of the pump field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' All of the above aims: short near-UV pulse generation directly from long IR pulses, generation in situ at the position on NPs, as well as spectral tunability, are met by this design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 3 20 -15 -10 -5 0 5 10 15 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='75 1 I/I0 ρ t (fs) ρres Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The schematic representation of the key idea of the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In a nanocomposite, the intensity inside the spherical nanoparticles (red curve) can be high compared to incident intensity (green curve) during the short time range when the plasmonic resonance (indicated by the yellow asterisk) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The resonance takes place when the relative plasma density (blue curve) crosses the resonant value 휌res indicated by the dashed black curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In section 2, we present the physical model used for simulation of nonlinear pulse propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In section 3, we show the results regarding the nonlinear dynamics and short pulse generation, both in time and spectral domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In section 4, we discuss the tunability of short pulses and the responsible mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The summary of the results is provided in the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Theoretical model The model used in this paper is based on the formalism described in our recent work (for details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' [25] and discussion therein), with important modifications pertinent to the time-dependent contribution of free-carrier plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We consider a composite consisting of a homogeneous host material and spherical NPs, randomly distributed in space, with diameter well below the light wavelength so that effective- medium theory can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The following effects are included into account: linear dispersion including intrinsic and scattering losses, second- and third-order optical nonlinearities, as well as photoionization accompanied by ionization losses and plasma dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' A unidirectional (1+1)D propagation equation [26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='27] is the most suitable for this kind of situations: 휕퐸(푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔) 휕푧 = −푖 [푛eff(휔) − 푛푔(휔0)]휔 푐 퐸(푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔) − 푖휔 2푐푛eff(휔0) 푃NL(푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (1) where 퐸(푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔) is the Fourier transform of the electric field 퐸(푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='푡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 푧 is the propagation coordinate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 푛eff(휔) is the refractive index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 푛푔(휔) is the group refractive index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔0 is a characteristic frequency of the pulse spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' and 푃NL(푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔) is the Fourier transform of the nonlinear part of the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' No slowly-varying envelope approximation is used, and 퐸(푧, 푡) represents the real-valued field including the carrier oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' This approach provides a unified treatment for a pulse with an arbitrary spectral content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The effective-mediumtheory allows to describe the nanocompositematerial as a homogenised medium with appropriately defined effective material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For low volume filling frac- tions of the spherical NPs 푓 and moderate scattering loss, the effective refractive index of a 20 10 0 10 20 40 20 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='75 1 E (GV/m) ρ t (fs) (a) 20 10 0 10 20 40 20 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='75 1 E (GV/m) ρ t (fs) (d) 20 10 0 10 20 40 20 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='75 1 E (GV/m) ρ t (fs) (g) 3 2 1 0 1 2 3 10 -8 -6 -4 -2 0 2 4 6 8 10 E (GV/m) t (fs) (c) 3 2 1 0 1 2 3 10 -8 -6 -4 -2 0 2 4 6 8 10 E (GV/m) t (fs) (f) 3 2 1 0 1 2 3 10 -8 -6 -4 -2 0 2 4 6 8 10 E (GV/m) t (fs) (i) 10-6 10-4 10-2 1 0 1 2 3 4 E (GV/m) ω/ω0 (b) 10-6 10-4 10-2 1 0 1 2 3 4 E (GV/m) ω/ω0 (e) 10-6 10-4 10-2 1 0 1 2 3 4 E (GV/m) ω/ω0 (h) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The temporal properties [left column, (a),(d),(g)], spectra of the average field Eav [middle column, (b),(e),(h)], and temporal profiles generated short pulses [right column, (c),(f),(i)] for propagation distances of 15 nm [top row, (a),(b),(c)], 45 nm [middle row, (d),(e),(f)], 105 nm [bottom row, (g),(h),(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Input pulse centered at 800 nm has a duration of 25 fs and intensity of 25 TW/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' A composite of AlN NPs with identical radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 nm and filling factor of 푓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='003 in SiO2 host is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In left column, composite-averaged electric field (green curve), local field inside the NPs (red curve), and relative plasma density (blue curve) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' composite is given by [27] 푛eff = √휖 ℎ + 3 2 푓 (휖푖 − 휖ℎ) 2휖ℎ + 휖푖 + 푖√휖ℎ푐 � 휖ℎ − 휖푖 2휖ℎ + 휖푖 �2 �푟NP휔 푐 �3 , (2) where 푟NP is the NPs radius, and 휖ℎ,푖 are the dielectric functions of the host and of the NPs, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We consider the situation when the presence of plasma leads to significant Drude-type modi- fication of the dielectric susceptibility of the NPs: 휖푖 → 휖푖 − 푁휌(푡)푒2 휖0푚푒휔(휔 + 푖휈) , (3) where 푁 is the density of the neutral atoms or molecules before the ionization, 휌(푡) is the time- dependent relative density of the plasma, 푚푒 is the effective electron mass in the conduction zone, and 휈 is the collision rate of the conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Note that we consider composites with ionization rate in host material lower than that in NPs and neglect the plasma contribution to 휖ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' One can see that the above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (3) combines quantities defined both in frequency (휔) and time (푡) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' This issue can be resolved by formally substituting 휔 → 푖휕푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Let us consider the ratio 푥 = 퐸loc/퐸av between the local field inside the NPs 퐸loc, and the average field in the composite 퐸av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For stationary materials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=', those without time-dependent parameters), it is given by: 푥(휔) = 3휖ℎ(휔) 2휖ℎ(휔) + 휖푖(휔) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (4) In a general time-dependent case, knowing 푥 allows to directly calculate the factor 1/(2휖ℎ(휔) + 휖푖(휔)) = 푥/(3휖ℎ) which appears in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (3) and (4), we obtain � 휕2 휕푡2 + 휈 휕 휕푡 � 퐸loc(푡) = 휅휌(푡)퐸loc(푡) + 3휖ℎ 2휖ℎ + 휖푖 � 휕2 휕푡2 + 휈 휕 휕푡 � 퐸av(푡), (5) 휕휌(푡) 휕푡 = Γ(퐸loc(푡)), (6) where Γ is the ionization rate (see more details below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We have neglected the frequency dependence of 휖ℎ and 휖푖 in this term (but not in linear polarization) for pump frequency well below the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For each position in 푧, 퐸av(푡) is a known function determined from the propagation equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' This system of equations is the key novel contribution of the presented model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' It allows to calculate the time-dependent ratio 푥 = 퐸loc/퐸av including the contribution of plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The second equation in the above system describes the photoionization inside of NPs, which is induced by the local field inside of the NPs 퐸loc(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Due to high relevance of the photoionization for the considered process, it is critically important to develop an accurate formalism for the ionization rate in agreement with the experiment for both of the NP materials considered in this paper: AlN and ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Two models for the photoionization were considered: one based on Ivanov-Yudin formalism [28] and one based on ADK formula [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In femtosecond regime the damage threshold (DT) is associated [30] with the intensity at which the plasma contribution would lead to strong backreflection in bulk (corresponding to bulk dielectric function near zero), which allows us to benchmark the models by the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For AlN, data regarding the DT is available for several values of the pulse duration [31, 32], suggesting not the typical 휏1/2 [33] but a 휏1/4 [34] law for increase of DT with pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Our calculations show that while Ivanov-Yudin model provides poor agreement with the experiment, the ADK formula is surprisingly accurate in predicting the DT for a range of the pulse durations, therefore ADK photoionization rate was used, with an insignificant phenomenological pre-factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For ZnO, less data regarding the DT is available [35], therefore for ZnO we augmented the ADK rate by a first-principle calculation of the ionization rate based on the numerical solution of the time-dependent 3D Schrodinger equation in single active electron approximation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In this approach the empirical pseudopotential method was used for the electron band structure of ZnO [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' These calculations provided outstanding agreement with the available experimental data, with numericalthreshold differing fromthe experimentalone by 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Thereforea pre-factor obtained from the time-dependent 3D Schrodinger equation was utilized for the photionization rate Γ(퐸loc(푡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The second- and third-order nonlinear processes can also be described in the framework of the effective-medium theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The expressions for the effective second- and third-order order susceptibility in a stationary medium look like [38] 휒(2) eff (휔1 = 휔2 + 휔3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔2, 휔3) = (1 − 푓 )휒(2) ℎ + 푓 푥(휔1)푥(휔2)푥(휔3)휒(2) 푖 , (7) 휒(3) eff (휔1 = 휔2 + 휔3 + 휔4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 휔2, 휔3, 휔4) = (1 − 푓 )휒(3) ℎ + 푓 푥(휔1)푥(휔2)푥(휔3)푥(휔4)휒(3) 푖 , (8) where 휒(2) ℎ and 휒(2) 푖 are the susceptibilities of host and NP materials, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Note that we neglected the frequency dependence of the bulk susceptibilities of host and NPs, which is a good assumption far from bulk resonances, as well as thermal effects which happen on picosecond time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' For the considered dynamic case, we derive the following expressions for the second- and third-order polarizations: 푃(2) (푡) = (1 − 푓 )휒(2) ℎ 퐸av(푡)2 + 푓 휒(2) 푖 푥퐸loc(푡)2, (9) 푃(3) (푡) = (1 − 푓 )휒(2) ℎ 퐸av(푡)3 + 푓 휒(3) 푖 푥퐸loc(푡)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (10) We solve the propagation equation by an extended split-step method, whereby each of the contributionsto the polarization is treated subsequently,which allows to reducethe accumulation of numerical error, using the Runge-Kutta method of the order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Fixed step of the grid both in time and in the propagation coordinate is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The appearance of numerical artifacts during the propagation is monitored by tracing the total pulse energy as well as the total energy absorbed at the boundaries of the numerical time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Numerical results In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2, the numerical results for 25-fs, 25 TW/cm2 pulses at 800 nm propagating in a composite of AlN particles (volume filling fraction 푓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='003) in SiO2 host are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We have used availableSellmeyer-typeexpressions to model the dispersion of both materials [39,40], phenomenological values of the nonlinear susceptibilites [41–43], and the bandgap of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='01 eV for the AlN [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Note that the fluence of above pulses is below the DT for fused silica of 1 J/cm2, suggesting that ionization will be predominantly happening in the AlN NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We note parenthetically that absence or presence of backreflection is determined by effective refractive index, therefore for AlN NP composite (as opposed to bulk AlN) due to low filling factor 푓 even significant levels of relative ionization in NPs will not lead to backreflection, and damage can be avoided even for 휌 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' From blue curves in left column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2, one can see that indeed high levels of relative ionization are reached during the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Slightly before the maximum of the pulse, the system passes through the plasmonic resonance, which manifests itself as a sharp peak of the field inside the NPs Eloc (red curve) as compared to the average field Eav (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' At later stages of propagation, the input pulse is modified and depleted in the center of the pulse due to photoionization, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In the spectral domain, at the initial stage of the propagation a pronounced peak is formed at roughly (but not exactly) double the input frequency 휔0, which later broadens and extends to higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In the right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2, we show the temporal profile corresponding to this higher-frequency spectral components, by leaving only the spectral range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='2휔0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5휔0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' It is important to note that we do not calculate the Fourier-limited pulse, rather, all the spectral phases which result from propagation are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' One can see that after only 35 nm of propagation, a well-isolated short pulse with a FWHM duration of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='9 fs and weak pedestal is formed, with energy efficiency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='2% (corresponding to the efficiency determined from the peak field ratio of roughly 14%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Subsequent propagation, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2(i), shows further efficiency increase, which is however accompanied by longer and less regular pulse shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Pulse tunability and generation mechanism We explored the possibility to influence the position of the spectral peak visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 2(b) by varying the pump pulse intensity and, correspondingly, the relative density of plasma after the pulse, 휌(+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 3(a) we show that for different pump intensities it is possible to shift the peak (and the corresponding short pulse) in a significant spectral range, from 410 to 545 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 3(b), the maximum wavelength of the peak is presented as a function of the relative density of plasma after the pulse, 휌(+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 400 420 440 460 480 500 520 540 560 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='9 λpeak (nm) ρ (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 1 350 400 450 500 550 600 I(λ) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' units) λ (nm) (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Spectra of generated short pulse for different intensities (a) and the dependence of the central wavelength on the after-pulse plasma density (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In (a), the intensities of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='045 TW/cm2, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='05 TW/cm2, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='075 TW/cm2, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='2 TW/cm2, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 TW/cm2, 17 TW/cm2, 25 TW/cm2 (from right to left) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In (b), by solid curve the analytical dependence given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (11) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 20 10 0 10 20 40 20 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='4 E (GV/m) ρ t (fs) (a) 2 1 0 1 2 5 10 15 20 25 30 35 40 45 E (GV/m) t (fs) (b) 10-6 10-4 10-2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 E (GV/m) ω/ω0 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The temporal properties (a), temporal profile of the generated short pulse (b), and spectrum (c) for propagation distance of 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Input pulse centered at 800 nm has a duration of 55 fs and intensity of 10 TW/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' A composite of ZnO NPs with identical radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 nm and filling factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='0003 in SiO2 host is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In (a), composite-averaged electric field (green curve), local field inside the NPs (red curve), and relative plasma density (blue curve) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The mechanism responsible for the generation of this peak is highly relevant to understand its tunability and further features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We note that it cannot be explained by the well-known plasma- induced blue shift of the spectrum, since such shift is proportional to the nonlinear phase accumulated during propagation, and therefore the position of the peak would be 푧-dependent, in contradiction to the numerical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Also, the energy of the peak grow quadratically with propagation length, which excludes amplification-like processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Rather, we speculate that a short burst at the plasmonic resonance contains many spectral components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' After the plasmonic resonance, the relative plasma density 휌(+∞) > 휌res corresponds to the plasmonic resonance at frequency 휔∗ > 휔0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Spectral components at or around this frequency could be preserved and grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' To confirm this conjecture, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 3(b) we plot the wavelength corresponding to the resonant frequency after the pulse, the latter being given by 휔∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='91 � 푁푒2 휖0푚푒[2휖ℎ + 휖푖(+∞)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (11) In order to fit the numerical data, we have introduced a prefactor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='21, which is justified by the fact that the peak is generated under highly dynamical conditions with plasma density quickly changing in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 3(b), the prediction given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (11) is shown by the red curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' An almost perfect agreement with numerical results is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Even with a fit parameter, such agreement is highly indicative that proposed mechanism indeed describes the peak generation in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' With the aim to investigate the influence of the material choice of the transient plasmonic resonance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 4 we show the numerical results for the 55-fs, 10 TW/cm2 pulses at 800 nm propagating in a composite of ZnO NPs (volume filling fraction 푓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='0003) in SiO2 host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Similar to the case of AlN NPs, phenomenological bulk material parameters were used [45–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' ZnO has a much lower bandgap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='37 eV, which has significant influence on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The dependence of the ionization rate on the intensity is smoother and does not have a strongly pronounced threshold-like character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Therefore the growth of the relative ionization, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 4(a), occurs slower, and the systems spends a longer time in the plasmonic resonance, as can be seen from comparison of local field (red curve) and average field (green curve) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Correspondingly, the generated pulse is longer with FWHM of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='5 fs, whereas the efficiency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='3% is comparable to AlN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Despite the quantitative differences to AlN case, the generation of the short pulse is based on the same mechanism, as can be seen from the spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 4(c) showing clear isolated feature around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='4휔0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We would like to stress that, despite the different ionization dynamics, for ZnO NPs the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' (11) provides accurate estimation of the peak spectral position using the same fitting factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We conclude that short pulse generation is possible for composites with host bandgap larger than inclusion bandgap, however, the latter should not be below roughly 4 eV for photoionization to be threshold-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' In addition, pedestal-free and sufficiently strong input pulses (intensity above 10 TW/cm2) are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Conclusion In conclusion, we have developed a model for simulation of nonlinear pulse propagation under the condition of rapid free carrier generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We showed that a transient plasmonic resonance in a nanoparticle composite can lead to a very short burst of the local field inside the NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' We predict a direct generation of tunable few-fs near-UV pulses from much longer near-IR pulses, with efficiencies in the range of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The generation mechanism is connected to growth of spectral components which are in plasmonic resonance after the pulse peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The above nonlinear dynamics is explored for two NP materials, AlN and ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Authors acknowledge financial support from European Union project H2020-MSCA-RISE- 2018-823897 "Atlantic".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' thanks Cluster of Excellence PhoenixD (EXC 2122, project ID 390833453) for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Support from the BNSF under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' KP-06-COST/7 is acknowledged (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=') Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E4T4oBgHgl3EQfWwwm/content/2301.05035v1.pdf'} +page_content=' Data availability.' metadata={'source': 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b/v9E2T4oBgHgl3EQfggeM/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:312504dd0608a8da0cc364040ae7663206e957a2f937d05dc25f9777d932f2bc +size 362326 diff --git a/wdAzT4oBgHgl3EQfCPqo/content/tmp_files/2301.00956v1.pdf.txt b/wdAzT4oBgHgl3EQfCPqo/content/tmp_files/2301.00956v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fd7396fa24090fa8c0387faaa5b2c36746cfbcd --- /dev/null +++ b/wdAzT4oBgHgl3EQfCPqo/content/tmp_files/2301.00956v1.pdf.txt @@ -0,0 +1,1157 @@ +arXiv:2301.00956v1 [gr-qc] 3 Jan 2023 +Shadows of quintessential dark energy black holes in the domain +of outer communication +Balendra Pratap Singha∗ +aDepartment of Applied Sciences and Engineering, +Tula’s Institute, Dehradun, Uttarakhand 248197, India +Abstract +The rotating black holes in the quintessential dark energy correspond to three horizons: inner, +outer, and quintessential horizon. The domain of outer communication is the region between outer +and quintessential horizon. Here, in this work we study the photon region and shadows of the +quintessential dark energy black holes when the observer stays statically in the domain of outer +communication. The quintessential dark energy black holes shadow characterizes by its mass (M), +spin parameter (a), quintessential dark energy parameter (ωq), and normalization factor (γ). The +dark energy parameter ωq can take values in between −1.1 < ωq < −1/3 and follows the equation +of state ωq=pressure(p)/energy density(ρq). This state parameter significantly affects the shape +and size of the black hole shadow. We generalize all the geodesic equations of motion for ωq and +obtain relation to visualize the black hole shadow by a static observer at any arbitrary distance in +the domain of outer communication. We analytically estimate the black hole shadow observables: +radius Rs, distortion parameter δs and the shadow area A. Using the numerical values of shadow +radius Rs and area A, we obtain the angular diameter of the black hole shadow. The angular +size of the M87 and Sgr A∗ black holes are +42 ± 3µas and 48.7 ± 7µas respectively as observe +by Event Horizon Telescope (EHT). In this case, the angular diameter of the black hole shadow +increases with the quintessence parameter ωq and takes values θd ≈ 20 ± 3o with the parameter +−0.66 ≤ ωq ≤ −0.62 for the static observer at ro = 5M in the domain of outer communication. +PACS numbers: +∗Electronic address: balendra29@gmail.com +1 + +I. +INTRODUCTION +The observations of type Ia supernovae (SNe Ia) lie between the red shift range 0.16 ≤ z ≤ +0.62, confirming that our Universe is going under the late time acceleration [1, 2]. According +to general relativity, this cosmic acceleration indicates that there exists some strange energy +component in the Universe which is called dark energy. The observational constrains over +the state parameter ωq provided by the large state structure of the Universe and the cosmic +microwave background (CMB) is −1.1 < ωq < −1/3 [3–5]. The hypothesis of dark energy +is compatible with the standard model of big bang cosmology (ΛCDM model) when the +dark energy state parameter is exactly equal to -1. +For this value of state parameter, +the dark energy is considered as cosmological constant, which is interestingly agreements +with the observations but still there is a possibility that some significant component of the +dark energy densities have state parameters other than -1 [6]. One of the simplest is the +quintessence dark energy model in which the dynamical scalar field is minimally coupled with +the gravity [6]. The quintessence dark energy is dynamic and time-varying which is different +from the cosmological constant model which does not change with time. Some researchers +also consider the quintessence dark energy as the fifth fundamental force responsible for the +expansion of the Universe [7–9]. +The asymptotic structure of the black hole gets modified in the presence of the quintessen- +tial dark energy. +The black hole spacetime remains no more asymptotically flat in +quintessence due to the cosmological horizon. The very first model of the black hole in +quintessence was presented by Kislev [10]. After that, several researchers intensively stud- +ied the properties of spherically symmetric black holes in quintessence dark energy [11–30]. +The Lovelock black holes in quintessence have been studied by [31, 32]. The study of Nar- +nia black holes in quintessence has been done by [33]. The authors of [34] discussed the +geodesics of the Hayward black hole in quintessence. Thermodynamics of the Bardeen black +hole in quintessential dark energy studied in [35]. The rotating counterpart of the spherically +symmetric black hole in quintessence obtained by [36] and [37]. The study of rotating anti- +de-sitter and rotating charged anti-de-sitter black holes in the presence of perfect fluid matter +have been intensively studied by [38] and [30]. The author of [39] extended Schwarzschild +black holes in quintessence up to D-dimensional spacetimes and studied gravitational lensing +and shadow properties. +2 + +The observational results from the Event Horizon Telescope (EHT) proved that black +holes are just not only theoretical concepts. The first picture of the Messier 87 (M87) black +hole revealed by the EHT group in 2019 [40–45]. Black holes are completely dark objects +but interestingly they cast a shadow [46]. The incoming photons towards the black hole +horizon which have quite large angular momentum fall inside the black hole event horizon +and create a dark spot, and photons that carries a little bit smaller angular momentum +form the photon region around the black hole [46]. This dark spot surrounded with the +bright photon rings is called a black hole shadow [47]. Casting shadow by the black hole +also verifies the existence of the event horizon [40]. Recently EHT collaborators published +the image of Sgr A∗ black hole shadow. This supermassive black hole lies at the heart of our +galaxy Milky way [48–53]. The first analytical study of this subject was done by Synge [54] +and Luminet [55]. Later Bardeen extended it for the rotating spacetime [56]. In the past +few years, several researchers analytically studied the black hole shadow for rotating and +non-rotating spacetimes.[57–79]. The optical properties of the rotating and non-rotating +black holes in the presence of the plasma medium have been studied by [80–82] This subject +has been extended for the higher dimensional spacetime by [83–87]. Various researchers +estimated black hole parameters using the properties of the black hole shadow [89–93]. +The primary goal of this work is to analytically study the angular size of the photon +regions for rotating black holes in quintessential dark energy in the domain of outer com- +munication. The rotating black hole in quintessence corresponds to three horizons: inner, +outer, and quintessential or cosmological event horizons [36]. The domain of outer com- +munication is the region where the observer statically stays in between the outer and the +cosmological horizon [47]. The quintessence field affects the shadow region, and the appear- +ance of the bright photons region varies as seen by the observer. The size of the black hole +shadow decreases and gets distorted with the state parameter ωq. The photon region will +appear deviated and more close to the event horizon in the presence of the quintessential +dark energy. +This paper is organized as follows: In Sec. II, we discuss the rotating and non-rotating +black hole metric with the quintessential dark energy. In Sec. III, we derive the effective +potential of the black hole and find the impact parameters. We analytically study the black +hole shadow in Sec. IV and for better understating of the effect of quintessential dark energy +on the black hole shadow, we determine the shadow observables and angular size in Sec. V. +3 + +2 +4 +6 +8 +10 +r +−0.25 +−0.20 +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +f(r) +Horizons of Quintessential Black Holes for ω += +− +2/3 +γ=0.08 +γ=0.09 +γ=0.10 +2.05 +2.10 +2.15 +2.20 +2.25 +2.30 +2.35 +r +−0.05 +−0.04 +−0.03 +−0.02 +−0.01 +0.00 +0.01 +0.02 +0.03 +0.04 +f(r) +Horizons of Quintessential Black Holes for ω += +− +1/3 +γ=0.08 +γ=0.09 +γ=0.10 +1 +2 +3 +4 +5 +r +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +f(r) +Horizons of Quintessential Black Holes for ω +q += +− +1 +γ=0.045 +γ=0.037 +γ=0.030 +FIG. 1: Plot showing the horizons of the non-rotating black holes . +Finally, we conclude our results in Sec. VI. +II. +BLACK HOLE METRIC +A. +Non-rotating black hole metric in the presence of quintessential dark energy +The spherically symmetric black hole metric with quintessential dark energy has been +derived by Kislev [10]. In Boyer Lindquist coordinates and natural units G = c = 1, the +black hole metric is given by +ds2 = −f(r)dt2 + f −1(r)dr2 + r2(dθ2 + sin2 θdφ2), +(1) +with +f(r) = 1 − 2M +r +− +γ +r3ωq+1, +(2) +4 + +ωq = −2/3 +ωq = −1/3 +ωq = −1 +γ +Inner Horizon Outer Horizon +γ +Inner Horizon Outer Horizon +γ +Horizon +0.030 +no horizon +no horizon +0.100 +2.763932 +7.236067 +0.080 2.173913 +0.037 +2.946526 +3.056142 +0.120 +4.000000 +4.000000 +0.900 2.197802 +0.045 +2.752125 +2.752125 +0.150 +no horizon +no horizon +0.100 2.222222 +TABLE I: The horizons of non-rotating black holes in quintessential dark energy +ωq = −2/3 +a +γ +Inner +Horizon +Outer +Horizon +Quintessential +Horizon +0.70 0.13 +0.283718 +3.590570 +3.590570 +0.70 0.11 +0.394402 +2.309770 +6.386736 +1.07 0.11 +1.263181 +1.263181 +6.564545 +1.15 0.10 No Horizon No Horizon +7.598965 +TABLE II: Table showing the numerical values for the horizons of the rotating black hole in +quintessential dark energy +where M is the black hole, γ is the normalization factor and ωq is the quintessential state +parameter. In Fig. 1, we plot the metric function f(r) with radial distance r for different +values of ωq and γ. For ωq = −2/3, the quintessential dark energy significantly affects the +black hole horizons. Non-degenerate and degenerate horizons exist for the corresponding +values of normalization parameter γ. +If we consider ωq = −1/3, the black hole metric +reduces for the Kottler spacetime also degenerate and non-degenerate horizons exist with +the corresponding values of γ. For ωq = −1, the black hole metric corresponds single horizon +(cf. Table I and Fig. 1). In the absence of the quintessential field the black hole metric (1) +5 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +r +−2 +−1 +0 +1 +2 +3 +4 +5 +Δ(r) +Horizons of the rotating black hole for ω +q += +− +2/3 +γ=0.10, a=1.15 +γ=0.11, a=1.07 +γ=0.11, a=0.70 +γ=0.13, a=0.70 +FIG. 2: Plot showing the horizons of the rotating black hole in quintessential dark energy. +reduces for the Schwarzschild spacetime. +B. +Rotating black holes in quintessential dark energy +The rotating black hole metric surrounded with the quintessential dark energy is +ds2 = − +� +1 − 2Mr + γr1−3ωq +Σ +� +dt2 + Σ +∆dr2 − 2a sin2 θ +�2Mr + γr1−3ωq +Σ +� +dφdt + Σdθ2 ++ sin2 θ +� +r2 + a2 + a2 sin2 θ +�2Mr + γr1−3ωq +Σ +�� +dφ2 , +(3) +where +∆(r) = r2 − 2Mr + a2 − γr1−3ωq, +(4) +and +Σ = r2 + a2 cos2 θ . +(5) +We get the horizons of the black hole by simply solving ∆(r) = 0. For ωq = −2/3, the +equation (4) reduces to the cubic polynomial equation, and we find the radius of the horizon +by solving +r2 − 2Mr + a2 − γr3 = 0, +(6) +6 + +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +r +10 +15 +20 +25 +30 +35 +V(r) + ω +q += +− +2/3 and a += +0.9 +γ=0.08 +γ=0.10 +γ=0.12 +γ=0.14 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +r +10 +12 +14 +16 +18 +20 +22 +24 +V(r) + ω +q += +− +1/3 and a += +0.9 +γ=0.08 +γ=0.10 +γ=0.12 +γ=0.14 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +r +10 +15 +20 +25 +30 +35 +V(r) + ω +q += +− +2/3 and a += +0.7 +γ=0.08 +γ=0.10 +γ=0.12 +γ=0.14 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +r +10 +12 +14 +16 +18 +20 +22 +24 +V(r) + ω +q += +− +1/3 and a += +0.7 +γ=0.08 +γ=0.10 +γ=0.12 +γ=0.14 +FIG. 3: Plot showing the variation of effective potential with r for different values of a, ωq and γ. +which also can be written as +γ(r − rin)(r − rout)(r − rq) = 0, +(7) +where rin and rout are the inner and outer horizon, and rq corresponds to the quintessential +horizon respectively. The rotating black hole in the quintessential dark energy corresponds +to three horizons which vary with the spin parameter a and the normalization parameter γ +as shown in Table (II). In Fig. 2, we show the variations of horizons with the corresponding +values of a and γ. For extremely rotating case a = 1.15, the black hole corresponds only +quintessential horizon (cf. Fig. 2 and Table (II)). +III. +EFFECTIVE POTENTIAL AND IMPACT PARAMETERS +The motion of particles and electromagnetic radiations around the rotating spacetime is +7 + +0.08 +0.09 +0.10 +0.11 +0.12 +0.13 +0.14 +γ +2.35 +2.40 +2.45 +2.50 +2.55 +r +cric +a += +0.9 +ω +q += +− +2/3 +ω +q += +− +1/3 +0.08 +0.09 +0.10 +0.11 +0.12 +0.13 +0.14 +γ +2.450 +2.475 +2.500 +2.525 +2.550 +2.575 +2.600 +2.625 +r +cric + a += +0. +7 +ω +q += +− +2/3 +ω +q += +− +1/3 +FIG. 4: Plot showing the variation of critical radius rcric with γ for different values of quintessential +parameter ωq and spin parameter a. +a = 0.9 +a = 0.7 +ωq = −2/3 ωq = −1/3 ωq = −2/3 ωq = −1/3 +γ +rcric +γ +rcric +γ +rcric +γ +rcric +0.08 2.51 0.08 2.57 0.08 2.57 0.08 2.63 +0.10 2.45 0.10 2.54 0.10 2.54 0.10 2.60 +0.12 2.42 0.12 2.48 0.12 2.48 0.12 2.54 +0.14 2.36 0.14 2.45 0.14 2.45 0.14 2.51 +TABLE III: Table showing the variation of rcric with a, ωq and γ +regular so the equations of motion of a particle around a black hole in quintessential dark +energy are completely integrable. In this section, we derive complete integral equations of +motion and study the effective potential of the black hole and calculate the impact parame- +ters. We use the Hamilton-Jacobi formalism to find the complete geodesic equations which +were originally derived by Carter [94–97]. The Hamilton-Jacobi equation reads +∂S +∂υ = −1 +2gµνpµpν, +(8) +where υ is the affine parameter and pµ is the conjugate momenta corresponding to the first +derivative of Jacobian action S with respect to the generalized coordinate. We choose a +8 + +separable solution for the Jacobean action in the given form +S = 1 +2m0 +2υ − Et + Lφ + Sr(r) + Sθ(θ), +(9) +where m0 is the rest mass of the test particle which is zero for the photon. +The black +hole metric is cyclic for time t and azimuthal φ coordinates. There exist two conserved +quantities corresponding to these cyclic coordinates E and L having units of energy and +angular momentum respectively. Using our separable solution defined in Eq. (9), we obtain +our complete equations of motion for a massive test particle in the first-order differential +form as +Σ dt +dυ = r2 + a2 +∆(r) +� +E(r2 + a2) − aL +� +− a(aE sin2 θ − L) , +(10) +Σdr +dυ = +� +R(r) , +(11) +Σdθ +dυ = +� +Θ(θ) , +(12) +Σdφ +dυ = +a +∆(r) +� +E(r2 + a2) − aL +� +− +� +aE − +L +sin2 θ +� +, +(13) +where +R(r) = +� +(r2 + a2)E − aL +�2 − ∆(r) +� +m0 +2r2 + (aE − L)2 + K +� +, +(14) +Θ(θ) = K − +� L2 +sin2 θ − a2E2 +� +cos2 θ . +(15) +Here, K is the Carter separable constant. The dynamics of a test particle around the black +hole can be obtained from the geodesic Eqs. (10)-(13). These geodesic equations reduce for +the Kerr black holes in the absence of the quintessential field γ = 0 [95]. The study of the +motion of the test particle around the rotating black hole requires two impact parameters. +Here, we are introducing two impact parameters η and ξ in terms of conserved quantities E, +L and K as +ξ = L/E, +η = K/E2. +(16) +The rest mass of the photon is zero, so for the case of null geodesics, we consider m0 = 0. +The radial equation of motion in terms of impact parameters reads +R(r) = 1 +E2 +� +[(r2 + a2) − aξ]2 − ∆(r)[(a − ξ)2 + η] +� +. +(17) +The formation of circular orbits around the black hole is an important feature for the study +of gravitational lensing and shadow properties. The idea of the formation of circular orbits +9 + +FIG. 5: Plot showing the shadows of the rotating black holes in quintessential dark energy for the +different values of the parameter γ, a. +can be obtained from the effective potential of the black hole. The circular photon orbits +may be categorized into two types: unstable or stable circular photon orbits. When a ray of +photon departs from the unstable circular orbit then it may fall into the black hole or reach +the infinite observer. No bound orbits exit around the unstable circular orbits. Instead of +that, many possible bound orbits may exist around the stable circular orbits. We use the +radial equation of motion (17) to find the expression of the effective potential. The radial +10 + +FIG. 6: Plot showing the shadows of the rotating black holes in quintessential dark energy for the +different values of the parameter ωq, a . +equation of motion can be rewritten in terms of effective potential as Veff as +�dr +dυ +�2 ++ Veff(r) = 0, +(18) +from the above equation (18), one can find the expression of the effective potential which +takes the following form +Veff = 1 +Σ2[((r2 + a2) − aξ)2 − ∆(r)((a − ξ)2 + η)]. +(19) +11 + +−0.66 +−0.65 +−0.64 +−0.63 +−0.62 +ω +q +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Shadow Radius (R +s +) + a += +1.0 +γ += +0.09 +γ += +0.10 +γ += +0.11 +γ += +0.12 +FIG. 7: Plot showing the variation of shadow radius Rs with the quintessential dark energy pa- +rameter ωq for different γ. +In Fig. (3), we plot the behaviour of effective potential with radial coordinate r for different +values of a, ωq and γ. One can observe from Fig. (3) that for a fixed value of a and ωq the +maximum value or peak value increases with the increasing value of γ. Another interesting +result one can observe is that the critical radius corresponding to the peak value of effective +potential shifts to the left which means the radii of the unstable circular orbits decreases +with the increasing values of γ (cf. Fig. (4) and Table III). The critical radius rcri of unstable +photon orbits corresponds to the local maximum of the effective potential while the critical +radius for the most stable photon orbits corresponds to the local minimum. Here we are +interested to find the most unstable photon orbits and for that, we must have to maximize +the effective potential which satisfies the following condition +Veff = ∂Veff +∂r += 0 +or +R = ∂R +∂r = 0, +(20) +by applying the above conditions (20), we obtain the individual expressions for impact +12 + +−0.66 +−0.65 +−0.64 +−0.63 +−0.62 +ω +q +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +Distortion (δ +s +) + a += +1.0 +γ += +0.09 +γ += +0.10 +γ += +0.11 +γ += +0.12 +FIG. 8: Plot showing the variation of the distortion parameter with the quintessential dark energy +parameter ωq for different parameter γ . +parameters +η = +1 +a2 (−3γωq + γ + 2(M − r)r3ωq) 2 +� +r3(−4 +� +r(r − 3M)2 − 4a2M +� +r6ωq) ++ 4γr3ωq � +3ωq +� +2a2 + r(r − 3M) +� ++ 2a2 + 3r(r − 3M) − 9γ2r (ωq + 1) 2� � +, +(21) +ξ = +1 +a (−3γωq + γ + 2(M − r)r3ωq) +� +2 +� +a2(M + r) + r2(r − 3M) +� +r3ωq ++ γ +� +−3 +� +a2 + r2� +ωq + a2 − 3r2� � +. +(22) +These expressions of η and ξ define the motion of photons around the black holes in +quintessential dark energy. These equations of impact parameters exactly reduce for the +Kerr black holes in the absence of the quintessence dark energy field [95]. +IV. +BLACK HOLE SHADOW +A black hole form shadow due to the infalling light rays inside its event horizon. The +light rays at the unstable equilibrium form orbit around it. We consider orthogonal tetrad to +13 + +define the celestial coordinate along the boundary of the black hole shadow. The orthonormal +tetrads are +e0 = (r2 + a2)∂t + a∂φ +√ +Σ∆ +, +e1 = +1 +√ +Σ +∂θ, +e2 = −∂φ + a sin2 θ∂t +√ +Σ∆ +, +e3 = − +� +∆ +Σ ∂r. +(23) +The tangent vector over a light ray can be defined as +˙λ(υ) = ˙t∂t + ˙r∂r + ˙θ∂θ + ˙φ∂φ, +(24) +and the tangent vector for the observers position is +˙λ(υ) = α(−e0 + sin Φ cos Ψe1 + sin Φ sin Ψe2 + cos Φe3). +(25) +The expression for the scalar factor α can be determined by comparing Eq. (24) and (25), +and using Eq. (10 - 13), which takes the following form +α = aL − (r2 + a2)E +√ +Σ∆ +. +(26) +Next, the celestial coordinates φ and ψ can be defined as +sin Ψ(rO, rp) = +� +ξ − a +� +(a − ξ)2 + η +������ +r=rO +, +(27) +sin Φ(rO, rp) = +�� +∆[(a − ξ)2 + η] +(r2 + a2 − aξ) +������ +r=rO +, +(28) +where the impact parameters ξ and η are functions of the radius of the unstable photon +orbits around the black hole. To visualize a shadow, we introduce the Cartesian coordinates +X and Y [72] +X(rO, rp) = −2 tan +�Φ(rO, rp) +2 +� +sin(Ψ(rO, rp)), +Y (rO, rp) = −2 tan +�Φ(rO, rp) +2 +� +cos(Ψ(rO, rp)), +(29) +which satisfies the following relation +X2 + Y 2 = 4 tan2 �Φ(ro, rp) +2 +� +. +(30) +The contour plot of X vs Y defines the photon region or shadow boundary of rotating black +hole in quintessential dark energy as seen by static observer at the distance rO = 5M in +14 + +ωq +γ = 0.09 +γ = 0.10 +Rs +δs +A +θd +Rs +δs +A +θd +-0.66 0.9376 0.07086 2.7617000 21.489792 0.8988 0.05900 2.53790873 20.600496 +-0.65 0.9662 0.07469 2.93281007 22.145304 0.9388 0.06100 2.76882836 21.517296 +-0.64 0.9894 0.08221 3.07534388 22.677048 0.9734 0.06754 2.97668279 22.310328 +-0.63 1.0090 0.08504 3.19839579 23.12628 0.9985 0.07036 3.13217494 22.88562 +-0.62 1.0250 0.09492 3.30063578 23.49301 1.0250 0.07374 3.30063578 23.49300 +ωq +γ = 0.11 +γ = 0.12 +Rs +δs +A +θd +Rs +δs +A +θd +-0.66 0.8359 0.08500 2.1951213 19.158828 0.7055 0.03883 1.5636657 +16.17006 +-0.65 0.8959 0.09100 2.52155791 20.534028 0.8194 0.04637 2.1093167 18.780648 +-0.64 0.9460 0.09700 2.81146153 21.68232 0.8973 0.05215 2.52944481 20.566116 +-0.63 0.9877 0.09900 3.06478477 22.638084 0.9584 0.06103 2.88564886 21.966528 +-0.62 1.0170 0.1000 3.24931472 23.30964 1.0010 0.06463 3.14787898 22.942920 +TABLE IV: Table Showing the black hole shadow observables: shadow radius Rs, distortion δs, +Area A and angular diameter θd for spin parameter a = 1.0 with different ωq and γ. +the domain of outer communication. We plot various figures for the black hole shadow in +quintessence for different values of spin parameter a, state parameter ωq and normalization +factor γ (cf. Fig. 5 and 6). The effective size of the black hole shadow decreases with ωq +(Fig. 5) while the shadow size increases for the increasing values of γ (Fig. 6). The next +important result one can observe from the shadow plots, the shape of the black hole shadow +distorted with the quintessence parameter ωq while for the increasing values of normalization +factor γ the black hole shadow appears more circular even for the very high values of spin +parameter which we have clearly shown in the first plot of the figure 5. +V. +OBSERVABLES +In this section we estimate observables: shadow radius Rs, distortion parameter δS re- +spectively, which is originally shared by Hioki and Maeda in [98]. The observable shadow +15 + +−0.66 +−0.65 +−0.64 +−0.63 +−0.62 +ω +q +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +3.25 +Area (A) + a += +1.0 +γ += +0.09 +γ += +0.10 +γ += +0.11 +γ += +0.12 +FIG. 9: Plot showing the variation of the black hole shadow area A with the quintessential dark +energy parameter ωq +radius Rs is given by +Rs = (Xt − Xr)2 + Y 2 +t +2|Xr − Xt| +, +(31) +where (Xt, Yt), (Xr, Yr) are the top most and right most co-ordinates of the X and Y axis +in the Fig. 5 and 6. Another observable which is distortion parameter reads in the form +δ = D +Rs +, +(32) +where D = |Xl − X +′ +l| with Xl and X +′ +l are the left most co-ordinates on the X axis from +where the black hole shadow and reference circle passes. +In Fig. 7 and 8, we plot the variation of shadow radius Rs and distortion parameter with +the quintessence parameter ωq for different values of normalization factor γ. The shadow +radius Rs increases with the ωq while it decreases for the increasing γ that we have shown in +the Fig. 7. The distortion parameter increases with ωq (cf. Fig. 8). The black hole shadow +appears more distorted in the presence of quintessential field. +The angular diameter θd of the black hole shadow can be estimated via +θd = 2 +ro +× +� +A +π , +(33) +16 + +−0.66 +−0.65 +−0.64 +−0.63 +−0.62 +ω +q +16 +17 +18 +19 +20 +21 +22 +23 +Angular Diameter + a += +1.0 +γ += +0.09 +γ += +0.10 +γ += +0.11 +γ += +0.12 +FIG. 10: +Plot showing the variation of angular diameter of the black hole shadow with the +quintessential dark energy parameter ωq. +where A is the area of the black hole shadow in quintessential dark energy. The variation +of area of the black hole shadow A with the quintessence parameter ωq have shown in the +Fig. (9) and Table IV. The shadow area A monotonically increase with the parameter ωq +while it decreases with the parameter γ (cf. Fig. 9 and Table IV). Using the Eq. (33), we +analytically estimate the angular diameter of the black hole shadow in quintessential dark +energy. The angular diameter of the black hole shadow varies from θd ≈ 21o to 23o with the +variation of ωq from −0.66 to −0.62 for the parameter γ = 0.09. Our results shows that the +angular diameter of the black hole shadow monotonically increases with the quintessence +dark energy parameter ωq. As we further increase the parameter γ from 0.09 to 0.12, the +angular diameter of the black hole shadow decreases, for γ = 0.12 it varies from θd ≈ 16o to +22o as clearly shown in the Fig. 10 and Table IV. +VI. +CONCLUSION +The observations from EHT show that the angular size of the M87∗ black hole shadow is +42±3µas, and very recent results confirm the size of the supermassive black hole Sgr∗ 48.7± +17 + +7µas. These astrophysical objects are very far from the earth so the observed size of these +black hole’s shadow are very small such as 16.4 megaparsec for M87 and 7.86 kiloparsec +for Sgr A∗. Here, we have analytically estimated the angular size of the quintessential dark +energy black holes in the domain of outer communication. First, we derive the null geodesic +equations of motion and obtained the effective potential for the black hole. After maximizing +the effective potential we find impact parameters. We consider our observer in between the +outer and cosmological horizon so we introduced corresponding celestial coordinates and +plotted various figures of black hole shadow for different values of quintessential dark energy +parameter ωq, normalization factor γ and spin parameter a. For the analytical study of +the shape and size of the black hole shadow image, we numerically studied the shadow +observables. In our study, we have found the size of the black hole shadow increases with +the quintessence dark energy parameter ωq while it decreases with the normalization factor +γ. +Next, the shape of the shadow, which is observable by the distortion parameter δs, +increases with the parameter ωq while it decreases with the parameter γ. The observable +shadow area A also corresponds to the same results as the shadow radius and using the +numerical results of the shadow area, we estimated the angular diameter of the quintessence +dark energy black holes in the domain of outer communication. The angular diameter of the +black hole shadow increases with the quintessence parameter ωq and varies from θd ≈ 16o +to 23o for −0.66 ≤ ωq ≤ −0.62 and 0.09 ≤ γ ≤ 0.12 with spin parameter a = 1.0 when the +observer stays at rest in between the outer and the cosmological horizon of the quintessential +dark energy black holes. +VII. +ACKNOWLEDGEMENT +The author would like to thank Rahul Kumar Walia for the fruitful discussion and Tula’s +Institute for providing research facilities. +[1] A. G. Riess et al., Astron. J. 116, 1009-1038 (1998). +[2] S. Perlmutter et al., Astrophys. J. 517, 565-586 (1999). +[3] N. Suzuki et al. Astro. Phys. J., 746:85, 2012. +[4] P. A. R. Ade et al., Astron. Astrophys. 594, A14 (2016). +18 + +[5] T. Chiba, A. De Felice and S. Tsujikawa, Phys. Rev. D 87, no.8, 083505 (2013). +[6] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582-1585 (1998). +[7] S. M. Carroll, Phys. Rev. Lett. 81, 3067-3070 (1998). +[8] M. Cicoli, F. G. Pedro and G. 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D 80, 024042 (2009), arXiv:0904.3575. +22 + diff --git a/wdAzT4oBgHgl3EQfCPqo/content/tmp_files/load_file.txt b/wdAzT4oBgHgl3EQfCPqo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f6c83efc7a253a3e6293871b4b4366d6a339243 --- /dev/null +++ b/wdAzT4oBgHgl3EQfCPqo/content/tmp_files/load_file.txt @@ -0,0 +1,1234 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf,len=1233 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='00956v1 [gr-qc] 3 Jan 2023 Shadows of quintessential dark energy black holes in the domain of outer communication Balendra Pratap Singha∗ aDepartment of Applied Sciences and Engineering, Tula’s Institute, Dehradun, Uttarakhand 248197, India Abstract The rotating black holes in the quintessential dark energy correspond to three horizons: inner, outer, and quintessential horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The domain of outer communication is the region between outer and quintessential horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Here, in this work we study the photon region and shadows of the quintessential dark energy black holes when the observer stays statically in the domain of outer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The quintessential dark energy black holes shadow characterizes by its mass (M), spin parameter (a), quintessential dark energy parameter (ωq), and normalization factor (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The dark energy parameter ωq can take values in between −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='1 < ωq < −1/3 and follows the equation of state ωq=pressure(p)/energy density(ρq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' This state parameter significantly affects the shape and size of the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We generalize all the geodesic equations of motion for ωq and obtain relation to visualize the black hole shadow by a static observer at any arbitrary distance in the domain of outer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We analytically estimate the black hole shadow observables: radius Rs, distortion parameter δs and the shadow area A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Using the numerical values of shadow radius Rs and area A, we obtain the angular diameter of the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The angular size of the M87 and Sgr A∗ black holes are 42 ± 3µas and 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='7 ± 7µas respectively as observe by Event Horizon Telescope (EHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In this case, the angular diameter of the black hole shadow increases with the quintessence parameter ωq and takes values θd ≈ 20 ± 3o with the parameter −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 ≤ ωq ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 for the static observer at ro = 5M in the domain of outer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' PACS numbers: ∗Electronic address: balendra29@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='com 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' INTRODUCTION The observations of type Ia supernovae (SNe Ia) lie between the red shift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='16 ≤ z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62, confirming that our Universe is going under the late time acceleration [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' According to general relativity, this cosmic acceleration indicates that there exists some strange energy component in the Universe which is called dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The observational constrains over the state parameter ωq provided by the large state structure of the Universe and the cosmic microwave background (CMB) is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='1 < ωq < −1/3 [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The hypothesis of dark energy is compatible with the standard model of big bang cosmology (ΛCDM model) when the dark energy state parameter is exactly equal to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' For this value of state parameter, the dark energy is considered as cosmological constant, which is interestingly agreements with the observations but still there is a possibility that some significant component of the dark energy densities have state parameters other than -1 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' One of the simplest is the quintessence dark energy model in which the dynamical scalar field is minimally coupled with the gravity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The quintessence dark energy is dynamic and time-varying which is different from the cosmological constant model which does not change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Some researchers also consider the quintessence dark energy as the fifth fundamental force responsible for the expansion of the Universe [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The asymptotic structure of the black hole gets modified in the presence of the quintessen- tial dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The black hole spacetime remains no more asymptotically flat in quintessence due to the cosmological horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The very first model of the black hole in quintessence was presented by Kislev [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' After that, several researchers intensively stud- ied the properties of spherically symmetric black holes in quintessence dark energy [11–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The Lovelock black holes in quintessence have been studied by [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The study of Nar- nia black holes in quintessence has been done by [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The authors of [34] discussed the geodesics of the Hayward black hole in quintessence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Thermodynamics of the Bardeen black hole in quintessential dark energy studied in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The rotating counterpart of the spherically symmetric black hole in quintessence obtained by [36] and [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The study of rotating anti- de-sitter and rotating charged anti-de-sitter black holes in the presence of perfect fluid matter have been intensively studied by [38] and [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The author of [39] extended Schwarzschild black holes in quintessence up to D-dimensional spacetimes and studied gravitational lensing and shadow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2 The observational results from the Event Horizon Telescope (EHT) proved that black holes are just not only theoretical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The first picture of the Messier 87 (M87) black hole revealed by the EHT group in 2019 [40–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Black holes are completely dark objects but interestingly they cast a shadow [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The incoming photons towards the black hole horizon which have quite large angular momentum fall inside the black hole event horizon and create a dark spot, and photons that carries a little bit smaller angular momentum form the photon region around the black hole [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' This dark spot surrounded with the bright photon rings is called a black hole shadow [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Casting shadow by the black hole also verifies the existence of the event horizon [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Recently EHT collaborators published the image of Sgr A∗ black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' This supermassive black hole lies at the heart of our galaxy Milky way [48–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The first analytical study of this subject was done by Synge [54] and Luminet [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Later Bardeen extended it for the rotating spacetime [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In the past few years, several researchers analytically studied the black hole shadow for rotating and non-rotating spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='[57–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The optical properties of the rotating and non-rotating black holes in the presence of the plasma medium have been studied by [80–82] This subject has been extended for the higher dimensional spacetime by [83–87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Various researchers estimated black hole parameters using the properties of the black hole shadow [89–93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The primary goal of this work is to analytically study the angular size of the photon regions for rotating black holes in quintessential dark energy in the domain of outer com- munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The rotating black hole in quintessence corresponds to three horizons: inner, outer, and quintessential or cosmological event horizons [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The domain of outer com- munication is the region where the observer statically stays in between the outer and the cosmological horizon [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The quintessence field affects the shadow region, and the appear- ance of the bright photons region varies as seen by the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The size of the black hole shadow decreases and gets distorted with the state parameter ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The photon region will appear deviated and more close to the event horizon in the presence of the quintessential dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' This paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' II, we discuss the rotating and non-rotating black hole metric with the quintessential dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' III, we derive the effective potential of the black hole and find the impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We analytically study the black hole shadow in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' IV and for better understating of the effect of quintessential dark energy on the black hole shadow, we determine the shadow observables and angular size in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 3 2 4 6 8 10 r −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='15 f(r) Horizons of Quintessential Black Holes for ω = − 2/3 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='35 r −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='04 f(r) Horizons of Quintessential Black Holes for ω = − 1/3 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 1 2 3 4 5 r −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 f(r) Horizons of Quintessential Black Holes for ω q = − 1 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='045 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='037 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='030 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 1: Plot showing the horizons of the non-rotating black holes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Finally, we conclude our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' BLACK HOLE METRIC A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Non-rotating black hole metric in the presence of quintessential dark energy The spherically symmetric black hole metric with quintessential dark energy has been derived by Kislev [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In Boyer Lindquist coordinates and natural units G = c = 1, the black hole metric is given by ds2 = −f(r)dt2 + f −1(r)dr2 + r2(dθ2 + sin2 θdφ2), (1) with f(r) = 1 − 2M r − γ r3ωq+1, (2) 4 ωq = −2/3 ωq = −1/3 ωq = −1 γ Inner Horizon Outer Horizon γ Inner Horizon Outer Horizon γ Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='030 no horizon no horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='763932 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='236067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='080 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='173913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='037 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='946526 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='056142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='120 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='000000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='000000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='900 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='197802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='045 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='752125 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='752125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='150 no horizon no horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='222222 TABLE I: The horizons of non-rotating black holes in quintessential dark energy ωq = −2/3 a γ Inner Horizon Outer Horizon Quintessential Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='283718 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='590570 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='590570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='394402 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='309770 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='386736 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='263181 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='263181 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='564545 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 No Horizon No Horizon 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='598965 TABLE II: Table showing the numerical values for the horizons of the rotating black hole in quintessential dark energy where M is the black hole, γ is the normalization factor and ωq is the quintessential state parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 1, we plot the metric function f(r) with radial distance r for different values of ωq and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' For ωq = −2/3, the quintessential dark energy significantly affects the black hole horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Non-degenerate and degenerate horizons exist for the corresponding values of normalization parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' If we consider ωq = −1/3, the black hole metric reduces for the Kottler spacetime also degenerate and non-degenerate horizons exist with the corresponding values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' For ωq = −1, the black hole metric corresponds single horizon (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In the absence of the quintessential field the black hole metric (1) 5 0 1 2 3 4 5 6 7 8 r −2 −1 0 1 2 3 4 5 Δ(r) Horizons of the rotating black hole for ω q = − 2/3 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10, a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='15 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11, a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11, a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='70 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='13, a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='70 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2: Plot showing the horizons of the rotating black hole in quintessential dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' reduces for the Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Rotating black holes in quintessential dark energy The rotating black hole metric surrounded with the quintessential dark energy is ds2 = − � 1 − 2Mr + γr1−3ωq Σ � dt2 + Σ ∆dr2 − 2a sin2 θ �2Mr + γr1−3ωq Σ � dφdt + Σdθ2 + sin2 θ � r2 + a2 + a2 sin2 θ �2Mr + γr1−3ωq Σ �� dφ2 , (3) where ∆(r) = r2 − 2Mr + a2 − γr1−3ωq, (4) and Σ = r2 + a2 cos2 θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (5) We get the horizons of the black hole by simply solving ∆(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' For ωq = −2/3, the equation (4) reduces to the cubic polynomial equation, and we find the radius of the horizon by solving r2 − 2Mr + a2 − γr3 = 0, (6) 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 r 10 15 20 25 30 35 V(r) ω q = − 2/3 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 r 10 12 14 16 18 20 22 24 V(r) ω q = − 1/3 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 r 10 15 20 25 30 35 V(r) ω q = − 2/3 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='7 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='5 r 10 12 14 16 18 20 22 24 V(r) ω q = − 1/3 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='7 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 3: Plot showing the variation of effective potential with r for different values of a, ωq and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' which also can be written as γ(r − rin)(r − rout)(r − rq) = 0, (7) where rin and rout are the inner and outer horizon, and rq corresponds to the quintessential horizon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The rotating black hole in the quintessential dark energy corresponds to three horizons which vary with the spin parameter a and the normalization parameter γ as shown in Table (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2, we show the variations of horizons with the corresponding values of a and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' For extremely rotating case a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='15, the black hole corresponds only quintessential horizon (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2 and Table (II)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' EFFECTIVE POTENTIAL AND IMPACT PARAMETERS The motion of particles and electromagnetic radiations around the rotating spacetime is 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='55 r cric a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9 ω q = − 2/3 ω q = − 1/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='450 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='475 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='525 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='550 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='575 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='600 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='625 r cric a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 7 ω q = − 2/3 ω q = − 1/3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 4: Plot showing the variation of critical radius rcric with γ for different values of quintessential parameter ωq and spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='7 ωq = −2/3 ωq = −1/3 ωq = −2/3 ωq = −1/3 γ rcric γ rcric γ rcric γ rcric 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='51 TABLE III: Table showing the variation of rcric with a, ωq and γ regular so the equations of motion of a particle around a black hole in quintessential dark energy are completely integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In this section, we derive complete integral equations of motion and study the effective potential of the black hole and calculate the impact parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We use the Hamilton-Jacobi formalism to find the complete geodesic equations which were originally derived by Carter [94–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The Hamilton-Jacobi equation reads ∂S ∂υ = −1 2gµνpµpν, (8) where υ is the affine parameter and pµ is the conjugate momenta corresponding to the first derivative of Jacobian action S with respect to the generalized coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We choose a 8 separable solution for the Jacobean action in the given form S = 1 2m0 2υ − Et + Lφ + Sr(r) + Sθ(θ), (9) where m0 is the rest mass of the test particle which is zero for the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The black hole metric is cyclic for time t and azimuthal φ coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' There exist two conserved quantities corresponding to these cyclic coordinates E and L having units of energy and angular momentum respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Using our separable solution defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (9), we obtain our complete equations of motion for a massive test particle in the first-order differential form as Σ dt dυ = r2 + a2 ∆(r) � E(r2 + a2) − aL � − a(aE sin2 θ − L) , (10) Σdr dυ = � R(r) , (11) Σdθ dυ = � Θ(θ) , (12) Σdφ dυ = a ∆(r) � E(r2 + a2) − aL � − � aE − L sin2 θ � , (13) where R(r) = � (r2 + a2)E − aL �2 − ∆(r) � m0 2r2 + (aE − L)2 + K � , (14) Θ(θ) = K − � L2 sin2 θ − a2E2 � cos2 θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (15) Here, K is the Carter separable constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The dynamics of a test particle around the black hole can be obtained from the geodesic Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (10)-(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' These geodesic equations reduce for the Kerr black holes in the absence of the quintessential field γ = 0 [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The study of the motion of the test particle around the rotating black hole requires two impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Here, we are introducing two impact parameters η and ξ in terms of conserved quantities E, L and K as ξ = L/E, η = K/E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (16) The rest mass of the photon is zero, so for the case of null geodesics, we consider m0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The radial equation of motion in terms of impact parameters reads R(r) = 1 E2 � [(r2 + a2) − aξ]2 − ∆(r)[(a − ξ)2 + η] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (17) The formation of circular orbits around the black hole is an important feature for the study of gravitational lensing and shadow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The idea of the formation of circular orbits 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 5: Plot showing the shadows of the rotating black holes in quintessential dark energy for the different values of the parameter γ, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' can be obtained from the effective potential of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The circular photon orbits may be categorized into two types: unstable or stable circular photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' When a ray of photon departs from the unstable circular orbit then it may fall into the black hole or reach the infinite observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' No bound orbits exit around the unstable circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Instead of that, many possible bound orbits may exist around the stable circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We use the radial equation of motion (17) to find the expression of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The radial 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 6: Plot showing the shadows of the rotating black holes in quintessential dark energy for the different values of the parameter ωq, a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' equation of motion can be rewritten in terms of effective potential as Veff as �dr dυ �2 + Veff(r) = 0, (18) from the above equation (18), one can find the expression of the effective potential which takes the following form Veff = 1 Σ2[((r2 + a2) − aξ)2 − ∆(r)((a − ξ)2 + η)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (19) 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 ω q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='00 Shadow Radius (R s ) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 7: Plot showing the variation of shadow radius Rs with the quintessential dark energy pa- rameter ωq for different γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (3), we plot the behaviour of effective potential with radial coordinate r for different values of a, ωq and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' One can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (3) that for a fixed value of a and ωq the maximum value or peak value increases with the increasing value of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Another interesting result one can observe is that the critical radius corresponding to the peak value of effective potential shifts to the left which means the radii of the unstable circular orbits decreases with the increasing values of γ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (4) and Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The critical radius rcri of unstable photon orbits corresponds to the local maximum of the effective potential while the critical radius for the most stable photon orbits corresponds to the local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Here we are interested to find the most unstable photon orbits and for that, we must have to maximize the effective potential which satisfies the following condition Veff = ∂Veff ∂r = 0 or R = ∂R ∂r = 0, (20) by applying the above conditions (20), we obtain the individual expressions for impact 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 ω q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 Distortion (δ s ) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 8: Plot showing the variation of the distortion parameter with the quintessential dark energy parameter ωq for different parameter γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' parameters η = 1 a2 (−3γωq + γ + 2(M − r)r3ωq) 2 � r3(−4 � r(r − 3M)2 − 4a2M � r6ωq) + 4γr3ωq � 3ωq � 2a2 + r(r − 3M) � + 2a2 + 3r(r − 3M) − 9γ2r (ωq + 1) 2� � , (21) ξ = 1 a (−3γωq + γ + 2(M − r)r3ωq) � 2 � a2(M + r) + r2(r − 3M) � r3ωq + γ � −3 � a2 + r2� ωq + a2 − 3r2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (22) These expressions of η and ξ define the motion of photons around the black holes in quintessential dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' These equations of impact parameters exactly reduce for the Kerr black holes in the absence of the quintessence dark energy field [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' BLACK HOLE SHADOW A black hole form shadow due to the infalling light rays inside its event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The light rays at the unstable equilibrium form orbit around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We consider orthogonal tetrad to 13 define the celestial coordinate along the boundary of the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The orthonormal tetrads are e0 = (r2 + a2)∂t + a∂φ √ Σ∆ , e1 = 1 √ Σ ∂θ, e2 = −∂φ + a sin2 θ∂t √ Σ∆ , e3 = − � ∆ Σ ∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (23) The tangent vector over a light ray can be defined as ˙λ(υ) = ˙t∂t + ˙r∂r + ˙θ∂θ + ˙φ∂φ, (24) and the tangent vector for the observers position is ˙λ(υ) = α(−e0 + sin Φ cos Ψe1 + sin Φ sin Ψe2 + cos Φe3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (25) The expression for the scalar factor α can be determined by comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (24) and (25), and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (10 - 13), which takes the following form α = aL − (r2 + a2)E √ Σ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (26) Next, the celestial coordinates φ and ψ can be defined as sin Ψ(rO, rp) = � ξ − a � (a − ξ)2 + η ������ r=rO , (27) sin Φ(rO, rp) = �� ∆[(a − ξ)2 + η] (r2 + a2 − aξ) ������ r=rO , (28) where the impact parameters ξ and η are functions of the radius of the unstable photon orbits around the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' To visualize a shadow, we introduce the Cartesian coordinates X and Y [72] X(rO, rp) = −2 tan �Φ(rO, rp) 2 � sin(Ψ(rO, rp)), Y (rO, rp) = −2 tan �Φ(rO, rp) 2 � cos(Ψ(rO, rp)), (29) which satisfies the following relation X2 + Y 2 = 4 tan2 �Φ(ro, rp) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (30) The contour plot of X vs Y defines the photon region or shadow boundary of rotating black hole in quintessential dark energy as seen by static observer at the distance rO = 5M in 14 ωq γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 Rs δs A θd Rs δs A θd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07086 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='7617000 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='489792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='8988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='05900 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='53790873 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='600496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07469 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='93281007 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='145304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='06100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='76882836 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='517296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08221 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07534388 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='677048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='06754 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='97668279 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='310328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='08504 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='19839579 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='9985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07036 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='13217494 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='88562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09492 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='30063578 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='49301 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='07374 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='30063578 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='49300 ωq γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 Rs δs A θd Rs δs A θd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} 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+page_content='06463 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='14787898 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='942920 TABLE IV: Table Showing the black hole shadow observables: shadow radius Rs, distortion δs, Area A and angular diameter θd for spin parameter a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 with different ωq and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' the domain of outer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We plot various figures for the black hole shadow in quintessence for different values of spin parameter a, state parameter ωq and normalization factor γ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The effective size of the black hole shadow decreases with ωq (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 5) while the shadow size increases for the increasing values of γ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The next important result one can observe from the shadow plots, the shape of the black hole shadow distorted with the quintessence parameter ωq while for the increasing values of normalization factor γ the black hole shadow appears more circular even for the very high values of spin parameter which we have clearly shown in the first plot of the figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' OBSERVABLES In this section we estimate observables: shadow radius Rs, distortion parameter δS re- spectively, which is originally shared by Hioki and Maeda in [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The observable shadow 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 ω q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='25 Area (A) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 9: Plot showing the variation of the black hole shadow area A with the quintessential dark energy parameter ωq radius Rs is given by Rs = (Xt − Xr)2 + Y 2 t 2|Xr − Xt| , (31) where (Xt, Yt), (Xr, Yr) are the top most and right most co-ordinates of the X and Y axis in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Another observable which is distortion parameter reads in the form δ = D Rs , (32) where D = |Xl − X ′ l| with Xl and X ′ l are the left most co-ordinates on the X axis from where the black hole shadow and reference circle passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 7 and 8, we plot the variation of shadow radius Rs and distortion parameter with the quintessence parameter ωq for different values of normalization factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The shadow radius Rs increases with the ωq while it decreases for the increasing γ that we have shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The distortion parameter increases with ωq (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The black hole shadow appears more distorted in the presence of quintessential field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The angular diameter θd of the black hole shadow can be estimated via θd = 2 ro × � A π , (33) 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 ω q 16 17 18 19 20 21 22 23 Angular Diameter a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='10 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 10: Plot showing the variation of angular diameter of the black hole shadow with the quintessential dark energy parameter ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' where A is the area of the black hole shadow in quintessential dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The variation of area of the black hole shadow A with the quintessence parameter ωq have shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (9) and Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The shadow area A monotonically increase with the parameter ωq while it decreases with the parameter γ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 9 and Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Using the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' (33), we analytically estimate the angular diameter of the black hole shadow in quintessential dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The angular diameter of the black hole shadow varies from θd ≈ 21o to 23o with the variation of ωq from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 for the parameter γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Our results shows that the angular diameter of the black hole shadow monotonically increases with the quintessence dark energy parameter ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' As we further increase the parameter γ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12, the angular diameter of the black hole shadow decreases, for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 it varies from θd ≈ 16o to 22o as clearly shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 10 and Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' CONCLUSION The observations from EHT show that the angular size of the M87∗ black hole shadow is 42±3µas, and very recent results confirm the size of the supermassive black hole Sgr∗ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='7± 17 7µas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' These astrophysical objects are very far from the earth so the observed size of these black hole’s shadow are very small such as 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='4 megaparsec for M87 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='86 kiloparsec for Sgr A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Here, we have analytically estimated the angular size of the quintessential dark energy black holes in the domain of outer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' First, we derive the null geodesic equations of motion and obtained the effective potential for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' After maximizing the effective potential we find impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' We consider our observer in between the outer and cosmological horizon so we introduced corresponding celestial coordinates and plotted various figures of black hole shadow for different values of quintessential dark energy parameter ωq, normalization factor γ and spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' For the analytical study of the shape and size of the black hole shadow image, we numerically studied the shadow observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' In our study, we have found the size of the black hole shadow increases with the quintessence dark energy parameter ωq while it decreases with the normalization factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Next, the shape of the shadow, which is observable by the distortion parameter δs, increases with the parameter ωq while it decreases with the parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The observable shadow area A also corresponds to the same results as the shadow radius and using the numerical results of the shadow area, we estimated the angular diameter of the quintessence dark energy black holes in the domain of outer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' The angular diameter of the black hole shadow increases with the quintessence parameter ωq and varies from θd ≈ 16o to 23o for −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='66 ≤ ωq ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='62 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='09 ≤ γ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='12 with spin parameter a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='0 when the observer stays at rest in between the outer and the cosmological horizon of the quintessential dark energy black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' ACKNOWLEDGEMENT The author would like to thank Rahul Kumar Walia for the fruitful discussion and Tula’s Institute for providing research facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [1] A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2, L14 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 930, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2, L15 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [52] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 930, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 2, L16 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Synge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 131, 463 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Luminet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 75, 228 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Bardeen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Black Holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Edited by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' DeWitt and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' DeWitt Gordon and Breach, New York, 1973, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [57] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Amarilla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Eiroa and G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Eiroa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' D 87, 044057 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [59] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Yumoto, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Nitta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Chiba and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Sugiyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' D 86, 103001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [60] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 27, 205006 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [64] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Takahashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Jap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 57, 273 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 20 [65] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Wei and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Cosmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 11, 063 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Abdujabbarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 344, 429 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [67] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Amarilla and E.' metadata={'source': 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Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Jing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Cosmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 1710, 051 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content='11907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' [78] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Kumar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Singh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' Ali and S.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfCPqo/content/2301.00956v1.pdf'} diff --git a/wdE2T4oBgHgl3EQfgwdJ/content/tmp_files/2301.03940v1.pdf.txt b/wdE2T4oBgHgl3EQfgwdJ/content/tmp_files/2301.03940v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..24d77453de7e268f5efedd443da1564bf51b7bbc --- /dev/null +++ b/wdE2T4oBgHgl3EQfgwdJ/content/tmp_files/2301.03940v1.pdf.txt @@ -0,0 +1,3382 @@ +Hadron Production and Propagation in Pion-Induced Reactions on Nuclei +R. Abou Yassine6,13, J. Adamczewski-Musch5, O. Arnold10,9, E.T. Atomssa13, M. Becker11, C. Behnke8, +J.C. Berger-Chen10,9, A. Blanco1, C. Blume8, M. B¨ohmer10, L. Chlad14,d, P. Chudoba14, I. Ciepa�l3, C. Deveaux11, +D. Dittert6, J. Dreyer7, E. Epple10,9, L. Fabbietti10,9, P. Fonte1,a, C. Franco1, J. Friese10, I. Fr¨ohlich8, J. F¨ortsch18, +T. Galatyuk6,5, J. A. Garz´on15, R. Gernh¨auser10, R. Greifenhagen7,c,†, M. Grunwald17, M. Gumberidze5, +S. Harabasz6,b, T. Heinz5, T. Hennino13, C. H¨ohne11,5, F. Hojeij13, R. Holzmann5, M. Idzik2, B. K¨ampfer7,c, +K-H. Kampert18, B. Kardan8, V. Kedych6, I. Koenig5, W. Koenig5, M. Kohls8, J. Kolas17, B. W. Kolb5, G. Korcyl4, +G. Kornakov17, R. Kotte7, W. Krueger6, A. Kugler14, T. Kunz10, R. Lalik4, K. Lapidus10,9, S. Linev5, F. Linz6,5, +L. Lopes1, M. Lorenz8, T. Mahmoud11, L. Maier10, A. Malige4, J. Markert5, S. Maurus10, V. Metag11, J. Michel8, +D.M. Mihaylov10,9, V. Mikhaylov14,e, A. Molenda2, C. M¨untz8, R. M¨unzer10,9, M. Nabroth8, L. Naumann7, +K. Nowakowski4, J. Orli´nski16, J.-H. Otto11, Y. Parpottas12, M. Parschau8, C. Pauly18, V. Pechenov5, +O. Pechenova5, K. Piasecki16, J. Pietraszko5, T. Povar18, A. Prozorov14,d, W. Przygoda4, K. Pysz3, B. Ramstein13, +N. Rathod17, P. Rodriguez-Ramos14,e, A. Rost6,5, A. Rustamov5, P. Salabura4, T. Scheib8, N. Schild6, +K. Schmidt-Sommerfeld10, H. Schuldes8, E. Schwab5, F. Scozzi6,13, F. Seck6, P. Sellheim8, J. Siebenson10, L. Silva1, +U. Singh4, J. Smyrski4, S. Spatarof, S. Spies8, M. Stefaniak17, H. Str¨obele8, J. Stroth8,5, P. Strzempek4, C. Sturm5, +K. Sumara4, O. Svoboda14, M. Szala8, P. Tlusty14, M. Traxler5, H. Tsertos12, O. Vazquez-Doce10,9, V. Wagner14, +A.A. Weber11, C. Wendisch5, M.G. Wiebusch5, J. Wirth10,9, H.P. Zbroszczyk17, E. Zherebtsova5,g, P. Zumbruch5 +(HADES collaboration) +C. Curceanug, K. Piscicchiah,g, A. Scordog +1LIP-Laborat´orio de Instrumenta¸c˜ao e F´ısica Experimental de Part´ıculas , 3004-516 Coimbra, Portugal +2AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krak´ow, Poland +3Institute of Nuclear Physics, Polish Academy of Sciences, 31342 Krak´ow, Poland +4Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059 Krak´ow, Poland +5GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany +6Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany +7Institut f¨ur Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany +8Institut f¨ur Kernphysik, Goethe-Universit¨at, 60438 Frankfurt, Germany +9Excellence Cluster ’Origin and Structure of the Universe’ , 85748 Garching, Germany +10Physik Department E62, Technische Universit¨at M¨unchen, 85748 Garching, Germany +11II.Physikalisches Institut, Justus Liebig Universit¨at Giessen, 35392 Giessen, Germany +12Frederick University, 1036 Nicosia, Cyprus +13Laboratoire de Physique des 2 infinis Ir`ene Joliot-Curie, Universit´e Paris-Saclay, CNRS-IN2P3. , F-91405 Orsay , France +14Nuclear Physics Institute, The Czech Academy of Sciences, 25068 Rez, Czech Republic +15LabCAF. F. F´ısica, Univ. de Santiago de Compostela, 15706 Santiago de Compostela, Spain +16Uniwersytet Warszawski - Instytut Fizyki Do´swiadczalnej, 02-093 Warszawa, Poland +17Warsaw University of Technology, 00-662 Warsaw, Poland +18Bergische Universit¨at Wuppertal, 42119 Wuppertal, Germany +a also at Coimbra Polytechnic - ISEC, Coimbra, Portugal +b also at Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Darmstadt, 64390 Darmstadt, Germany +c also at Technische Universit¨at Dresden, 01062 Dresden, Germany +d also at Charles University, Faculty of Mathematics and Physics, 12116 Prague, Czech Republic +e also at Czech Technical University in Prague, 16000 Prague, Czech Republic +f also at Dipartimento di Fisica and INFN, Universit`a di Torino, 10125 Torino, Italy +g also at University of Wroc�law, 50-204 Wroc�law, Poland +hINFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy +iCENTRO FERMI - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, 00184 Rome, Italy +† Deceased. +(Dated: January 11, 2023) +Hadron production (π±, proton, Λ, K0 +S, K±) in π− + C and π− + W collisions is investigated +at an incident pion beam momentum of 1.7 GeV/c. +This comprehensive set of data measured +with HADES at SIS18/GSI significantly extends the existing world data on hadron production in +pion induced reactions and provides a new reference for models that are commonly used for the +interpretation of heavy-ion collisions. The measured inclusive differential production cross-sections +are compared with state-of-the-art transport model (GiBUU, SMASH) calculations. The (semi-) +exclusive channel π− + A → Λ + K0 +S + X, in which the kinematics of the strange hadrons are +correlated, is also investigated and compared to a model calculation. Agreement and remaining +tensions between data and the current version of the considered transport models are discussed. +arXiv:2301.03940v1 [nucl-ex] 10 Jan 2023 + +2 +PACS numbers: 25.80.Hp, 13.75.Jz, 13.75.Gx +I. +INTRODUCTION +The finite expectation values of various quark and +gluon operators characterising the QCD vacuum are +modified already at nuclear saturation density. +As a +consequence, various in-medium modifications of hadron +properties are predicted [1–6]. Of particular interest for +our understanding of neutron stars, such as their masses, +radii, stability properties, and tidal deformability, are +hadrons containing strange quarks in particular in the +context of the hyperon puzzle [7–10]. The presence of +hyperons in neutron stars would soften the equation of +state which is difficult to reconcile with the observation +of large neutron star masses ≥ 2 M⊙. +Experimentally, in-medium properties of hadrons at nu- +clear saturation density can be studied by colliding +photon-, proton-, or pion-beams with nuclear targets, +for reviews see [11, 12]. The experimental challenge is +to select those secondary hadrons which have stayed in- +side the nucleus long enough to experience a modifica- +tion of their properties. Ideally, the hadron of interest is +formed by the incoming beam particle on the surface of +the nucleus with a subsequent long flight path through +the nucleus. +Hence the energy and momentum of the +projectile must be appropriately chosen. Pion-induced +reactions are advantageous compared to proton-induced +reactions, because the inelastic π + A cross section at +low energies is much larger than the p + A one and the +momentum to energy ratio is favorable for the forma- +tion of ”slow” hadrons which propagate through the nu- +clear medium with low probability for secondary interac- +tions. The study of hadrons in nuclear matter provides an +intermediate step between hadron formation in vacuum +[13–15] and in a hot and dense system. Such an inter- +mediate step proved to be useful for the interpretation +of in-medium hadron properties deduced from heavy-ion +collisions [16–23]. Yet, data on pion induced reactions +on nuclear targets at low energies are extremely rare and +mainly focus on kaons [24]. This work presents the in- +clusive spectra of π±, proton, Λ, K0 +S and K± measured +in π− + C and π− + W reactions at a pion-beam mo- +mentum of 1.7 GeV/c. This comprehensive hadron set +significantly extends the existing world data on hadron +production in pion induced reactions at energies of a few +GeV and provides a unique testing ground for different +transport models. As a light (C) and a heavy (W) nuclear +target was used, our data allow to differentiate between +small and large scale medium effects. +In addition to the study of inclusive particle production +the semi-exclusive π + A → Λ + K0 +S + X channel was +measured, in which the correlation between the kinemat- +ics of the two strange hadrons can be exploited. +The single and two-strange-particle (double-)differential +spectra are compared with two state-of-the-art transport +models (GiBUU [25] and SMASH [26]), and it is shown +that for most of the observables a satisfactory description +is still lacking. +This paper is organized as follows; In Sec. II we describe +the experimental setup. Sec. III contains the details of +our data and the comparison with models of the inclusive +π±, proton, Λ, K0 +S and K± spectra. Sec. IV presents the +details and results of the semi-exclusive analysis of the +π− + A → Λ + K0 +S + X channel. +We summarize and +conclude in Sec. V. +II. +EXPERIMENT +The experimental data were measured with the versa- +tile High Acceptance Di-Electon Spectrometer (HADES) +at the SIS18 synchrotron at GSI Helmholtzzentrum in +Darmstadt, Germany [27]. +At this facility, beams can +be prepared with kinetic energies between 1-2 AGeV for +nuclei, up to 4.5 GeV for protons and 0.5-2 GeV for sec- +ondary pions. HADES consists of six identical sectors +surrounding the target area covering polar angles from +18° to 85°. The azimuthal coverage varies from 65 % to +90 %. Each of the six sectors consists of a Ring Imag- +ing CHerenkov (RICH) detector, followed by Multi-Wire- +Drift Chambers (MDCs), two in front of and two behind a +toroidal superconducting magnet, which enable the mea- +surement of the momentum and the specific energy loss, +dE/dx, of charged particles. The Multiplicity and Elec- +tron Trigger Array (META) is composed of two differ- +ent time-of-flight detectors (TOF and RPC) and cov- +ers the polar angle ranges of 44° < ΘT OF < 88° and +12° < ΘRP C < 45°. The META is also used to provide +the First Level Trigger (LVL1) signal. The measurements +were conducted in 2014 employing a momentum of the +secondary pion beam of pπ− = 1.7 GeV/c, impinging on +two nuclear targets (carbon (C) and tungsten (W)). The +pions were produced in interactions of nitrogen ions with +a 10 cm thick beryllium (Be) target. +After extraction +from the SIS18 synchrotron the fully stripped ions had +an intensity of ≈ 1010 during the spills of 2s duration. +Behind the secondary production target, a chicane leads +the π beam to the HADES target. Since the momentum +spread of the secondary pions accepted by the chicane +is about 8%, the latter is equipped with a tracking sys- +tem that allows for the measurement of the momentum +of each secondary π−. This dedicated CERBEROS [28] +setup consists of position sensitive silicon strip sensors +with a high rate stability and has a momentum resolu- +tion of ∆p/p < 0.5%. The secondary beam had an av- +erage beam intensity of Iπ− ≈ 3 × 105 π−/ spill with an +extension at the target focal point of δx ≈ 1 cm (rms) +in agreement with simulations. The pion beam line is +equipped with a mono-crystalline diamond T0 detector +with a timing resolution of στ < 250 ps. Both carbon +and tungsten targets consisted of 3 discs with a diameter + +3 +of 12 mm and thickness of 7.2 mm and 2.4 mm, respec- +tively. During the π− campaign the interaction trigger +LVL1 is defined by requiring the registration of at least +two hits in the META and one hit in the T0 detector. In +total, 1.3 × 108 π− + C and 1.7 × 108 π− + W interac- +tions were recorded. Charged particle trajectories were +reconstructed using the hits measured in the MDCs. The +resulting tracks were subjected to several selections based +on quality parameters delivered by a Runge-Kutta track +fitting algorithm. Their momentum resolution (∆p/p) is +approximately 3% [27]. +III. +INCLUSIVE DATA ANALYSIS +In this section we present the analysis of the inclusive +(double-)differential production cross-section of π±, pro- +ton, Λ and K0 +S. To provide a more complete picture of +strange hadron production, the (double-)differential pro- +duction cross-section of K+ and K− taken from [29] are +presented as well. The obtained differential cross-sections +are compared with two state-of-the-art transport models, +the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) [25] +model and the Simulating Many Accelerated Strongly- +Interacting Hadrons (SMASH) [26] model. +A. +Event selection and particle identification +Only events with a reconstructed primary vertex (PV) +in the target region are considered in the analysis. The +identification of charged particles is based on momentum +and time-of-flight measurements by exploiting the rela- +tion p/ +� +p2 + m2 +0 = β, with m0 being the nominal mass +of π+, π− or proton [30, 31]. The energy loss measured +in the MDCs is used only in the semi-exclusive analysis +discussed in Section IV. +1. +Charged pions and protons +The charged pions are identified by a window of a ±2σ +selection around the pion peak in the β distributions in +slices of p, separately for TOF and RPC. To reduce the +systematic uncertainty of the momentum reconstruction +and of the PID, the momentum of the charged pions +was restricted to pπ± < 1000 MeV/c. Using full-scale +detector-response Geant simulations as a reference, an +average π± purity of 95% and 88% was found for the +π−+C and π−+W reactions, respectively. In order to en- +sure that the efficiency correction takes into account the +effects of residual impurities from misidentification, those +pT −y bins were excluded from the analysis for which the +purity in experiment and simulation deviated by more +than ±5%. +Note, that the mass resolution was found +to be in agreement between simulation and experiment +within 8%. The π± yield was obtained by integrating the +mass distributions for the different pT −y bins. The total +1110 1120 1130 1140 +] +2 +c + [MeV/ +− +π +p +M +0 +1000 +2 +c +Counts / 1.4 MeV/ +X +Λ + +→ + + C +− +π +X +Λ + +→ + + W +− +π +(a) + < 600 +T + p +≤ +500 + y < 0.75 +≤ +0.6 +500 +550 +600 +] +2 +c + [MeV/ +− +π ++ +π +M +0 +1000 +2 +c +Counts / 2.5 MeV/ +X +s +0 + K +→ + + C +− +π +X +s +0 + K +→ + + W +− +π +(b) + < 600 +T + p +≤ +500 + y < 1.0 +≤ +0.8 +FIG. 1. (Color online) Invariant mass distributions of pπ− +(a) and π+π− pairs (b) in π− + C (open points) and π− + W +(solid points) collisions for the representative phase space bin +given in the legend. Lines are fits to the data, see text for +details. +number of reconstructed π+ and π− within the HADES +acceptance in π− + C is N π+ +C += (11.4 ± 0.003) × 106 and +N π− +C += (27.6 ± 0.005) × 106, and in π− + W collisions +N π+ +W = (9.0±0.003)×106 and N π− +W = (23.3±0.005)×106, +respectively. +Similar to the charged pions, the protons were identi- +fied by a ±2σ window around the nominal β vs. p corre- +lation. By integrating the measured mass distributions +the proton yield was extracted for each pT − y bin. On +the basis of full-scale Geant simulations the proton purity +was found to be above 99% for both colliding systems. +The total number of reconstructed protons within the +HADES acceptance is equal to N p +C = (30.5±0.006)×106 +and N p +W = (56.1 ± 0.007) × 106 in π− + C and π− + W +collisions, respectively. +2. +Λ and K0 +S +The +inclusive +production +of +the +neutral +strange +hadrons, Λ and K0 +S, was investigated via their charged +decay channels Λ → π−p (BR ≈ 63.9% [32]) and K0 +S → +π+π− (BR ≈ 69.2% [32]). It has to be noted that the re- +constructed Λ yield contains also a contribution from the +(slightly heavier) Σ0 hyperon, which is decaying electro- +magnetically (almost) exclusively into a Λ together with +a photon. Hence, ”Λ yield” has to be understood as that +of Λ + Σ0 throughout the paper. +Each daughter particle was identified applying a β vs. +momentum cut of +���p/ +� +p2 + m2 +0 − β +��� < 0.2 and the in- +variant mass of the Λ (K0 +S) candidates was calculated +using the nominal masses for the selected daughter par- +ticles. To maximize the signal-to-background ratio (S/B) +of both neutral strange hadrons and to minimize the con- +tribution by off-target reactions, additional topological +cuts were applied. +The position of the PV was calcu- + +4 +lated event-by-event by taking the point of closest ap- +proach (PCA) of the reconstructed Λ or K0 +S trajectories +and the beam axis. +The secondary decay vertex (SV) +corresponds to the PCA of the daughter tracks. Three +additional topological cuts were employed to enhance the +Λ (K0 +S) signal and reduce the combinatorial background: +i) the z coordinate of the SV has to be downstream with +respect to the PV (zP V < zSV ), ii) the distance of closest +approach (DCA) between the decay particle trajectories +and the PV has to fulfill the following conditions: dp > 5 +mm and dπ− > 18 mm for the Λ decays and dπ± > 4.5 +mm for the K0 +S decays. iii) the DCA between the trajec- +tories of the two decay particles has to be smaller than +10 mm for the Λ decays and 6 mm for the K0 +S decays. +Figure 1 shows an example of the resulting invariant +mass distributions for Λ (panel (a)) and K0 +s (panel (b)) +for a selected phase-space bin. For each pT − y bin the +Λ signal in the invariant mass distributions was mod- +elled by the sum of two Gaussians, and the background +by a third degree polynomial. The signal width was in +this case calculated by evaluating the weighted average +of the widths of the two Gaussian. +The K0 +S invariant +mass was fitted with a single Gaussian and a third-order +polynomial. The particle yields were obtained by inte- +grating the signal functions within a ±3σ region. The +mass and resolution are found to be µΛ = 1114.7 MeV/c, +σΛ = 2.3 MeV/c,, respectively µK0 +S = 495.7 MeV/c and +σK0 +S = 6.95 MeV/c and the agreement between exper- +iment and simulation is better than 7% over the whole +phase space. Typical signal-to-background ratios are 8.6 +for Λ and 2.1 for K0 +S candidates, respectively. The total +numbers of reconstructed Λ and K0 +s within the HADES +acceptance in π− + C collisions correspond to NΛ(C) = +(66.2 ± 0.3) × 103 and NK0 +S(C) = (58.6 ± 0.4) × 103, and +in π− + W collisions to NΛ(W) = (79.9 ± 0.3) × 103 and +NK0 +S(W) = (64.1 ± 0.3) × 103. +B. +Double-differential cross-sections +The obtained double-differential inclusive yields of the +five species π+, π−, p, Λ, K0 +S were corrected for the losses +due to inefficiencies of the reconstruction and to limited +acceptance. The average combined acceptance and effi- +ciency of π+(π−) is 50% (40%) for both collision systems, +while the average combined proton acceptance and effi- +ciency is around 56% (50%) for π− + C(W) collisions. +For Λ and K0 +S the average efficiency is 3.8% and 6.3%, +respectively. +The validity of the efficiency correction based on the sim- +ulated detector response of HADES was cross-checked by +means of an additional data sample recorded for pions +with a momentum of pπ− = 0.69 GeV/c impinging on +a solid 12 × 44 mm2 polyethylene (C2H4) target which +allowed to carry out the analysis of the exclusive elastic +interaction channel, π− + p → π− + p [33]. By exploiting +the kinematic constraints of the elastic reaction, it was +0 +200 +400 +600 800 +] +c + [MeV/ +T +p +4 +10 +10 +10 +16 +10 +22 +10 +28 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d + y < 0.1 +≤ +0.0 + y < 0.2 +≤ +0.1 + y < 0.3 +≤ +0.2 + y < 0.4 +≤ +0.3 + y < 0.5 +≤ +0.4 + y < 0.6 +≤ +0.5 + y < 0.7 +≤ +0.6 + y < 0.8 +≤ +0.7 + y < 0.9 +≤ +0.8 + y < 1.0 +≤ +0.9 +X ++ +π + +→ + + C +− +π +(a) +18 +10 +× +0 +10 +× +200 +400 +600 +800 +] +c + [MeV/ +T +p +4 +10 +16 +22 +28 + y < 1.1 +≤ +1.0 + y < 1.2 +≤ +1.1 + y < 1.3 +≤ +1.2 + y < 1.4 +≤ +1.3 + y < 1.5 +≤ +1.4 + y < 1.6 +≤ +1.5 + y < 1.7 +≤ +1.6 + y < 1.8 +≤ +1.7 + y < 1.9 +≤ +1.8 +X ++ +π + +→ + + W +− +π +(b) +17 +10 +× +0 +10 +× +FIG. 2. (Color online) Differential π+ cross-sections in subse- +quent rapidity intervals in the laboratory frame (see legend). +The left panel corresponds to π− + C reactions, while the +right panel to π− + W reactions. For a better representation, +the spectra are scaled by consecutive factors of 10 for each +rapidity bin (100 for 0 ≤ y < 0.1). The combined statistical +and systematic uncertainty and the normalization error are +smaller than the symbol size. The dashed curves correspond +to Boltzmann fits (see text for details). +possible to extract a data-driven detector efficiency map. +It was found that both, experimental and simulated, ef- +ficiencies are consistent within 3%. This uncertainty was +accounted for in the systematic error evaluation. To ob- +tain the absolute cross-sections, the corrected yields were +normalized to the total number of beam particles and the +target density. The normalization error due to the uncer- +tainty on the beam intensity on the target was estimated +to be about 15%. +The resulting double-differential cross-sections for π+ +emission in π− + C (Fig. 2 (a)) and π− + W (Fig. 2 (b)) +collisions are shown for 19 (18) rapidity intervals subdi- +viding the range 0 < y < 1.9 (1.8). Analogously to the +π+, the π− results are presented in Fig. 3 for 18 rapidity +intervals subdividing the range 0.1 < y < 1.9. The sys- +tematic uncertainty was obtained by varying the selection +in the velocity vs. momentum plane between ± 1.5σ, 2σ +and 2.5σ. +For the protons the resulting double-differential cross- +sections in π− + C (Fig. 4 (a)) and π− + W (Fig. 4 (b)) +collisions are shown for 12 rapidity intervals subdividing +the range 0 < y < 1.2. The systematic uncertainty was +extracted using the same variations employed in the pion + +5 +0 +200 +400 +600 800 +] +c + [MeV/ +T +p +4 +10 +10 +10 +16 +10 +22 +10 +28 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d + y < 0.2 +≤ +0.1 + y < 0.3 +≤ +0.2 + y < 0.4 +≤ +0.3 + y < 0.5 +≤ +0.4 + y < 0.6 +≤ +0.5 + y < 0.7 +≤ +0.6 + y < 0.8 +≤ +0.7 + y < 0.9 +≤ +0.8 + y < 1.0 +≤ +0.9 +X +− +π + +→ + + C +− +π +(a) +17 +10 +× +0 +10 +× +200 +400 +600 +800 +] +c + [MeV/ +T +p +4 +10 +16 +22 +28 + y < 1.1 +≤ +1.0 + y < 1.2 +≤ +1.1 + y < 1.3 +≤ +1.2 + y < 1.4 +≤ +1.3 + y < 1.5 +≤ +1.4 + y < 1.6 +≤ +1.5 + y < 1.7 +≤ +1.6 + y < 1.8 +≤ +1.7 + y < 1.9 +≤ +1.8 +X +− +π + +→ + + W +− +π +(b) +17 +10 +× +0 +10 +× +FIG. 3. (Color online) π− double-differential cross-sections +in subsequent rapidity intervals (see legend). The left panel +corresponds to π− + C reactions, while the right panel to +π− +W reactions. For a better representation, each spectrum +is scaled by consecutive factors of 10 for each rapidity range +(100 for 0.1 ≤ y < 0.2). The combined statistical and sys- +tematic uncertainty and the normalization error are smaller +than the symbol size. In the lower rapidity region (y ≲ 0.8), +the inelastic (low pT ) and (quasi-)elastically scattered (high +pT ) π− contribute to the transverse momentum spectra. The +dashed curves correspond to Boltzmann fits, while the solid +curves represent the combined Boltzmann and Gaussian fits +(see text for details). +analysis. The resulting double-differential cross-sections +for Λ in π− + C (Fig. 5 (a)) and π− + W (Fig. 5 (b)) +collisions are shown in Fig. 5 for 7 rapidity intervals sub- +dividing the range 0 < y < 1.05. Figure 6 depicts the +analog for the K0 +S with 8 rapidity intervals in the range +0 < y < 1.6. The systematic uncertainties were obtained +by varying the criteria on the decay topology within 20%. +The errors in Figs. 2 - 6 represent the quadratic sum of +the statistical and systematic, uncertainties and the nor- +malization error and are usually smaller than the symbol +size. +C. +pT -integrated cross-sections +The respective pT integrated cross-section per ra- +pidity +bin +was +calculated +in +the +following +way; +The integration of the measured cross-sections was +complemented with extrapolations in the low- and +high-pT +regions +not +covered +by +HADES +by +em- +500 +1000 +] +c + [MeV/ +T +p +10 +4 +10 +7 +10 +10 +10 +13 +10 +16 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +) +0 +10 +× + y < 0.10 ( +≤ +0.00 +) +1 +10 +× + y < 0.20 ( +≤ +0.10 +) +2 +10 +× + y < 0.30 ( +≤ +0.20 +) +3 +10 +× + y < 0.40 ( +≤ +0.30 +) +4 +10 +× + y < 0.50 ( +≤ +0.40 + pX +→ + + C +− +π +(a) +500 +1000 +] +c + [MeV/ +T +p +10 +4 +7 +10 +13 +16 +) +5 +10 +× + y < 0.60 ( +≤ +0.50 +) +6 +10 +× + y < 0.70 ( +≤ +0.60 +) +7 +10 +× + y < 0.80 ( +≤ +0.70 +) +8 +10 +× + y < 0.90 ( +≤ +0.80 +) +9 +10 +× + y < 1.00 ( +≤ +0.90 + pX +→ + + W +− +π +(b) +FIG. 4. +(Color online) Double-differential proton cross- +sections in different rapidity intervals (see legend). The rep- +resentation is analogous to Fig. 2. +0 +200 400 600 800 +] +c + [MeV/ +T +p +2 +10 +6 +10 +10 +10 +14 +10 +18 +10 +21 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +(a) +X +0 +0 +Λ + +→ + + C +− +π +) +0 +10 +× + y < 0.15 ( +≤ +0.00 +) +2 +10 +× + y < 0.30 ( +≤ +0.15 +) +4 +10 +× + y < 0.45 ( +≤ +0.30 +) +6 +10 +× + y < 0.60 ( +≤ +0.45 +200 400 600 800 +] +c + [MeV/ +T +p +10 +6 +11 +16 +21 +(b) +X +0 +0 +Λ + +→ + + W +− +π +) +8 +10 +× + y < 0.75 ( +≤ +0.60 +) +10 +10 +× + y < 0.90 ( +≤ +0.75 +) +12 +10 +× + y < 1.05 ( +≤ +0.90 +FIG. 5. (Color online) Double-differential Λ cross-sections in +different rapidity intervals (see legend). The representation is +analogous to Fig. 2. +ploying +a +Boltzmann +fit +to +the +measured +distri- +butions. +The +function +reads +d2N/(dpT dy) += +C(y) pT +� +p2 +T + m2 +0 +exp +� +− +� +p2 +T + m2 +0/TB(y) +� +, where +C(y) denotes a scaling factor, m0 is again the respec- +tive nominal mass and TB(y) stands for the inverse-slope +parameter. The relatively modest modifications of the + +6 +0 +200 400 600 800 +] +c + [MeV/ +T +p +2 +10 +7 +10 +12 +10 +17 +10 +22 +10 +24 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +(a) + X +S +0 + K +→ + + C +− +π +) +0 +10 +× + y < 0.20 ( +≤ +0.00 +) +2 +10 +× + y < 0.40 ( +≤ +0.20 +) +4 +10 +× + y < 0.60 ( +≤ +0.40 +) +6 +10 +× + y < 0.80 ( +≤ +0.60 +200 400 600 800 +] +c + [MeV/ +T +p +10 +6 +11 +16 +21 +24 +(b) + X +S +0 + K +→ + + W +− +π +) +8 +10 +× + y < 1.00 ( +≤ +0.80 +) +10 +10 +× + y < 1.20 ( +≤ +1.00 +) +12 +10 +× + y < 1.40 ( +≤ +1.20 +) +14 +10 +× + y < 1.60 ( +≤ +1.40 +FIG. 6. (Color online) Double-differential K0 +S cross-sections +in different rapidity intervals (see legend). The representation +is analogous to Fig. 2. +spectra by the Coulomb field of the nucleus [34] are +small compared to the applied systematic errors. +For +the negatively charged pions the extrapolation is more +complex, since also (quasi)-elastically scattered π− con- +tribute. Hence, in addition to the Boltzmann fit for the +inelastic reactions (low pT ), a Gaussian fit was used for +the elastic events (high pT ). However, for y ≲ 0.8 the +part of the pT distribution corresponding to the (quasi)- +elastically scattered π− is outside of the HADES accep- +tance, and hence only the inelastic part can be extrapo- +lated. In order to extract the inelastic yield over the en- +tire covered rapidity range, all measured data points were +summed up in the inelastic range up to pT = 390 MeV/c +for y ≲ 0.8. On the other hand, the pT coverage for the +protons is larger, and the enhancement due to the (quasi- +)elastic reaction channel is less pronounced. Therefore, +no Gaussian fit is needed for the extrapolation. +As +demonstrated in Figs. 2 - 6 the fits based on an exponen- +tial function describe the experimental data with reason- +able agreement, which is in line with simulation studies +with our event generator Pluto [35] in which the Fermi +motion inside the nucleus was taken into account [31]. +The extrapolation of the π+, π−, p, Λ and K0 +S yields over +the entire pT range allowed to extract the rapidity distri- +butions shown in Figs. 14 - 16. The integrated differential +production cross-sections ∆σ, in the rapidity ranges cov- +ered by HADES (0 ≤ y < 1.05 for Λ, 0 ≤ y < 1.6 for +K0 +S, 0 ≤ y < 1.9 (1.8) for π+ and 0 ≤ y < 0.9 for p), in +π−+C (W) reactions are listed in Tab. I. The uncertainty +of the Boltzmann or combined Boltzmann and Gaussian +extrapolation is taken into account in the systematic er- +ror estimate. The error values shown correspond to the +statistical (first), systematic (second) and normalization +TABLE I. Target, particle species and cross-section integrated +inside the rapidity range covered by HADES. Error values +shown are statistical (first), systematic (second) and normal- +ization (third). +Target Particle +y range +∆σ [µb] +C +Λ +0.0 - 1.05 (4.3 ± 0.02 ± 0.13 ± 0.65) × 103 +C +K0 +S +0.0 - 1.6 +(2.0 ± 0.01 ± 0.08 ± 0.3) × 103 +C +π+ +0.0 - 1.9 +(44 ± 0.01 ± 1.3 ± 6.6) × 103 +C +p +0.0 - 1.0 +(133 ± 0.02 ± 21 ± 20) × 103 +W +Λ +0.0 - 1.05 +(30 ± 0.13+0.68 +−1.1 ± 4.5) × 103 +W +K0 +S +0.0 - 1.6 +(13 ± 0.06+0.3 +−0.28 ± 2) × 103 +W +π+ +0.0 - 1.8 +(153 ± 0.05+4.6 +−5.6 ± 23) × 103 +W +p +0.0 - 0.9 +(156 ± 0.02 ± 56 ± 23) × 104 +TABLE II. As in Table I but for π−. +Target +Particle +y range +∆σ [µb] +C +π−(tot) +0.1 - 0.9 +(57 ± 0.01+1.7 +−1.9 ±8.6) × 103 +C +π−(inelastic) 0.1 - 1.9 (94 ± 0.02+2.8 +−3 +±14.1) × 103 +W +π−(tot) +0.1 - 0.8 (214 ± 0.06 ± 6.5 ±32) × 103 +W +π−(inelastic) 0.1 - 1.9 (348 ± 0.08 ± 11 ±52) × 103 +(third) contribution. Moreover, the integrated differen- +tial inelastic (total) production cross-sections ∆σ for π− +(0.1 ≤ y < 1.9 (0.9/0.8)) in both collision systems inside +the covered rapidity range are given in Tab. II. +D. +Comparison to transport model calculations +Figures 7 - 16 show the comparison of the measured +differential cross-sections as a function of transverse mo- +mentum pT as well as rapidity y with the hadronic trans- +port models GiBUU (v2017) [25] and SMASH (v1.6) +[26]. Both models are run without the inclusion of in- +medium potentials for strange hadrons. +The produc- +tion mechanisms employed in these transport models dif- +fer. In GiBUU, hadron production channels are directly +parameterized based on measured cross-sections. +De- +pending on the production channels SMASH uses an ex- +plicit treatment with intermediate baryon resonances or +parametrizations similar to the GiBUU model. The ele- +mentary strange hadron production channels are listed +in Tab. III. +The corresponding cross-section (σfit) is +given for each channel at the incident pion momentum of +1.7 GeV/c, which was extracted by applying the cross- +section parametrization given in [36, 37], to interpolate +the experimental data to the given beam momentum. In +addition, the cross-sections implemented in GiBUU and +SMASH are listed. In all the following figures, the re- +sults of the GiBUU calculation are represented by solid +curves, while the ones of SMASH are depicted by long- +dashed curves. The upper panels present the comparison + +7 +0 +200 +400 600 +800 + [MeV/c] +T +p +0 +50 +100 +(Sim-Exp)/Exp [%] +200 +400 600 +800 + [MeV/c] +T +p +0 +50 +100 +SMASH x 0.5 +0 +200 +400 600 +800 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +) +0 +10 +× + y < 0.1 ( +≤ +0.0 +) +1 +10 +× + y < 0.6 ( +≤ +0.5 +) +2 +10 +× + y < 1.1 ( +≤ +1.0 +X ++ +π + +→ + + C +− +π +(a) +200 +400 +600 +800 +1 +10 +2 +3 +4 +5 +6 +7 + GiBUU + SMASH +X ++ +π + +→ + + W +− +π +(b) +FIG. 7. +(Color online) Upper panel: +(Double-)differential +cross-sections of π+ as a function of the transverse momen- +tum pT in π− + C (a) and π− + W (b) reactions compared +with GiBUU (solid curves) and SMASH (long-dashed curves) +for different rapidity intervals (see legend). The combined, +statistical and systematic error is represented by the lines, +while the normalization error is indicated by a box. Both er- +rors are smaller than the symbol size. Lower panel: Relative +deviations between experimental data and the two transport +model calculations. For better visibility the deviations to the +SMASH calculation are scaled with the factor 0.5. +of the experimental with the model data in a logarithmic +scale, while the lower panels show the deviation between +the measured and simulated distributions expressed as +the relative difference normalized to experimental cross- +section ((Sim-Exp)/Exp) in linear scale. +1. +Pions and protons +Considering first π+, Fig. +7 shows the comparison +between the measured differential cross-sections as a +function of transverse momentum pT +with GiBUU +(solid curve) and SMASH (long-dashed curve) results +for low (0.0 − 0.1), intermediate (0.5 − 0.6) and high +(1.0 − 1.1) rapidity regions in π− + C (Fig. 7 (a)) +and π− + W (Fig. 7 (b)) collisions, respectively. +In +general, both models describe the shapes of the pT +distribution for π+ similarly well, with differences of +mostly less than 50%. The yields from the models are +systematically higher than those in the experimental +data by about 25%, with deviations as large as a +factor of 2 (3) at low and high pT in the heavy target +case for GiBUU (SMASH) data. +The π+ production +cross-section as function of rapidity is included in +Fig. 14 below, together with the model data. +The +model calculations differ by up to 50% over the whole +0 +200 +400 600 +800 + [MeV/c] +T +p +0 +100 +200 +300 +(Sim-Exp)/Exp [%] +200 +400 600 +800 + [MeV/c] +T +p +0 +100 +200 +300 +0 +200 +400 600 +800 +2 +10 +4 +10 +6 +10 +8 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +) +0 +10 +× + y < 0.2 ( +≤ +0.1 +) +4 +10 +× + y < 0.6 ( +≤ +0.5 +) +9 +10 +× + y < 1.1 ( +≤ +1.0 +X +− +π + +→ + + C +− +π +(a) +200 +400 +600 +800 +2 +4 +6 +8 + GiBUU + SMASH +X +− +π + +→ + + W +− +π +(b) +FIG. 8. +(Color online) Comparison of the π− differential +cross-sections as a function of the transverse momentum with +GiBUU (solid curves) and SMASH (long-dashed curves). The +representation is analogous to Fig. 7. +considered rapidity range for the heavy target case and +only at forward rapidities for the light target case. The +relative differences with respect to the experimental data +stay below 100% in the former and 50% in the latter case. +The π− differential cross-sections as a function of pT +are compared to the GiBUU (solid curve) and SMASH +(long-dashed curve) calculations for low (0.1 − 0.2), +intermediate (0.5 − 0.6) and high rapidity (1.0 − 1.1) +regions in π− + C (Fig. 8 (a)) and π− + W (Fig. 8 (b)) +collisions, respectively. The general features are similar +to the ones observed for π+ production. +However, +there is in addition the (quasi-)elastic process which +contributes to the measured π− cross-section. +The +corresponding enhancement is visible in the high-pT +region and more pronounced in the model results than +in the experimental data by a factor of two for SMASH +and three for GiBUU. +Not only the inelastic but also the (quasi-)elastic +reactions contribute to the measured π− cross-section. +In particular, in the high-pT region, corresponding to +the (quasi-)elastic scattering events, both theoretical +predictions +significantly +overshoot +the +experimental +data. +The comparison of the π− cross-section as a +function of rapidity with the models is shown in Fig. 15. +Both models reproduce the experimental data within +30% for the small target nucleus. In the tungsten case +the cross section found by the models is by a factor of +two higher than the experimental data. +For technical reasons, protons are only compared to the +GiBUU calculations. Figure 9 shows the proton differen- + +8 +500 +1000 + [MeV/c] +T +p +0 +100 +200 +(Sim-Exp)/Exp [%] +500 +1000 + [MeV/c] +T +p +0 +100 +200 +500 +1000 +2 +10 +4 +10 +6 +10 +8 +10 +10 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +) +0 +10 +× + y < 0.2 ( +≤ +0.1 +) +2 +10 +× + y < 0.6 ( +≤ +0.4 +) +4 +10 +× + y < 0.9 ( +≤ +0.8 + pX +→ + + C +− +π +(a) +500 +1000 +2 +4 +6 +8 +10 + GiBUU + pX +→ + + W +− +π +(b) +FIG. 9. (Color online) Comparison of the proton differential +cross-sections as a function of the transverse momentum with +GiBUU (solid curves). +The representation is analogous to +Fig. 7. +tial cross-sections as a function of pT compared with the +predictions, for low (0.1 − 0.2), intermediate (0.4 − 0.6) +and high (0.8−0.9) rapidity regions in π−+C (panel (a)) +and π− +W (panel (b)) collisions, respectively. For both +colliding systems, the proton yield is overestimated by +the GiBUU model, most pronounced at high pT where +it is higher by a factor of roughly 2.0 (1.6) in the case +of carbon (tungsten). Note that GiBUU does not form +composite objects, hence a part of the proton excess is +due the neglected binding of protons in light nuclei. A +hint at the expected enhancement due to elastic events is +visible in the model data in the lowest rapidity bin, but in +a region which is not covered by the experimental data. +The experimental proton cross-section as a function of +rapidity is presented in Fig. 14 together with the GiBUU +calculations, which overshoots the data by a factor of 3 +(2) only near target rapidity in the carbon (tungsten) +case. +2. +Strange hadrons +In Fig. 10 the experimental pT distributions of Λ are +compared with the models for low (0.0 − 0.15), medium +(0.45−0.5) and high (0.9−1.05) rapidities. Similar shapes +and absolute cross-sections are observed for GiBUU and +SMASH. However, the values predicted by the models +are systematically below the measured ones for both col- +lision systems, except for the high rapidity interval. +Fig. 14 shows different rapidity distributions for the Λ +production off C (panel (a)) and W (panel (b)) targets. +While in case of carbon most of the yield is inside the ra- +0 +200 400 600 800 + [MeV/c] +T +p +50 +− +0 +50 +100 +(Sim-Exp)/Exp [%] +x 0.5 +200 400 600 800 + [MeV/c] +T +p +50 +0 +50 +100 +x 0.5 +0 +200 400 600 800 +1 +2 +10 +4 +10 +6 +10 +8 +10 +10 +10 +12 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +(a) +) +0 +10 +× + y < 0.15 ( +≤ +0.00 +) +6 +10 +× + y < 0.60 ( +≤ +0.45 +) +12 +10 +× + y < 1.05 ( +≤ +0.90 +X +0 +0 +Λ + +→ + + C +− +π +200 400 600 800 +1 +2 +5 +8 +11 +12 +(b) + GiBUU + SMASH +X +0 +0 +Λ + +→ + + W +− +π +FIG. 10. +(Color online) Comparison of the Λ differential +cross-sections as a function of the transverse momentum with +GiBUU (solid curves) and SMASH (long-dashed curves). The +representation is analogous to Fig. 7. Lower panel: Devia- +tions between transport models and data. For better visibility +the deviations in the forward bin are scaled with the factor +0.5. +pidity range covered by HADES, the Λ hyperons experi- +ence backward scattering in tungsten. Also here the data +of the transport models do not agree well with the ex- +perimental distributions. Both models predict a double- +hump structure for the lighter target, not seen in the ex- +perimental data. The calculated cross section in π− + C +(π− + W) undershoots the data by up to 50 % (60 %). +For the heavier target both models show similar distri- +butions, again a double-hump structure, contrary to the +experimental data and underestimate the cross-section. +Summarizing, a precise theoretical description of the +double-differential Λ production cross-sections is missing. +For the K0 +S, the comparison of the differential cross- +section as a function of pT is depicted in Fig. 11 for +backward (0.15 − 0.30), middle (0.45 − 0.60) and for- +ward (0.75 − 0.90) rapidity. For the GiBUU model an +overall good agreement of the shape and cross-section is +observed in both collision systems with minor deviations +for pT ≥ 600 MeV/c. +SMASH overshoots the experi- +mental data over the entire pT range in both collision +systems. +In Fig. 16, the K0 +S rapidity distribution for +π− + C (panel (a)) and π− + W (panel (b)) collisions is +shown. The two experimental distributions have different +shapes. Similar to the Λ, they are shifted to backward +rapidity in reactions with the heavier target. The result +of the GiBUU model is consistent with the experimental +data also as function of rapidity over (almost) the entire +range. SMASH overestimates the cross-section over the +entire rapidity range by a factor of 2 (4) for reactions +with the Carbon (Tungsten) target. + +9 +0 +200 400 600 800 + [MeV/c] +T +p +0 +100 +(Sim-Exp)/Exp [%] +SMASH x 0.5 +SMASH x 0.5 +SMASH x 0.5 +200 400 600 800 + [MeV/c] +T +p +0 +100 +SMASH x 0.5 +SMASH x 0.5 +SMASH x 0.5 +0 +200 400 600 800 +1 +3 +10 +6 +10 +9 +10 +12 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +(a) +) +2 +10 +× + y < 0.30 ( +≤ +0.15 +) +6 +10 +× + y < 0.60 ( +≤ +0.45 +) +10 +10 +× + y < 0.90 ( +≤ +0.75 + X +S +0 + K +→ + + C +− +π +200 400 600 800 +1 +3 +6 +9 +12 +(b) + GiBUU + SMASH + X +S +0 + K +→ + + W +− +π +FIG. 11. +(Color online) Comparison of the K0 +S differential +cross-sections as a function of the transverse momentum with +GiBUU (solid curves) and SMASH (long-dashed curves). The +representation is analogous to Fig. 7. Lower panel: Deviation +of the transport models calculations to the experimental data +as a function of rapidity. For better visibility the deviations +in the SMASH case are scaled with the factor 0.5. +0 +200 +400 +600 + [MeV/c] +T +p +100 +− +50 +− +0 +50 +(Sim-Exp)/Exp [%] +200 +400 +600 + [MeV/c] +T +p +100 +50 +0 +50 +SMASH x 0.5 +0 +200 +400 +600 +1 +2 +10 +4 +10 +6 +10 +8 +10 +9 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +) +0 +10 +× + y < 0.1 ( +≤ +0.0 +) +2 +10 +× + y < 0.6 ( +≤ +0.5 +) +4 +10 +× + y < 1.1 ( +≤ +1.0 +X ++ + K +→ + + C +− +π +(a) +200 +400 +600 +1 +2 +4 +6 +8 +9 + GiBUU + SMASH +X ++ + K +→ + + W +− +π +(b) +FIG. 12. (Color online) Comparison of the K+ differential +cross-sections [29] as a function of the transverse momentum +to GiBUU (solid curves) and SMASH (long-dashed curves). +The representation is analogous to Fig. 7. The deviations to +SMASH in the lower right panel are scaled with the factor +0.5. +100 +200 +300 +400 + [MeV/c] +T +p +50 +− +0 +50 +(Sim-Exp)/Exp [%] +100 +200 +300 +400 + [MeV/c] +T +p +50 +0 +50 +100 +200 +300 +400 +1 +− +10 +1 +10 +2 +10 +3 +10 +4 +10 +)] +c + MeV/ +y +∆ +b/( +µ +) [ +y +d +T +p +/(d +σ +2 +d +) +0 +10 +× + y < 0.5 ( +≤ +0.2 +) +1 +10 +× + y < 0.7 ( +≤ +0.5 +) +2 +10 +× + y < 1.0 ( +≤ +0.7 +X +− + K +→ + + C +− +π +(a) +100 +200 +300 +400 +1 +1 +10 +2 +3 +4 + GiBUU + SMASH +X +− + K +→ + + W +− +π +(b) +FIG. 13. (Color online) Comparison of the K− differential +cross-sections [29] as a function of the transverse momentum +to GiBUU (solid curves) and SMASH (long-dashed curves). +The representation is analogous to Fig. 7. +Both models are also compared with the recently pub- +lished differential K+ production cross-sections obtained +for the same collision systems [29]. In Fig. 12 the K+ +differential cross-section as a function of pT is shown +for backward (0.0 − 0.1), middle (0.5 − 0.6) and forward +(1.0−1.1) rapidity. GiBUU underestimates the K+ cross- +section in π− + C (panel (a)) and π− + W (panel (b)) +collisions over the entire pT and rapidity range by up +to 50 %. +Except for the region close to target rapid- +ity, the SMASH results exceed the experimental cross- +section in both nuclear reactions by up to 80%. The K+ +cross-section is presented as a function of the rapidity in +Fig. 16 together with the results of the model calcula- +tions. GiBUU describes the data rather well with devi- +ations of only 20% to 60%, whereas SMASH exhibits a +different shape with agreement near target rapidity and +a deviation of up to a factor of 5 at the highest measured +rapidity. The model calculations of K+ and K0 +S produc- +tion shown in Fig. 16 are significantly different: SMASH +finds very similar shapes and sizes of the two cross sec- +tions resulting in an almost constant K+/K0 +S cross sec- +tion ratio (close to unity) as a function of rapidity. The +GiBUU ratios, however, increase significantly from close +to unity near target rapidity to 10 at high rapidity. This +trend is also seen in the experimental data. +The set of kaons are completed with the comparison for +charged antikaons [29]. Figure 13 presents the differential +K− cross-sections as a function of pT for three measured +rapidity intervals, 0.2 − 0.5, 0.5 − 0.7 and 0.7 − 1.0. For +both colliding systems, GiBUU reproduces the shape of +the experimental spectra rather well. The cross-section +is slightly underestimated for low pT in π− + C collisions + +10 +0.5 +− +0 +0.5 +1 +1.5 +y +0 +100 +(Sim-Exp)/Exp [%] +x 0.5 +0.5 +− +0 +0.5 +1 +1.5 +y +0 +100 +0.5 +− +0 +0.5 +1 +1.5 +4 +10 +6 +10 +8 +10 +10 +10 +y] +∆ +b/ +µ +/dy [ +σ +d +X +Λ + +→ + + C +− +π +X ++ +π + +→ + + C +− +π +5) +× + pX ( +→ + + C +− +π + + GiBUU + SMASH +y +(a) +0.5 +− +0 +0.5 +1 +1.5 +4 +6 +8 +10 +X +Λ + +→ + + W +− +π +X ++ +π + +→ + + W +− +π +5) +× + pX ( +→ + + W +− +π + +y +(b) +FIG. 14. (Color online) Upper panel: Cross-section of Λ (or- +ange points), π+ (green squares) and p (red triangle) as a +function of rapidity in π− + C (a) and π− + W (b) reactions +compared with the transport models, GiBUU (solid curve) +and SMASH (long-dashed curve). The shaded bands denote +the systematic errors. The open boxes indicate the normal- +ization error. The statistical uncertainties are smaller than +the symbol size. Lower panel: Deviations of the three trans- +port models from the measured cross-section of Λ (π±, p) as +a function of rapidity. For better visibility the deviations for +protons from GiBUU are scaled with factor 0.5. +(panel (a)) and π− + W (panel (b)) reactions, except for +low rapidities in the latter reaction. On the other hand, +SMASH underestimates the differential cross-section al- +most over the entire pT range for the lighter nucleus, +while the shape agrees rather well. Also the model re- +sults for the antikaon cross-section as a function of rapid- +ity is investigated in Fig. 16. GiBUU slightly underesti- +mates the K− production cross-section off carbon, while +the production cross-section off tungsten is slightly over- +estimated. Both shapes are rather well reproduced by +GiBUU. For the heavier nucleus, SMASH is able to re- +produce the experimental data. Only minor deviations +are observed for low rapidity. In general, the experimen- +tal data and GiBUU are almost consistent. +In summary, neither GiBUU nor SMASH can precisely +describe simultaneously the cross-sections as function of +transverse momentum and rapidity in terms of shape and +absolute yield of the presented comprehensive hadron set. +IV. +(SEMI-) EXCLUSIVE DATA ANALYSIS +At the pion beam momentum of 1.7 GeV/c, which is +studied here, strangeness production occurs mainly in +first-chance π− + N collisions with a kaon and a Λ (or +Σ) in the final state. +In addition, several other semi- +inclusive channels contribute as well (see Tab. III). +Although GiBUU describes the inclusive K0 +S data rea- +0.5 +− +0 +0.5 +1 +1.5 +y +0 +50 +100 +(Sim-Exp)/Exp [%] +0.5 +− +0 +0.5 +1 +1.5 +y +0 +50 +100 +0.5 +− +0 +0.5 +1 +1.5 +5 +10 +6 +10 +7 +10 +y] +∆ +b/ +µ +/dy [ +σ +d +X +− +π + +→ + + C +− +π +(tot) + +X +− +π + +→ + + C +− +π +(inelastic) (x 0.5) + GiBUU + SMASH + +(a) +0.5 +− +0 +0.5 +1 1.5 +5 +6 +7 +X +− +π + +→ + + W +− +π +(tot) +X +− +π + +→ + + W +− +π +(inelastic) (x 0.5) +(b) +FIG. 15. (Color online) Comparison of the total (triangles) +and inelastic (crosses) π− differential cross-sections as a func- +tion of rapidity with GiBUU (solid curves) and SMASH (long- +dashed curves). The representation is analogous to Fig. 14. +0.5 +− +0 +0.5 +1 +1.5 +y +0 +100 +200 +300 +(Sim-Exp)/Exp [%] +0 +0.5 +1 +1.5 +y +0 +100 +200 +300 +0.5 +− +0 +0.5 +1 +1.5 +3 +10 +5 +10 +7 +10 +9 +10 +10 +10 +y] +∆ +b/ +µ +/dy [ +σ +d +X +s +0 + K +→ + + C +− +π +X ++ + K +→ + + C +− +π +X +− + K +→ + + C +− +π + GiBUU + SMASH +2 + 10 +× +1 + 10 +× +(a) +0.5 +− +0 +0.5 +1 +1.5 +3 +5 +7 +9 +10 +X +s +0 + K +→ + + W +− +π +X ++ + K +→ + + W +− +π +X +− + K +→ + + W +− +π + +2 + 10 +× +1 + 10 +× +(b) +FIG. 16. (Color online) Comparison of the K0 +S (violet rect- +angles), K+ [29] (red triangles) and K− [29] (green stars) +cross-sections as a function of rapidity with GiBUU (solid +curves) and SMASH (long-dashed curves). The representa- +tion is analogous to Fig. 14. +sonably well, the agreement with inclusive Λ and K+ +data is not satisfactory. +Therefore, more information +was gained by also analysing the (semi-)exclusive channel +π−+A → Λ+K0 +S+X for both colliding systems, allowing +a comparison of the data on associated strangeness pro- +duction to model calculations. The corresponding final +states were reconstructed via the weak charged decays of +the Λ and the K0 +S inside the HADES acceptance. The fol- +lowing final states were analysed: Λ+K0 +S, Λ+K0 +S +π0,−, +Σ0 + K0 +S and Σ0 + K0 +S + π0,−. These include contribu- + +11 +Channel +pthr +σfit +σGiBUU σSMASH +π− + p +[GeV/c] +[mb] +[mb] +[mb] +ΛK0 +0.896 +0.177 +0.067 +0.163 +Σ0K0 +1.031 +0.146 +0.132 +0.105 +Σ−K+ +1.035 +0.150 +0.156 +0.130 +Λπ0K0 +1.140 +0.118 +0.110 +0.074 +Λπ−K+ +1.144 +0.079 +0.091 +0.149 +Σ+π−K0 +1.290 +0.014 +0.015 +0.005 +Σ0π0K0 +1.286 +0.034 +0.030 +0.136 +Σ0π−K+ +1.290 +0.022 +0.021 +0.269 +Σ−π+K0 +1.305 +0.037 +0.030 +0.201 +Σ−π0K+ +1.290 +0.019 +0.015 +0.102 +pK0K− +1.290 +0.007 +0.011 +0.003 +nK+K− +1.495 +0.023 +0.022 +0.024 +nφ +1.559 +0.027 +0.020 +- +Λπ+π−K0 +1.423 +0.003 +- +- +Λπ0π−K+ +1.407 +0.002 +- +- +Σ+π0π−K0 +1.564 +≈ 0 +- +- +Σ+π−π−K+ +1.568 +≈ 0 +- +- +Σ0π−π+K0 +1.580 +≈ 0 +- +- +Σ−π+π0K0 +1.580 +≈ 0 +- +- +Σ−π+π−K+ +1.580 +≈ 0 +- +- +π− + n +Σ−K0 +1.038 +< 0.049 +0.458 +0.273 +Σ−π0K0 +1.296 +< 0.042 +0.036 +0.505 +Σ−π−K+ +1.290 +< 0.070 +0.025 +1.035 +TABLE III. The production channels of Λ and K0 in elemen- +tary π−N reactions together with the corresponding thresh- +old momenta of the incident pions. The cross-section σfit at +pπ− = 1.7 GeV/c represents the value obtained from a fit ac- +cording to the parametrisation given in [36, 37] to experimen- +tal data at several beam momenta. Also listed are σGiBUU, +where the parametrisations were evaluated at the proper in- +cident pion momenta, and σSMASH, where the cross-sections +were extracted in elementary mode. Channels not included +in the models are labeled with ”-”. +tions from the production of Σ−K0 +S with the subsequent +strong conversion process of Σ−N → Λ(Σ0)N. +A. +Event hypothesis and constraints +Considering the decay patterns of Λ → pπ− and +K0 +S → π+π−, two positively and two negatively charged +tracks were required as a minimal event selection crite- +rion. Due to the limited acceptance for events with four +charged particles in HADES, a different particle identi- +fication based on probability and event hypothesis was +employed. All negatively charged particles were assumed +to be π− originating from strange particle decays, and +an additional cut on the reconstructed mass, as calu- +lated from the momentum and velocity measurement, of +mπ− > 80 MeV/c2 was applied. For the remaining two +positively charged particles, a likelihood method was em- +ployed, selecting the best matching candidate for a pro- +ton, based on the difference to the theoretical values of +its velocity and energy loss dE/dx in the MDCs. In ad- +dition, the proton candidate had to fulfill a mass cut of +800 < mp [MeV/c2] < 1400. Finally, the remaining pos- +itively charged particle was accepted as a π+ if its mass +fulfilled the condition of 80 < mπ+ [MeV/c2] < 400. +To resolve the ambiguity of the negative pion origi- +nating from the different sources, the possible combi- +nations Λ1(p, π− +1 )K0 +S,1(π+π− +2 ) and Λ2(p, π− +2 )K0 +S,2(π+π− +1 ) +are formed. Only the combination with the best match- +ing of the invariant mass of pπ− pairs (Mpπ−) to the +nominal Λ mass and of the invariant mass of π+π− pairs +(Mπ+π−) to the nominal K0 +S mass was considered for the +further analysis. +The plot of the corresponding corre- +lations is shown in Fig. 17. This selection does not in- +troduce any bias as the invariant masses of the rejected +combination do not fit the Λ and K0 +S hypotheses. +The final data sample was selected using a two- +dimensional elliptical (TDE) area around on the invariant +mass correlation with half-axes of ±3σ: +� +� +� +� +�∆MΛ − µΛ +3 · σΛ +�2 ++ +� +∆MK0 +S − µK0 +S +3 · σK0 +S +�2 +≤ 1, +(1) +where σΛ(K0 +S) denotes the width, µΛ(K0 +S) the offset and +∆MΛ(K0 +S) the difference of the invariant mass to the nom- +inal mass. The width σΛ (σK0 +S) was extracted by fitting +the invariant-mass distribution Mpπ− (Mπ+π−), which +has been pre-selected to be within a ±3¯σK0 +S (±3¯σΛ) win- +dow around the invariant mass Mπ+π− (Mpπ−) with ¯σK0 +S +(¯σΛ) obtained beforehand in the inclusive analyses. The +invariant mass distributions were modeled with a Gaus- +sian for the signal and a second-order polynomial for +the background. This choice ensures a minimal loss of +signal, while obtaining a data sample with a signal-to- +background ratio between S/B = 1.3 and 5.45. +To reject the remaining background after the TDE +selection, a sideband subtraction was employed. Since +the selection of the semi-exclusive ΛK0 +S channel is based +on the correlation of invariant mass spectra, a simple +one-dimensional sideband is not applicable separately for +each particle. To extract a suitable sample containing +enough statistics to describe the background in the sig- +nal area, a TDE cut of 4σΛ(K0 +S)−15σΛ(K0 +S) was applied to +the invariant mass correlation, indicated by the black el- +lipse in Fig. 17. This sideband sample thus accounts for +the kinematic correlation of the Λ and K0 +S. For the side- +band subtraction, the sideband sample has to be scaled +to the background contribution after the TDE selection. +The corresponding scaling factor was extracted in the fol- +lowing way. The total Λ and K0 +S signal was obtained by +fitting both invariant mass distributions before the TDE +selection. +Since, after the TDE selection, the total Λ +and K0 +S signal stays the same, but the underlying back- +ground is altered and thus cannot be well described by + +12 +] +2 +c + [MeV/ +PDG +Λ + - M +-π +p/ +M +40 +− +20 +− +0 +20 +40 +] +2 +c + [MeV/ +PDG +S +0 +K + - M +-π +/ ++ +π +M +50 +− +0 +50 +Counts +1 +10 +2 +10 +FIG. 17. (Color Online) Yield distribution in the plane of in- +variant mass of π+π− pairs vs the invariant mass of pπ− pairs, +both subtracted by their nominal mother particle mass.. The +grey shaded area indicates the 3σ TDE cut, while the black +ellipse represents the lower boundary for the two-dimensional +side-band, spanning from 4σ - 15σ. A clear peak at the origin +is visible, with a low background contribution. +any fitting procedure, the background contribution was +estimated by subtracting the combined Λ and K0 +S signal +from the total yield of invariant mass distributions. The +sideband sample was scaled to the estimated background +after the TDE selection and the obtained distribution +was then subtracted from all spectra fulfilling the TDE +selection. The kinematic distributions obtained after the +subtraction are used for the kinematic investigations and +comparisons performed later-on. Figures 18 and 19 show +the transverse momentum (Fig. 18 (a) and Fig. 19 (a)) +and rapidity (Figs. 18 (b) and 19 (b)) distributions for +K0 +S and Λ inside the HADES acceptance for the C target +(purple circles) and the W target (orange stars) without +any corrections for reconstruction efficiency. Therefore, +the simulated kinematic distributions by GiBUU have +been convoluted with the acceptance and efficiency of +HADES to allow for a direct comparison. +B. +Systematics +To estimate the systematic error introduced by the de- +scribed analysis procedure all applied cuts were varied +and their impact on the final spectra was investigated. +As the exclusive data was not corrected for efficiency and +acceptance effects, the impact on the shape of the distri- +bution and not on the yield was studied. In this way the +whole analysis procedure was performed with another cut +set and then compared to the shape of the nominal cut +set by calculating the difference in each point, after per- +forming a χ2 minimization. +In total eight different variations have been considered: +±1σ variation for the extraction of the particle invariant +mass widths, ±0.5σ for the TDE cut, 5 < σΛ(K0 +S) < 15 +and 4 < σΛ(K0 +S) < 10 for the sideband region, and the +signal yield was taken solely from the K0 +S or Λ. +The +signal to background ratio for the carbon target for the +nominal cut set is 2.4 and varies systematically from 1.8 +to 5.0. For the tungsten target the corresponding value +is 3.1 and the systematics is found to be in between 2.0 +to 8.2. The same procedure was performed for the sim- +ulation, where the combined variations are smaller than +the line width. +C. +Comparison to transport models +As GiBUU allows to reconstruct the particle history we +restrict the theory comparison to this model. Figures 18 +and 19 show the transverse momentum and rapidity dis- +tributions for K0 +S and Λ inside the HADES acceptance +for the C target (purple circles) and the W target (orange +stars) compared to results from GiBUU. The experimen- +tal statistical errors are indicated by the error bars, while +in the simulation the statistical errors are negligible. The +systematic errors are indicated by the shaded boxes. The +systematic study revealed that in simulations they are +smaller then the width of the line. For this comparison +we focus on the shape of the spectra, therefore the simu- +lated distributions have been scaled to the experimental +distributions by means of a global χ2/NDF minimization +procedure. The experimental pT spectrum for the K0 +S in +Fig. 18 (a) for the heavy target is rather symmetric with +a maximum around 300 MeV/c. GiBUU however pre- +dicts a distribution that is shifted to lower pT , peaking at +200 MeV/c, with χ2/d.o.f. = 28.2. For the lighter target, +the maximum of the experimental distribution is shifted +to higher pT around 400 MeV/c featuring an asymmetric +shape, while GiBUU predicts a more symmetric shape, +with a lower maximum and a lower cut-off of the dis- +tribution and a corresponding χ2/d.o.f. = 28.5. In both +cases the experimental shape cannot be reproduced. The +rapidity distribution of K0 +S (Fig. 18 (b)) for the heavy +system (W) is rather symmetric with a maximum at +about 0.7 and a shift to lower rapidies with respect to +the smaller colliding system. This is well reproduced by +the GiBUU model as reflected by a χ2/d.o.f. = 1.8 and +points to K0 scattering inside the heavy nucleus, also +seen in the inclusive spectra. In case of the lighter system +(C), where the distribution is shifted to higher rapidi- +ties, GiBUU can reproduce the data qualitatively, with +a slightly smaller maximum and a χ2/d.o.f. of 2.7. +The transverse momentum distributions of the Λ hy- +perons are shown in Fig. 19 (a). Both experimental dis- +tributions have rather similar shapes, although in π− +C +reactions the distribution is shifted to higher pT . In both +cases GiBUU is able to reproduce the pT dependence +very well in the low pT region, with a slight systematic + +13 +0 +200 400 600 800 +] +c + [MeV/ +T +p +0 +200 +400 +600 +800 +1000 +Counts +(a) ++ X +S +0 + + K +Λ + +→ + + C/W +− +π +in HADES acc. +0.5 +1 +1.5 +y +0 +200 +400 +600 +800 +1000 + C HADES + C GiBUU + W HADES + W GiBUU +s +0 +K +(b) +FIG. 18. +(Color Online) Transverse momentum and ra- +pidity distributions of K0 +S in the (semi-)exclusive channel +π− + A → Λ(Σ0) + K0 +S + X without reconstruction efficiency +correction inside the HADES acceptance together with the +GiBUU predictions. (a) Transverse momentum spectra of the +K0 +S for the lighter carbon target (purple circles) and the heav- +ier tungsten nuclei (orange stars). For the experimental data +the statistical errors are indicated by error bars and system- +atic errors indicated by shaded boxes, while the systematic +errors for the simulation are smaller than the width of the +drawn line. +(b) Rapidity distribution for the K0 +S with the +same convention as for the transverse momentum. +0 +200 400 600 800 +] +c + [MeV/ +T +p +0 +200 +400 +600 +800 +1000 +Counts +(a) ++ X +S +0 + + K +Λ + +→ + + C/W +− +π +in HADES acc. +0.5 +1 +y +0 +200 +400 +600 +800 +1000 + C HADES + C GiBUU + W HADES + W GiBUU +Λ +(b) +FIG. 19. As Fig. 18 but for Λ. +shift towards the high pT region and a corresponding +χ2/d.o.f. of 2.7 and 1.8 for carbon and tungsten, respec- +tively. The situation changes for the rapidity distribu- +tions. For the heavier target a maximum around 0.4 is +observed. +GiBUU can predict the shape qualitatively +with χ2/d.o.f. = 2.8, while the deviations in the lighter +system increase, reflected in χ2/d.o.f. = 4.1. In general, +the rapidity distributions of both particles in both nu- +clear systems are qualitatively reproduced, where again +a backward shift is observed, pointing to scattering in- +side the heavy nucleus. +In case of the transverse mo- +mentum distribution of K0 +S, the results of GiBUU show +larger deviations while for the Λs they are qualitatively +reproduced. If one considers the global χ2, the results +for the heavier system are slightly better with a χ2 of +9.38 compared to the lighter one with 10.26. Neverthe- +less, a satisfactory description of all kinematic observable +simultaneously in both systems is not achieved, which is +consistent with the results of, the inclusive analysis of +strange hadrons above. +The (semi-) exclusive data might be the ideal tool to test +the implementation of interaction potentials in transport +models simultaneously for kaons and hyperons in the fu- +ture, especially in light of the new constraints on these in- +teractions extracted from femtoscopy measurements [38– +40]. +V. +SUMMARY AND CONCLUSION +We presented the inclusive differential cross-sections +as a function of transverse momentum pT and rapid- +ity y for π+, π−, p, Λ and K0 +S measured in π− + C +and π− + W reactions at an incident pion momentum +of pπ− = 1.7 GeV/c within the rapidity range covered by +the HADES detector. The presented data significantly +extend the world data base on hadron production in pion- +induced reactions on nuclear targets. +Scattering effects are observed, shifting the maximum of +the π+, π−, Λ and K0 +S rapidity distributions to smaller +rapidities in the heavier target. +The pT and rapidity spectra have been compared to two +state-of-the-art transport models, GiBUU and SMASH. +To provide a more complete picture of the (strange) me- +son production, the inclusive double-differential produc- +tion cross-section of K± measured in the same reactions +system, taken from [29], were compared with theory as +well. In both transport models presented, no in-medium +potentials for the KN or ΛN interactions were included. +Concerning the phase space distributions of π+ in π−+C +reactions, GiBUU describes (almost) the experimental +data in terms of the shape and absolute cross-section, +whereas in π− + W reactions the cross-section is signif- +icantly overestimated with deviations up to factor of 2. +SMASH overshoots the experimental data in both col- +liding systems with deviations as large as a factor of 3. +Similar to the π+, both models overestimate the π− dif- +ferential cross-sections. Hence, the description of rescat- +tering and/or absorption effects seems to be particularly +insufficient, as the model predictions deviate significantly +stronger for the heavier target (W) and for the (quasi)- +elastically scattered π−. GiBUU is also not able to de- +scribe the (quasi)-elastically scattered protons. +While +the results of GiBUU for K0 +S and K− are rather consis- +tent with our data, the cross-sections of Λ and K+ are +under-estimated. +In general, due to the imperfect de- +scription of all observable of the comprehensive hadron +set (π±, Λ, K0 +S and K±), an improvement of these models +becomes desirable, especially with regard to the interpre- +tation of heavy-ion data. + +14 +Furthermore, the phase-space distribution in the (semi- +)exclusive channel π− + A → Λ + K0 +S + X was investi- +gated and a comparison to the GiBUU model was done. +It was found that GiBUU cannot describe all the corre- +lated kinematic observable simultaneously, in particular +the calculation for the K0 +S transverse momentum distri- +bution is not well reproduced. +The +HADES +Collaboration +thanks +T. +Gaitanos, +M. +Bleicher, +J. +Steinheimer, +H. +Elfner, +J. +Stau- +denmaier +and +V. +Steinberg +for +elucidating +dis- +cussions. +We +gratefully +acknowledge +the +sup- +port given by the following institutions and agen- +cies: +SIP +JUC +Cracow, +Cracow +(Poland), +Na- +tional Science Center, 2016/23/P/ST2/040 POLONEZ, +2017/25/N/ST2/00580, 2017/26/M/ST2/00600; WUT +Warszawa (Poland) No: 2020/38/E/ST2/00019 (NCN), +IDUB-POB-FWEiTE-3; +TU +Darmstadt, +Darmstadt +(Germany), +DFG GRK 2128, +DFG CRC-TR 211, +BMBF:05P18RDFC1, HFHF, ELEMENTS:500/10.006, +VH-NG-823,GSI F&E, ExtreMe Matter Institute EMMI +at GSI Darmstadt; Goethe-University, Frankfurt (Ger- +many), BMBF:05P12RFGHJ, GSI F&E, HIC for FAIR +(LOEWE), ExtreMe Matter Institute EMMI at GSI +Darmstadt; TU M¨unchen, Garching (Germany), MLL +M¨unchen, DFG EClust 153, GSI TMLRG1316F, BmBF +05P15WOFCA, SFB 1258, +DFG FAB898/2-2; +JLU +Giessen, +Giessen (Germany), +BMBF:05P12RGGHM; +IJCLab Orsay, Orsay (France), CNRS/IN2P3, P2IO +Labex, +France; +NPI +CAS, +Rez, +Rez +(Czech +Re- +public), MSMT LM2018112, LTT17003, MSMT OP +VVV CZ.02.1.01/0.0/0.0/18 046/0016066; IDUB-POB- +FWEiTE-3. +The following colleagues from Russian institutes did con- +tribute to the results presented in this publication, but +are not listed as authors following the decision of the +HADES Collaboration Board on March 23, 2022: +A. +Belyaev, O. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Wiebusch5, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Wirth10,9, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Zbroszczyk17, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Zherebtsova5,g, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Zumbruch5 (HADES collaboration) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Curceanug, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Piscicchiah,g, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Scordog 1LIP-Laborat´orio de Instrumenta¸c˜ao e F´ısica Experimental de Part´ıculas ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 3004-516 Coimbra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Portugal 2AGH University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Faculty of Physics and Applied Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 30-059 Krak´ow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Poland 3Institute of Nuclear Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 31342 Krak´ow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Poland 4Smoluchowski Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Jagiellonian University of Cracow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 30-059 Krak´ow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Poland 5GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 64291 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Germany 6Technische Universit¨at Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 64289 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} 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9Excellence Cluster ’Origin and Structure of the Universe’ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Germany 10Physik Department E62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Germany 11II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='Physikalisches Institut, Justus Liebig Universit¨at Giessen, 35392 Giessen, Germany 12Frederick University, 1036 Nicosia, Cyprus 13Laboratoire de Physique des 2 infinis Ir`ene Joliot-Curie, Universit´e Paris-Saclay, CNRS-IN2P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' , F-91405 Orsay , France 14Nuclear Physics Institute, The Czech Academy of Sciences, 25068 Rez, Czech Republic 15LabCAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' F´ısica, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' de Santiago de Compostela,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 15706 Santiago de Compostela,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Spain 16Uniwersytet Warszawski - Instytut Fizyki Do´swiadczalnej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 02-093 Warszawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Poland 17Warsaw University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 00-662 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Poland 18Bergische Universit¨at Wuppertal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 42119 Wuppertal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Germany a also at Coimbra Polytechnic - ISEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Coimbra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Portugal b also at Helmholtz Research Academy Hesse for FAIR (HFHF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Campus Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 64390 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Germany c also at Technische Universit¨at Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 01062 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Germany d also at Charles University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Faculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 12116 Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Czech Republic e also at Czech Technical University in Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 16000 Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Czech Republic f also at Dipartimento di Fisica and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Universit`a di Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 10125 Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Italy g also at University of Wroc�law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 50-204 Wroc�law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Poland hINFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Laboratori Nazionali di Frascati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 00044 Frascati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Italy iCENTRO FERMI - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 00184 Rome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Italy † Deceased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Dated: January 11, 2023) Hadron production (π±, proton, Λ, K0 S, K±) in π− + C and π− + W collisions is investigated at an incident pion beam momentum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This comprehensive set of data measured with HADES at SIS18/GSI significantly extends the existing world data on hadron production in pion induced reactions and provides a new reference for models that are commonly used for the interpretation of heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The measured inclusive differential production cross-sections are compared with state-of-the-art transport model (GiBUU, SMASH) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The (semi-) exclusive channel π− + A → Λ + K0 S + X, in which the kinematics of the strange hadrons are correlated, is also investigated and compared to a model calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Agreement and remaining tensions between data and the current version of the considered transport models are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='03940v1 [nucl-ex] 10 Jan 2023 2 PACS numbers: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='Hp, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='Jz, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='Gx I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' INTRODUCTION The finite expectation values of various quark and gluon operators characterising the QCD vacuum are modified already at nuclear saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' As a consequence, various in-medium modifications of hadron properties are predicted [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Of particular interest for our understanding of neutron stars, such as their masses, radii, stability properties, and tidal deformability, are hadrons containing strange quarks in particular in the context of the hyperon puzzle [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The presence of hyperons in neutron stars would soften the equation of state which is difficult to reconcile with the observation of large neutron star masses ≥ 2 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Experimentally, in-medium properties of hadrons at nu- clear saturation density can be studied by colliding photon-, proton-, or pion-beams with nuclear targets, for reviews see [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The experimental challenge is to select those secondary hadrons which have stayed in- side the nucleus long enough to experience a modifica- tion of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Ideally, the hadron of interest is formed by the incoming beam particle on the surface of the nucleus with a subsequent long flight path through the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Hence the energy and momentum of the projectile must be appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Pion-induced reactions are advantageous compared to proton-induced reactions, because the inelastic π + A cross section at low energies is much larger than the p + A one and the momentum to energy ratio is favorable for the forma- tion of ”slow” hadrons which propagate through the nu- clear medium with low probability for secondary interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The study of hadrons in nuclear matter provides an intermediate step between hadron formation in vacuum [13–15] and in a hot and dense system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Such an inter- mediate step proved to be useful for the interpretation of in-medium hadron properties deduced from heavy-ion collisions [16–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Yet, data on pion induced reactions on nuclear targets at low energies are extremely rare and mainly focus on kaons [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This work presents the in- clusive spectra of π±, proton, Λ, K0 S and K± measured in π− + C and π− + W reactions at a pion-beam mo- mentum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This comprehensive hadron set significantly extends the existing world data on hadron production in pion induced reactions at energies of a few GeV and provides a unique testing ground for different transport models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' As a light (C) and a heavy (W) nuclear target was used, our data allow to differentiate between small and large scale medium effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In addition to the study of inclusive particle production the semi-exclusive π + A → Λ + K0 S + X channel was measured, in which the correlation between the kinemat- ics of the two strange hadrons can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The single and two-strange-particle (double-)differential spectra are compared with two state-of-the-art transport models (GiBUU [25] and SMASH [26]), and it is shown that for most of the observables a satisfactory description is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This paper is organized as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' II we describe the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' III contains the details of our data and the comparison with models of the inclusive π±, proton, Λ, K0 S and K± spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' IV presents the details and results of the semi-exclusive analysis of the π− + A → Λ + K0 S + X channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' We summarize and conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' EXPERIMENT The experimental data were measured with the versa- tile High Acceptance Di-Electon Spectrometer (HADES) at the SIS18 synchrotron at GSI Helmholtzzentrum in Darmstadt, Germany [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' At this facility, beams can be prepared with kinetic energies between 1-2 AGeV for nuclei, up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 GeV for protons and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5-2 GeV for sec- ondary pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' HADES consists of six identical sectors surrounding the target area covering polar angles from 18° to 85°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The azimuthal coverage varies from 65 % to 90 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Each of the six sectors consists of a Ring Imag- ing CHerenkov (RICH) detector, followed by Multi-Wire- Drift Chambers (MDCs), two in front of and two behind a toroidal superconducting magnet, which enable the mea- surement of the momentum and the specific energy loss, dE/dx, of charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The Multiplicity and Elec- tron Trigger Array (META) is composed of two differ- ent time-of-flight detectors (TOF and RPC) and cov- ers the polar angle ranges of 44° < ΘT OF < 88° and 12° < ΘRP C < 45°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The META is also used to provide the First Level Trigger (LVL1) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The measurements were conducted in 2014 employing a momentum of the secondary pion beam of pπ− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c, impinging on two nuclear targets (carbon (C) and tungsten (W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The pions were produced in interactions of nitrogen ions with a 10 cm thick beryllium (Be) target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' After extraction from the SIS18 synchrotron the fully stripped ions had an intensity of ≈ 1010 during the spills of 2s duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Behind the secondary production target, a chicane leads the π beam to the HADES target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Since the momentum spread of the secondary pions accepted by the chicane is about 8%, the latter is equipped with a tracking sys- tem that allows for the measurement of the momentum of each secondary π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This dedicated CERBEROS [28] setup consists of position sensitive silicon strip sensors with a high rate stability and has a momentum resolu- tion of ∆p/p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The secondary beam had an av- erage beam intensity of Iπ− ≈ 3 × 105 π−/ spill with an extension at the target focal point of δx ≈ 1 cm (rms) in agreement with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The pion beam line is equipped with a mono-crystalline diamond T0 detector with a timing resolution of στ < 250 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both carbon and tungsten targets consisted of 3 discs with a diameter 3 of 12 mm and thickness of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 mm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 mm, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' During the π− campaign the interaction trigger LVL1 is defined by requiring the registration of at least two hits in the META and one hit in the T0 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In total, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 × 108 π− + C and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 × 108 π− + W interac- tions were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Charged particle trajectories were reconstructed using the hits measured in the MDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The resulting tracks were subjected to several selections based on quality parameters delivered by a Runge-Kutta track fitting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Their momentum resolution (∆p/p) is approximately 3% [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' INCLUSIVE DATA ANALYSIS In this section we present the analysis of the inclusive (double-)differential production cross-section of π±, pro- ton, Λ and K0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To provide a more complete picture of strange hadron production, the (double-)differential pro- duction cross-section of K+ and K− taken from [29] are presented as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The obtained differential cross-sections are compared with two state-of-the-art transport models, the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) [25] model and the Simulating Many Accelerated Strongly- Interacting Hadrons (SMASH) [26] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Event selection and particle identification Only events with a reconstructed primary vertex (PV) in the target region are considered in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The identification of charged particles is based on momentum and time-of-flight measurements by exploiting the rela- tion p/ � p2 + m2 0 = β, with m0 being the nominal mass of π+, π− or proton [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The energy loss measured in the MDCs is used only in the semi-exclusive analysis discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Charged pions and protons The charged pions are identified by a window of a ±2σ selection around the pion peak in the β distributions in slices of p, separately for TOF and RPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To reduce the systematic uncertainty of the momentum reconstruction and of the PID, the momentum of the charged pions was restricted to pπ± < 1000 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Using full-scale detector-response Geant simulations as a reference, an average π± purity of 95% and 88% was found for the π−+C and π−+W reactions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In order to en- sure that the efficiency correction takes into account the effects of residual impurities from misidentification, those pT −y bins were excluded from the analysis for which the purity in experiment and simulation deviated by more than ±5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Note, that the mass resolution was found to be in agreement between simulation and experiment within 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The π± yield was obtained by integrating the mass distributions for the different pT −y bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The total 1110 1120 1130 1140 ] 2 c [MeV/ − π p M 0 1000 2 c Counts / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 MeV/ X Λ → + C − π X Λ → + W − π (a) < 600 T p ≤ 500 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 500 550 600 ] 2 c [MeV/ − π + π M 0 1000 2 c Counts / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 MeV/ X s 0 K → + C − π X s 0 K → + W − π (b) < 600 T p ≤ 500 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Invariant mass distributions of pπ− (a) and π+π− pairs (b) in π− + C (open points) and π− + W (solid points) collisions for the representative phase space bin given in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lines are fits to the data, see text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' number of reconstructed π+ and π− within the HADES acceptance in π− + C is N π+ C = (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='003) × 106 and N π− C = (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='005) × 106, and in π− + W collisions N π+ W = (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='003)×106 and N π− W = (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='005)×106, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Similar to the charged pions, the protons were identi- fied by a ±2σ window around the nominal β vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' p corre- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' By integrating the measured mass distributions the proton yield was extracted for each pT − y bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' On the basis of full-scale Geant simulations the proton purity was found to be above 99% for both colliding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The total number of reconstructed protons within the HADES acceptance is equal to N p C = (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='006)×106 and N p W = (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='007) × 106 in π− + C and π− + W collisions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Λ and K0 S The inclusive production of the neutral strange hadrons, Λ and K0 S, was investigated via their charged decay channels Λ → π−p (BR ≈ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9% [32]) and K0 S → π+π− (BR ≈ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2% [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' It has to be noted that the re- constructed Λ yield contains also a contribution from the (slightly heavier) Σ0 hyperon, which is decaying electro- magnetically (almost) exclusively into a Λ together with a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Hence, ”Λ yield” has to be understood as that of Λ + Σ0 throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Each daughter particle was identified applying a β vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' momentum cut of ���p/ � p2 + m2 0 − β ��� < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 and the in- variant mass of the Λ (K0 S) candidates was calculated using the nominal masses for the selected daughter par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To maximize the signal-to-background ratio (S/B) of both neutral strange hadrons and to minimize the con- tribution by off-target reactions, additional topological cuts were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The position of the PV was calcu- 4 lated event-by-event by taking the point of closest ap- proach (PCA) of the reconstructed Λ or K0 S trajectories and the beam axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The secondary decay vertex (SV) corresponds to the PCA of the daughter tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Three additional topological cuts were employed to enhance the Λ (K0 S) signal and reduce the combinatorial background: i) the z coordinate of the SV has to be downstream with respect to the PV (zP V < zSV ), ii) the distance of closest approach (DCA) between the decay particle trajectories and the PV has to fulfill the following conditions: dp > 5 mm and dπ− > 18 mm for the Λ decays and dπ± > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 mm for the K0 S decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' iii) the DCA between the trajec- tories of the two decay particles has to be smaller than 10 mm for the Λ decays and 6 mm for the K0 S decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Figure 1 shows an example of the resulting invariant mass distributions for Λ (panel (a)) and K0 s (panel (b)) for a selected phase-space bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For each pT − y bin the Λ signal in the invariant mass distributions was mod- elled by the sum of two Gaussians, and the background by a third degree polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The signal width was in this case calculated by evaluating the weighted average of the widths of the two Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The K0 S invariant mass was fitted with a single Gaussian and a third-order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The particle yields were obtained by inte- grating the signal functions within a ±3σ region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The mass and resolution are found to be µΛ = 1114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 MeV/c, σΛ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 MeV/c,, respectively µK0 S = 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 MeV/c and σK0 S = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='95 MeV/c and the agreement between exper- iment and simulation is better than 7% over the whole phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Typical signal-to-background ratios are 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 for Λ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 for K0 S candidates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The total numbers of reconstructed Λ and K0 s within the HADES acceptance in π− + C collisions correspond to NΛ(C) = (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3) × 103 and NK0 S(C) = (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4) × 103, and in π− + W collisions to NΛ(W) = (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3) × 103 and NK0 S(W) = (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3) × 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Double-differential cross-sections The obtained double-differential inclusive yields of the five species π+, π−, p, Λ, K0 S were corrected for the losses due to inefficiencies of the reconstruction and to limited acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The average combined acceptance and effi- ciency of π+(π−) is 50% (40%) for both collision systems, while the average combined proton acceptance and effi- ciency is around 56% (50%) for π− + C(W) collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For Λ and K0 S the average efficiency is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The validity of the efficiency correction based on the sim- ulated detector response of HADES was cross-checked by means of an additional data sample recorded for pions with a momentum of pπ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='69 GeV/c impinging on a solid 12 × 44 mm2 polyethylene (C2H4) target which allowed to carry out the analysis of the exclusive elastic interaction channel, π− + p → π− + p [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' By exploiting the kinematic constraints of the elastic reaction, it was 0 200 400 600 800 ] c [MeV/ T p 4 10 10 10 16 10 22 10 28 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 X + π → + C − π (a) 18 10 × 0 10 × 200 400 600 800 ] c [MeV/ T p 4 10 16 22 28 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 X + π → + W − π (b) 17 10 × 0 10 × FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Differential π+ cross-sections in subse- quent rapidity intervals in the laboratory frame (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The left panel corresponds to π− + C reactions, while the right panel to π− + W reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For a better representation, the spectra are scaled by consecutive factors of 10 for each rapidity bin (100 for 0 ≤ y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The combined statistical and systematic uncertainty and the normalization error are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The dashed curves correspond to Boltzmann fits (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' possible to extract a data-driven detector efficiency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' It was found that both, experimental and simulated, ef- ficiencies are consistent within 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This uncertainty was accounted for in the systematic error evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To ob- tain the absolute cross-sections, the corrected yields were normalized to the total number of beam particles and the target density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The normalization error due to the uncer- tainty on the beam intensity on the target was estimated to be about 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The resulting double-differential cross-sections for π+ emission in π− + C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2 (a)) and π− + W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2 (b)) collisions are shown for 19 (18) rapidity intervals subdi- viding the range 0 < y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Analogously to the π+, the π− results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 3 for 18 rapidity intervals subdividing the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 < y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The sys- tematic uncertainty was obtained by varying the selection in the velocity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' momentum plane between ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5σ, 2σ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the protons the resulting double-differential cross- sections in π− + C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 4 (a)) and π− + W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 4 (b)) collisions are shown for 12 rapidity intervals subdividing the range 0 < y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The systematic uncertainty was extracted using the same variations employed in the pion 5 0 200 400 600 800 ] c [MeV/ T p 4 10 10 10 16 10 22 10 28 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 X − π → + C − π (a) 17 10 × 0 10 × 200 400 600 800 ] c [MeV/ T p 4 10 16 22 28 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 X − π → + W − π (b) 17 10 × 0 10 × FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) π− double-differential cross-sections in subsequent rapidity intervals (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The left panel corresponds to π− + C reactions, while the right panel to π− +W reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For a better representation, each spectrum is scaled by consecutive factors of 10 for each rapidity range (100 for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ≤ y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The combined statistical and sys- tematic uncertainty and the normalization error are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In the lower rapidity region (y ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8), the inelastic (low pT ) and (quasi-)elastically scattered (high pT ) π− contribute to the transverse momentum spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The dashed curves correspond to Boltzmann fits, while the solid curves represent the combined Boltzmann and Gaussian fits (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The resulting double-differential cross-sections for Λ in π− + C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 5 (a)) and π− + W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 5 (b)) collisions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 5 for 7 rapidity intervals sub- dividing the range 0 < y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Figure 6 depicts the analog for the K0 S with 8 rapidity intervals in the range 0 < y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The systematic uncertainties were obtained by varying the criteria on the decay topology within 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The errors in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2 - 6 represent the quadratic sum of the statistical and systematic, uncertainties and the nor- malization error and are usually smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' pT -integrated cross-sections The respective pT integrated cross-section per ra- pidity bin was calculated in the following way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The integration of the measured cross-sections was complemented with extrapolations in the low- and high-pT regions not covered by HADES by em- 500 1000 ] c [MeV/ T p 10 4 10 7 10 10 10 13 10 16 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='10 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ) 1 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='20 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='10 ) 2 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='30 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='20 ) 3 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='40 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='30 ) 4 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='50 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='40 pX → + C − π (a) 500 1000 ] c [MeV/ T p 10 4 7 10 13 16 ) 5 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='50 ) 6 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='70 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ) 7 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='80 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='70 ) 8 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='80 ) 9 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90 pX → + W − π (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Double-differential proton cross- sections in different rapidity intervals (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The rep- resentation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 0 200 400 600 800 ] c [MeV/ T p 2 10 6 10 10 10 14 10 18 10 21 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d (a) X 0 0 Λ → + C − π ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='15 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ) 2 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='30 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='15 ) 4 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='30 ) 6 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45 200 400 600 800 ] c [MeV/ T p 10 6 11 16 21 (b) X 0 0 Λ → + W − π ) 8 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ) 10 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75 ) 12 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Double-differential Λ cross-sections in different rapidity intervals (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' ploying a Boltzmann fit to the measured distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The function reads d2N/(dpT dy) = C(y) pT � p2 T + m2 0 exp � − � p2 T + m2 0/TB(y) � , where C(y) denotes a scaling factor, m0 is again the respec- tive nominal mass and TB(y) stands for the inverse-slope parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The relatively modest modifications of the 6 0 200 400 600 800 ] c [MeV/ T p 2 10 7 10 12 10 17 10 22 10 24 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d (a) X S 0 K → + C − π ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='20 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ) 2 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='40 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='20 ) 4 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='40 ) 6 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='80 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 200 400 600 800 ] c [MeV/ T p 10 6 11 16 21 24 (b) X S 0 K → + W − π ) 8 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='80 ) 10 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='20 ( ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ) 12 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='40 ( ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='20 ) 14 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ( ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='40 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Double-differential K0 S cross-sections in different rapidity intervals (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' spectra by the Coulomb field of the nucleus [34] are small compared to the applied systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the negatively charged pions the extrapolation is more complex, since also (quasi)-elastically scattered π− con- tribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Hence, in addition to the Boltzmann fit for the inelastic reactions (low pT ), a Gaussian fit was used for the elastic events (high pT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' However, for y ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 the part of the pT distribution corresponding to the (quasi)- elastically scattered π− is outside of the HADES accep- tance, and hence only the inelastic part can be extrapo- lated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In order to extract the inelastic yield over the en- tire covered rapidity range, all measured data points were summed up in the inelastic range up to pT = 390 MeV/c for y ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' On the other hand, the pT coverage for the protons is larger, and the enhancement due to the (quasi- )elastic reaction channel is less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Therefore, no Gaussian fit is needed for the extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' As demonstrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2 - 6 the fits based on an exponen- tial function describe the experimental data with reason- able agreement, which is in line with simulation studies with our event generator Pluto [35] in which the Fermi motion inside the nucleus was taken into account [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The extrapolation of the π+, π−, p, Λ and K0 S yields over the entire pT range allowed to extract the rapidity distri- butions shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14 - 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The integrated differential production cross-sections ∆σ, in the rapidity ranges cov- ered by HADES (0 ≤ y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05 for Λ, 0 ≤ y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 for K0 S, 0 ≤ y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8) for π+ and 0 ≤ y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 for p), in π−+C (W) reactions are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The uncertainty of the Boltzmann or combined Boltzmann and Gaussian extrapolation is taken into account in the systematic er- ror estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The error values shown correspond to the statistical (first), systematic (second) and normalization TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Target, particle species and cross-section integrated inside the rapidity range covered by HADES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Error values shown are statistical (first), systematic (second) and normal- ization (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Target Particle y range ∆σ [µb] C Λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='65) × 103 C K0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3) × 103 C π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='01 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) × 103 C p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 (133 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='02 ± 21 ± 20) × 103 W Λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05 (30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='68 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5) × 103 W K0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 (13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='28 ± 2) × 103 W π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 (153 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ± 23) × 103 W p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (156 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='02 ± 56 ± 23) × 104 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' As in Table I but for π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Target Particle y range ∆σ [µb] C π−(tot) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='01+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) × 103 C π−(inelastic) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='02+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 −3 ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1) × 103 W π−(tot) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 (214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='06 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ±32) × 103 W π−(inelastic) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (348 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='08 ± 11 ±52) × 103 (third) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Moreover, the integrated differen- tial inelastic (total) production cross-sections ∆σ for π− (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ≤ y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8)) in both collision systems inside the covered rapidity range are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Comparison to transport model calculations Figures 7 - 16 show the comparison of the measured differential cross-sections as a function of transverse mo- mentum pT as well as rapidity y with the hadronic trans- port models GiBUU (v2017) [25] and SMASH (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both models are run without the inclusion of in- medium potentials for strange hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The produc- tion mechanisms employed in these transport models dif- fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In GiBUU, hadron production channels are directly parameterized based on measured cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' De- pending on the production channels SMASH uses an ex- plicit treatment with intermediate baryon resonances or parametrizations similar to the GiBUU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The ele- mentary strange hadron production channels are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The corresponding cross-section (σfit) is given for each channel at the incident pion momentum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c, which was extracted by applying the cross- section parametrization given in [36, 37], to interpolate the experimental data to the given beam momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In addition, the cross-sections implemented in GiBUU and SMASH are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In all the following figures, the re- sults of the GiBUU calculation are represented by solid curves, while the ones of SMASH are depicted by long- dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The upper panels present the comparison 7 0 200 400 600 800 [MeV/c] T p 0 50 100 (Sim-Exp)/Exp [%] 200 400 600 800 [MeV/c] T p 0 50 100 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 0 200 400 600 800 1 10 2 10 3 10 4 10 5 10 6 10 7 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ) 1 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ) 2 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ( ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 X + π → + C − π (a) 200 400 600 800 1 10 2 3 4 5 6 7 GiBUU SMASH X + π → + W − π (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Upper panel: (Double-)differential cross-sections of π+ as a function of the transverse momen- tum pT in π− + C (a) and π− + W (b) reactions compared with GiBUU (solid curves) and SMASH (long-dashed curves) for different rapidity intervals (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The combined, statistical and systematic error is represented by the lines, while the normalization error is indicated by a box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both er- rors are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lower panel: Relative deviations between experimental data and the two transport model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For better visibility the deviations to the SMASH calculation are scaled with the factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' of the experimental with the model data in a logarithmic scale, while the lower panels show the deviation between the measured and simulated distributions expressed as the relative difference normalized to experimental cross- section ((Sim-Exp)/Exp) in linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Pions and protons Considering first π+, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7 shows the comparison between the measured differential cross-sections as a function of transverse momentum pT with GiBUU (solid curve) and SMASH (long-dashed curve) results for low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1), intermediate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) and high (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1) rapidity regions in π− + C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7 (a)) and π− + W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7 (b)) collisions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In general, both models describe the shapes of the pT distribution for π+ similarly well, with differences of mostly less than 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The yields from the models are systematically higher than those in the experimental data by about 25%, with deviations as large as a factor of 2 (3) at low and high pT in the heavy target case for GiBUU (SMASH) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The π+ production cross-section as function of rapidity is included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14 below, together with the model data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The model calculations differ by up to 50% over the whole 0 200 400 600 800 [MeV/c] T p 0 100 200 300 (Sim-Exp)/Exp [%] 200 400 600 800 [MeV/c] T p 0 100 200 300 0 200 400 600 800 2 10 4 10 6 10 8 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ) 4 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ) 9 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ( ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 X − π → + C − π (a) 200 400 600 800 2 4 6 8 GiBUU SMASH X − π → + W − π (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the π− differential cross-sections as a function of the transverse momentum with GiBUU (solid curves) and SMASH (long-dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' considered rapidity range for the heavy target case and only at forward rapidities for the light target case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The relative differences with respect to the experimental data stay below 100% in the former and 50% in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The π− differential cross-sections as a function of pT are compared to the GiBUU (solid curve) and SMASH (long-dashed curve) calculations for low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2), intermediate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) and high rapidity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1) regions in π− + C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 8 (a)) and π− + W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 8 (b)) collisions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The general features are similar to the ones observed for π+ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' However, there is in addition the (quasi-)elastic process which contributes to the measured π− cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The corresponding enhancement is visible in the high-pT region and more pronounced in the model results than in the experimental data by a factor of two for SMASH and three for GiBUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Not only the inelastic but also the (quasi-)elastic reactions contribute to the measured π− cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In particular, in the high-pT region, corresponding to the (quasi-)elastic scattering events, both theoretical predictions significantly overshoot the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The comparison of the π− cross-section as a function of rapidity with the models is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both models reproduce the experimental data within 30% for the small target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In the tungsten case the cross section found by the models is by a factor of two higher than the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For technical reasons, protons are only compared to the GiBUU calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Figure 9 shows the proton differen- 8 500 1000 [MeV/c] T p 0 100 200 (Sim-Exp)/Exp [%] 500 1000 [MeV/c] T p 0 100 200 500 1000 2 10 4 10 6 10 8 10 10 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ) 2 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 ) 4 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 pX → + C − π (a) 500 1000 2 4 6 8 10 GiBUU pX → + W − π (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the proton differential cross-sections as a function of the transverse momentum with GiBUU (solid curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' tial cross-sections as a function of pT compared with the predictions, for low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2), intermediate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) and high (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9) rapidity regions in π−+C (panel (a)) and π− +W (panel (b)) collisions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For both colliding systems, the proton yield is overestimated by the GiBUU model, most pronounced at high pT where it is higher by a factor of roughly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) in the case of carbon (tungsten).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Note that GiBUU does not form composite objects, hence a part of the proton excess is due the neglected binding of protons in light nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' A hint at the expected enhancement due to elastic events is visible in the model data in the lowest rapidity bin, but in a region which is not covered by the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The experimental proton cross-section as a function of rapidity is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14 together with the GiBUU calculations, which overshoots the data by a factor of 3 (2) only near target rapidity in the carbon (tungsten) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Strange hadrons In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 10 the experimental pT distributions of Λ are compared with the models for low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='15), medium (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5) and high (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='9−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05) rapidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Similar shapes and absolute cross-sections are observed for GiBUU and SMASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' However, the values predicted by the models are systematically below the measured ones for both col- lision systems, except for the high rapidity interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14 shows different rapidity distributions for the Λ production off C (panel (a)) and W (panel (b)) targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' While in case of carbon most of the yield is inside the ra- 0 200 400 600 800 [MeV/c] T p 50 − 0 50 100 (Sim-Exp)/Exp [%] x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 200 400 600 800 [MeV/c] T p 50 0 50 100 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 0 200 400 600 800 1 2 10 4 10 6 10 8 10 10 10 12 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d (a) ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='15 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='00 ) 6 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45 ) 12 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='05 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90 X 0 0 Λ → + C − π 200 400 600 800 1 2 5 8 11 12 (b) GiBUU SMASH X 0 0 Λ → + W − π FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the Λ differential cross-sections as a function of the transverse momentum with GiBUU (solid curves) and SMASH (long-dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lower panel: Devia- tions between transport models and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For better visibility the deviations in the forward bin are scaled with the factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' pidity range covered by HADES, the Λ hyperons experi- ence backward scattering in tungsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Also here the data of the transport models do not agree well with the ex- perimental distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both models predict a double- hump structure for the lighter target, not seen in the ex- perimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The calculated cross section in π− + C (π− + W) undershoots the data by up to 50 % (60 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the heavier target both models show similar distri- butions, again a double-hump structure, contrary to the experimental data and underestimate the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Summarizing, a precise theoretical description of the double-differential Λ production cross-sections is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the K0 S, the comparison of the differential cross- section as a function of pT is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 11 for backward (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='30), middle (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60) and for- ward (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90) rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the GiBUU model an overall good agreement of the shape and cross-section is observed in both collision systems with minor deviations for pT ≥ 600 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' SMASH overshoots the experi- mental data over the entire pT range in both collision systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 16, the K0 S rapidity distribution for π− + C (panel (a)) and π− + W (panel (b)) collisions is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The two experimental distributions have different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Similar to the Λ, they are shifted to backward rapidity in reactions with the heavier target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The result of the GiBUU model is consistent with the experimental data also as function of rapidity over (almost) the entire range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' SMASH overestimates the cross-section over the entire rapidity range by a factor of 2 (4) for reactions with the Carbon (Tungsten) target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 9 0 200 400 600 800 [MeV/c] T p 0 100 (Sim-Exp)/Exp [%] SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 200 400 600 800 [MeV/c] T p 0 100 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 0 200 400 600 800 1 3 10 6 10 9 10 12 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d (a) ) 2 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='30 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='15 ) 6 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='60 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45 ) 10 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='90 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='75 X S 0 K → + C − π 200 400 600 800 1 3 6 9 12 (b) GiBUU SMASH X S 0 K → + W − π FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the K0 S differential cross-sections as a function of the transverse momentum with GiBUU (solid curves) and SMASH (long-dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lower panel: Deviation of the transport models calculations to the experimental data as a function of rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For better visibility the deviations in the SMASH case are scaled with the factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 0 200 400 600 [MeV/c] T p 100 − 50 − 0 50 (Sim-Exp)/Exp [%] 200 400 600 [MeV/c] T p 100 50 0 50 SMASH x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 0 200 400 600 1 2 10 4 10 6 10 8 10 9 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ) 2 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ) 4 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 ( ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 X + K → + C − π (a) 200 400 600 1 2 4 6 8 9 GiBUU SMASH X + K → + W − π (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the K+ differential cross-sections [29] as a function of the transverse momentum to GiBUU (solid curves) and SMASH (long-dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The deviations to SMASH in the lower right panel are scaled with the factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 100 200 300 400 [MeV/c] T p 50 − 0 50 (Sim-Exp)/Exp [%] 100 200 300 400 [MeV/c] T p 50 0 50 100 200 300 400 1 − 10 1 10 2 10 3 10 4 10 )] c MeV/ y ∆ b/( µ ) [ y d T p /(d σ 2 d ) 0 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 ) 1 10 × y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 ) 2 10 × y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 ( ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 X − K → + C − π (a) 100 200 300 400 1 1 10 2 3 4 GiBUU SMASH X − K → + W − π (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the K− differential cross-sections [29] as a function of the transverse momentum to GiBUU (solid curves) and SMASH (long-dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both models are also compared with the recently pub- lished differential K+ production cross-sections obtained for the same collision systems [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 12 the K+ differential cross-section as a function of pT is shown for backward (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1), middle (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='6) and forward (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1) rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' GiBUU underestimates the K+ cross- section in π− + C (panel (a)) and π− + W (panel (b)) collisions over the entire pT and rapidity range by up to 50 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Except for the region close to target rapid- ity, the SMASH results exceed the experimental cross- section in both nuclear reactions by up to 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The K+ cross-section is presented as a function of the rapidity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 16 together with the results of the model calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' GiBUU describes the data rather well with devi- ations of only 20% to 60%, whereas SMASH exhibits a different shape with agreement near target rapidity and a deviation of up to a factor of 5 at the highest measured rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The model calculations of K+ and K0 S produc- tion shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 16 are significantly different: SMASH finds very similar shapes and sizes of the two cross sec- tions resulting in an almost constant K+/K0 S cross sec- tion ratio (close to unity) as a function of rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The GiBUU ratios, however, increase significantly from close to unity near target rapidity to 10 at high rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This trend is also seen in the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The set of kaons are completed with the comparison for charged antikaons [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Figure 13 presents the differential K− cross-sections as a function of pT for three measured rapidity intervals, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For both colliding systems, GiBUU reproduces the shape of the experimental spectra rather well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The cross-section is slightly underestimated for low pT in π− + C collisions 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 100 (Sim-Exp)/Exp [%] x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 4 10 6 10 8 10 10 10 y] ∆ b/ µ /dy [ σ d X Λ → + C − π X + π → + C − π 5) × pX ( → + C − π GiBUU SMASH y (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 4 6 8 10 X Λ → + W − π X + π → + W − π 5) × pX ( → + W − π y (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Upper panel: Cross-section of Λ (or- ange points), π+ (green squares) and p (red triangle) as a function of rapidity in π− + C (a) and π− + W (b) reactions compared with the transport models, GiBUU (solid curve) and SMASH (long-dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The shaded bands denote the systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The open boxes indicate the normal- ization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The statistical uncertainties are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lower panel: Deviations of the three trans- port models from the measured cross-section of Λ (π±, p) as a function of rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For better visibility the deviations for protons from GiBUU are scaled with factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (panel (a)) and π− + W (panel (b)) reactions, except for low rapidities in the latter reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' On the other hand, SMASH underestimates the differential cross-section al- most over the entire pT range for the lighter nucleus, while the shape agrees rather well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Also the model re- sults for the antikaon cross-section as a function of rapid- ity is investigated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' GiBUU slightly underesti- mates the K− production cross-section off carbon, while the production cross-section off tungsten is slightly over- estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both shapes are rather well reproduced by GiBUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the heavier nucleus, SMASH is able to re- produce the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Only minor deviations are observed for low rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In general, the experimen- tal data and GiBUU are almost consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In summary, neither GiBUU nor SMASH can precisely describe simultaneously the cross-sections as function of transverse momentum and rapidity in terms of shape and absolute yield of the presented comprehensive hadron set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (SEMI-) EXCLUSIVE DATA ANALYSIS At the pion beam momentum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c, which is studied here, strangeness production occurs mainly in first-chance π− + N collisions with a kaon and a Λ (or Σ) in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In addition, several other semi- inclusive channels contribute as well (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Although GiBUU describes the inclusive K0 S data rea- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 50 100 (Sim-Exp)/Exp [%] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 5 10 6 10 7 10 y] ∆ b/ µ /dy [ σ d X − π → + C − π (tot) X − π → + C − π (inelastic) (x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5) GiBUU SMASH (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 5 6 7 X − π → + W − π (tot) X − π → + W − π (inelastic) (x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the total (triangles) and inelastic (crosses) π− differential cross-sections as a func- tion of rapidity with GiBUU (solid curves) and SMASH (long- dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representation is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 100 200 300 (Sim-Exp)/Exp [%] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 3 10 5 10 7 10 9 10 10 10 y] ∆ b/ µ /dy [ σ d X s 0 K → + C − π X + K → + C − π X − K → + C − π GiBUU SMASH 2 10 × 1 10 × (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 3 5 7 9 10 X s 0 K → + W − π X + K → + W − π X − K → + W − π 2 10 × 1 10 × (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color online) Comparison of the K0 S (violet rect- angles), K+ [29] (red triangles) and K− [29] (green stars) cross-sections as a function of rapidity with GiBUU (solid curves) and SMASH (long-dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The representa- tion is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' sonably well, the agreement with inclusive Λ and K+ data is not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Therefore, more information was gained by also analysing the (semi-)exclusive channel π−+A → Λ+K0 S+X for both colliding systems, allowing a comparison of the data on associated strangeness pro- duction to model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The corresponding final states were reconstructed via the weak charged decays of the Λ and the K0 S inside the HADES acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The fol- lowing final states were analysed: Λ+K0 S, Λ+K0 S +π0,−, Σ0 + K0 S and Σ0 + K0 S + π0,−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' These include contribu- 11 Channel pthr σfit σGiBUU σSMASH π− + p [GeV/c] [mb] [mb] [mb] ΛK0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='163 Σ0K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='105 Σ−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='130 Λπ0K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='074 Λπ−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='149 Σ+π−K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='005 Σ0π0K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='136 Σ0π−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='269 Σ−π+K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='201 Σ−π0K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='102 pK0K− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='003 nK+K− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='024 nφ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='020 Λπ+π−K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='003 Λπ0π−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='002 Σ+π0π−K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='564 ≈ 0 Σ+π−π−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='568 ≈ 0 Σ0π−π+K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='580 ≈ 0 Σ−π+π0K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='580 ≈ 0 Σ−π+π−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='580 ≈ 0 π− + n Σ−K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='038 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='273 Σ−π0K0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='296 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='505 Σ−π−K+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='290 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='035 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The production channels of Λ and K0 in elemen- tary π−N reactions together with the corresponding thresh- old momenta of the incident pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The cross-section σfit at pπ− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c represents the value obtained from a fit ac- cording to the parametrisation given in [36, 37] to experimen- tal data at several beam momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Also listed are σGiBUU, where the parametrisations were evaluated at the proper in- cident pion momenta, and σSMASH, where the cross-sections were extracted in elementary mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Channels not included in the models are labeled with ”-”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' tions from the production of Σ−K0 S with the subsequent strong conversion process of Σ−N → Λ(Σ0)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Event hypothesis and constraints Considering the decay patterns of Λ → pπ− and K0 S → π+π−, two positively and two negatively charged tracks were required as a minimal event selection crite- rion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Due to the limited acceptance for events with four charged particles in HADES, a different particle identi- fication based on probability and event hypothesis was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' All negatively charged particles were assumed to be π− originating from strange particle decays, and an additional cut on the reconstructed mass, as calu- lated from the momentum and velocity measurement, of mπ− > 80 MeV/c2 was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the remaining two positively charged particles, a likelihood method was em- ployed, selecting the best matching candidate for a pro- ton, based on the difference to the theoretical values of its velocity and energy loss dE/dx in the MDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In ad- dition, the proton candidate had to fulfill a mass cut of 800 < mp [MeV/c2] < 1400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Finally, the remaining pos- itively charged particle was accepted as a π+ if its mass fulfilled the condition of 80 < mπ+ [MeV/c2] < 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To resolve the ambiguity of the negative pion origi- nating from the different sources, the possible combi- nations Λ1(p, π− 1 )K0 S,1(π+π− 2 ) and Λ2(p, π− 2 )K0 S,2(π+π− 1 ) are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Only the combination with the best match- ing of the invariant mass of pπ− pairs (Mpπ−) to the nominal Λ mass and of the invariant mass of π+π− pairs (Mπ+π−) to the nominal K0 S mass was considered for the further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The plot of the corresponding corre- lations is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This selection does not in- troduce any bias as the invariant masses of the rejected combination do not fit the Λ and K0 S hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The final data sample was selected using a two- dimensional elliptical (TDE) area around on the invariant mass correlation with half-axes of ±3σ: � � � � �∆MΛ − µΛ 3 · σΛ �2 + � ∆MK0 S − µK0 S 3 · σK0 S �2 ≤ 1, (1) where σΛ(K0 S) denotes the width, µΛ(K0 S) the offset and ∆MΛ(K0 S) the difference of the invariant mass to the nom- inal mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The width σΛ (σK0 S) was extracted by fitting the invariant-mass distribution Mpπ− (Mπ+π−), which has been pre-selected to be within a ±3¯σK0 S (±3¯σΛ) win- dow around the invariant mass Mπ+π− (Mpπ−) with ¯σK0 S (¯σΛ) obtained beforehand in the inclusive analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The invariant mass distributions were modeled with a Gaus- sian for the signal and a second-order polynomial for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This choice ensures a minimal loss of signal, while obtaining a data sample with a signal-to- background ratio between S/B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To reject the remaining background after the TDE selection, a sideband subtraction was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Since the selection of the semi-exclusive ΛK0 S channel is based on the correlation of invariant mass spectra, a simple one-dimensional sideband is not applicable separately for each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To extract a suitable sample containing enough statistics to describe the background in the sig- nal area, a TDE cut of 4σΛ(K0 S)−15σΛ(K0 S) was applied to the invariant mass correlation, indicated by the black el- lipse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This sideband sample thus accounts for the kinematic correlation of the Λ and K0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the side- band subtraction, the sideband sample has to be scaled to the background contribution after the TDE selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The corresponding scaling factor was extracted in the fol- lowing way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The total Λ and K0 S signal was obtained by fitting both invariant mass distributions before the TDE selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Since, after the TDE selection, the total Λ and K0 S signal stays the same, but the underlying back- ground is altered and thus cannot be well described by 12 ] 2 c [MeV/ PDG Λ M π p/ M 40 − 20 − 0 20 40 ] 2 c [MeV/ PDG S 0 K M π / + π M 50 − 0 50 Counts 1 10 2 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color Online) Yield distribution in the plane of in- variant mass of π+π− pairs vs the invariant mass of pπ− pairs, both subtracted by their nominal mother particle mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='. The grey shaded area indicates the 3σ TDE cut, while the black ellipse represents the lower boundary for the two-dimensional side-band, spanning from 4σ - 15σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' A clear peak at the origin is visible, with a low background contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' any fitting procedure, the background contribution was estimated by subtracting the combined Λ and K0 S signal from the total yield of invariant mass distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The sideband sample was scaled to the estimated background after the TDE selection and the obtained distribution was then subtracted from all spectra fulfilling the TDE selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The kinematic distributions obtained after the subtraction are used for the kinematic investigations and comparisons performed later-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Figures 18 and 19 show the transverse momentum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 18 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 19 (a)) and rapidity (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 18 (b) and 19 (b)) distributions for K0 S and Λ inside the HADES acceptance for the C target (purple circles) and the W target (orange stars) without any corrections for reconstruction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Therefore, the simulated kinematic distributions by GiBUU have been convoluted with the acceptance and efficiency of HADES to allow for a direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Systematics To estimate the systematic error introduced by the de- scribed analysis procedure all applied cuts were varied and their impact on the final spectra was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' As the exclusive data was not corrected for efficiency and acceptance effects, the impact on the shape of the distri- bution and not on the yield was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In this way the whole analysis procedure was performed with another cut set and then compared to the shape of the nominal cut set by calculating the difference in each point, after per- forming a χ2 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In total eight different variations have been considered: ±1σ variation for the extraction of the particle invariant mass widths, ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5σ for the TDE cut, 5 < σΛ(K0 S) < 15 and 4 < σΛ(K0 S) < 10 for the sideband region, and the signal yield was taken solely from the K0 S or Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The signal to background ratio for the carbon target for the nominal cut set is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 and varies systematically from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the tungsten target the corresponding value is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1 and the systematics is found to be in between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The same procedure was performed for the sim- ulation, where the combined variations are smaller than the line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Comparison to transport models As GiBUU allows to reconstruct the particle history we restrict the theory comparison to this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Figures 18 and 19 show the transverse momentum and rapidity dis- tributions for K0 S and Λ inside the HADES acceptance for the C target (purple circles) and the W target (orange stars) compared to results from GiBUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The experimen- tal statistical errors are indicated by the error bars, while in the simulation the statistical errors are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The systematic errors are indicated by the shaded boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The systematic study revealed that in simulations they are smaller then the width of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For this comparison we focus on the shape of the spectra, therefore the simu- lated distributions have been scaled to the experimental distributions by means of a global χ2/NDF minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The experimental pT spectrum for the K0 S in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 18 (a) for the heavy target is rather symmetric with a maximum around 300 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' GiBUU however pre- dicts a distribution that is shifted to lower pT , peaking at 200 MeV/c, with χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the lighter target, the maximum of the experimental distribution is shifted to higher pT around 400 MeV/c featuring an asymmetric shape, while GiBUU predicts a more symmetric shape, with a lower maximum and a lower cut-off of the dis- tribution and a corresponding χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In both cases the experimental shape cannot be reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The rapidity distribution of K0 S (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 18 (b)) for the heavy system (W) is rather symmetric with a maximum at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 and a shift to lower rapidies with respect to the smaller colliding system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' This is well reproduced by the GiBUU model as reflected by a χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 and points to K0 scattering inside the heavy nucleus, also seen in the inclusive spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In case of the lighter system (C), where the distribution is shifted to higher rapidi- ties, GiBUU can reproduce the data qualitatively, with a slightly smaller maximum and a χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The transverse momentum distributions of the Λ hy- perons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 19 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Both experimental dis- tributions have rather similar shapes, although in π− +C reactions the distribution is shifted to higher pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In both cases GiBUU is able to reproduce the pT dependence very well in the low pT region, with a slight systematic 13 0 200 400 600 800 ] c [MeV/ T p 0 200 400 600 800 1000 Counts (a) + X S 0 + K Λ → + C/W − π in HADES acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 y 0 200 400 600 800 1000 C HADES C GiBUU W HADES W GiBUU s 0 K (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (Color Online) Transverse momentum and ra- pidity distributions of K0 S in the (semi-)exclusive channel π− + A → Λ(Σ0) + K0 S + X without reconstruction efficiency correction inside the HADES acceptance together with the GiBUU predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (a) Transverse momentum spectra of the K0 S for the lighter carbon target (purple circles) and the heav- ier tungsten nuclei (orange stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the experimental data the statistical errors are indicated by error bars and system- atic errors indicated by shaded boxes, while the systematic errors for the simulation are smaller than the width of the drawn line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' (b) Rapidity distribution for the K0 S with the same convention as for the transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 0 200 400 600 800 ] c [MeV/ T p 0 200 400 600 800 1000 Counts (a) + X S 0 + K Λ → + C/W − π in HADES acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='5 1 y 0 200 400 600 800 1000 C HADES C GiBUU W HADES W GiBUU Λ (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 18 but for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' shift towards the high pT region and a corresponding χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8 for carbon and tungsten, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The situation changes for the rapidity distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' For the heavier target a maximum around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='4 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' GiBUU can predict the shape qualitatively with χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='8, while the deviations in the lighter system increase, reflected in χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In general, the rapidity distributions of both particles in both nu- clear systems are qualitatively reproduced, where again a backward shift is observed, pointing to scattering in- side the heavy nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In case of the transverse mo- mentum distribution of K0 S, the results of GiBUU show larger deviations while for the Λs they are qualitatively reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' If one considers the global χ2, the results for the heavier system are slightly better with a χ2 of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='38 compared to the lighter one with 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Neverthe- less, a satisfactory description of all kinematic observable simultaneously in both systems is not achieved, which is consistent with the results of, the inclusive analysis of strange hadrons above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The (semi-) exclusive data might be the ideal tool to test the implementation of interaction potentials in transport models simultaneously for kaons and hyperons in the fu- ture, especially in light of the new constraints on these in- teractions extracted from femtoscopy measurements [38– 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' SUMMARY AND CONCLUSION We presented the inclusive differential cross-sections as a function of transverse momentum pT and rapid- ity y for π+, π−, p, Λ and K0 S measured in π− + C and π− + W reactions at an incident pion momentum of pπ− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='7 GeV/c within the rapidity range covered by the HADES detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The presented data significantly extend the world data base on hadron production in pion- induced reactions on nuclear targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Scattering effects are observed, shifting the maximum of the π+, π−, Λ and K0 S rapidity distributions to smaller rapidities in the heavier target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The pT and rapidity spectra have been compared to two state-of-the-art transport models, GiBUU and SMASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' To provide a more complete picture of the (strange) me- son production, the inclusive double-differential produc- tion cross-section of K± measured in the same reactions system, taken from [29], were compared with theory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In both transport models presented, no in-medium potentials for the KN or ΛN interactions were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Concerning the phase space distributions of π+ in π−+C reactions, GiBUU describes (almost) the experimental data in terms of the shape and absolute cross-section, whereas in π− + W reactions the cross-section is signif- icantly overestimated with deviations up to factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' SMASH overshoots the experimental data in both col- liding systems with deviations as large as a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Similar to the π+, both models overestimate the π− dif- ferential cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Hence, the description of rescat- tering and/or absorption effects seems to be particularly insufficient, as the model predictions deviate significantly stronger for the heavier target (W) and for the (quasi)- elastically scattered π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' GiBUU is also not able to de- scribe the (quasi)-elastically scattered protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' While the results of GiBUU for K0 S and K− are rather consis- tent with our data, the cross-sections of Λ and K+ are under-estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' In general, due to the imperfect de- scription of all observable of the comprehensive hadron set (π±, Λ, K0 S and K±), an improvement of these models becomes desirable, especially with regard to the interpre- tation of heavy-ion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 14 Furthermore, the phase-space distribution in the (semi- )exclusive channel π− + A → Λ + K0 S + X was investi- gated and a comparison to the GiBUU model was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' It was found that GiBUU cannot describe all the corre- lated kinematic observable simultaneously, in particular the calculation for the K0 S transverse momentum distri- bution is not well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The HADES Collaboration thanks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Gaitanos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Bleicher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Steinheimer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Elfner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Stau- denmaier and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Steinberg for elucidating dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' We gratefully acknowledge the sup- port given by the following institutions and agen- cies: SIP JUC Cracow, Cracow (Poland), Na- tional Science Center, 2016/23/P/ST2/040 POLONEZ, 2017/25/N/ST2/00580, 2017/26/M/ST2/00600;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' WUT Warszawa (Poland) No: 2020/38/E/ST2/00019 (NCN), IDUB-POB-FWEiTE-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' TU Darmstadt, Darmstadt (Germany), DFG GRK 2128, DFG CRC-TR 211, BMBF:05P18RDFC1, HFHF, ELEMENTS:500/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='006, VH-NG-823,GSI F&E, ExtreMe Matter Institute EMMI at GSI Darmstadt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Goethe-University, Frankfurt (Ger- many), BMBF:05P12RFGHJ, GSI F&E, HIC for FAIR (LOEWE), ExtreMe Matter Institute EMMI at GSI Darmstadt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' TU M¨unchen, Garching (Germany), MLL M¨unchen, DFG EClust 153, GSI TMLRG1316F, BmBF 05P15WOFCA, SFB 1258, DFG FAB898/2-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' JLU Giessen, Giessen (Germany), BMBF:05P12RGGHM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' IJCLab Orsay, Orsay (France), CNRS/IN2P3, P2IO Labex, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' NPI CAS, Rez, Rez (Czech Re- public), MSMT LM2018112, LTT17003, MSMT OP VVV CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content='0/18 046/0016066;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' IDUB-POB- FWEiTE-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' The following colleagues from Russian institutes did con- tribute to the results presented in this publication, but are not listed as authors following the decision of the HADES Collaboration Board on March 23, 2022: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Belyaev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Fateev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Golubeva, F.' metadata={'source': 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Ladygin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lebedev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Morozov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Petukhov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Reshetin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Sadovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} 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Meissner, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' A489 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Rapp and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Wambach, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 25, 1 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Friman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Hohne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Knoll, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Leupold, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Randrup, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Rapp, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Senger, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 814, 11 ((2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lonardoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lovato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Gandolfi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Pederiva, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} +page_content=' 114, 092301 (2015).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfgwdJ/content/2301.03940v1.pdf'} diff --git a/xtE2T4oBgHgl3EQfhQc5/content/tmp_files/2301.03945v1.pdf.txt b/xtE2T4oBgHgl3EQfhQc5/content/tmp_files/2301.03945v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5363189812d2bd5db40487b917f161a0be6aa11c --- /dev/null +++ b/xtE2T4oBgHgl3EQfhQc5/content/tmp_files/2301.03945v1.pdf.txt @@ -0,0 +1,1118 @@ +1 + +Spatially indirect interfacial excitons in n-ZnO/p- +GaN heterostructures +Simran, Subhabrata Dhar +Department of Physics, IIT Bombay, Mumbai-Maharashtra-400076, India. +E-mail: dhar@phy.iitb.ac.in +Abstract + Electroluminescence properties of epitaxially grown n-ZnO/p-GaN pn-heterojunctions are investigated as functions of +applied bias and temperature. The study reveals the existence of indirect interfacial excitons at sufficiently low +temperatures. Electroluminescence feature associated with these excitons redshifts with increasing forward bias. It has +been found that the binding energy of these entities can be controlled through applied forward bias and can even be made +higher than that of the excitons in ZnO bulk (60 meV). However, formation of these excitons becomes unsustainable when +either the applied bias or the temperature crosses a threshold. This has been explained in terms of leakage and thermal +escape of electrons (holes) into GaN (ZnO) side. Calculations for the band diagram and the binding energy of these spatially +indirect electron-hole coulomb-coupled entities are carried out. Theoretical results are found to explain the experimental +findings quite well. + +ZnO has direct band gap (Eg) of 3.37 eV and large exciton +binding energy of 60meV, which make the semiconductor +an attractive candidate for optoelectronis1,2. However, in spite +of intense research for the past several years, controllable and +reproducible p-type doping in ZnO is yet to be +accomplished3,4. Heterojunctions of n-ZnO with other p-type +layers can be an alternative to exploit all the merits of ZnO +while bypassing the p-type doping challenge1,5–9. In this +regard, GaN, which has a very similar lattice structure as ZnO +with lattice mismatch <1.8% along c-direction10 and where +controllable p-type doping is possible11–13, can arguably be a +wonderful choice. Electroluminescence (EL) from n-ZnO/p- +GaN heterostructures grown by different techniques has +indeed been reported by several groups14–18. However, a +detailed and systematic study to understand EL in this system +is still lacking. +ZnO and GaN have type-II band alignment19–21. The two +materials also have significantly different spontaneous +polarizations along the c-direction, which results in a net +positive charge accumulation at the interface, when a (0001) +ZnO film is grown on (0001) GaN. This polarization field +may cause quantum confinement of the conduction band +electrons in the ZnO side of the junction. The combination of +the band alignment and the polarization charge accumulation +at the interface can lead to a unique type of bound indirect +excitons formation at the interface, whose electron part is +quantum confined along c-direction in the ZnO side, while +the hole part stays at the GaN side of the junction. Like the +interlayer excitons observed in homo-/hetero-bilayers of +transition metal dichalcogenides (TMDs)22–26, these spatially +indirect excitons should also have planer character with a +finite c-directional dipole moment. Here, the added advantage +is the possibility to bring the plane of electron motion +sufficiently close to that of the hole through applied forward +bias. Certain important properties of these excitons, such as +the transition energy, binding energy and g-factor can thus be +modified through applied field. Due to the spatial separation +between the electron and hole, these excitons are expected to +show long lifetime, which could be useful for the +development of exciton based logic circuits in the future23. +Note that interlayer excitons in TMD heterostructures are +found to show lifetime exceeding nanoseconds24,25 and +electrically tuneable resonance26. It should be further +mentioned that Bose-Einstein condensation of these entities +is theoretically possible as they possess finite perpendicular +dipole moment 27,28. A search for such excitons in ZnO/GaN +heterojunction could thus be highly exciting. +Here, we investigate the behaviour of electroluminescence +(EL) as a function of applied bias at different temperatures in +pn-heterojunctions consisting of c-oriented n-ZnO layer +grown epitaxially on c-GaN/sapphire template. In certain +samples, an EL peak is found at low temperatures, which +shows a redshift with increasing forward bias. Our study +assigns this peak to spatially indirect interfacial excitons. +Binding energy of these entities can be significantly increased +by changing the applied bias and can even be made larger than +that of ZnO bulk. Formation of such indirect electron-hole + +2 + +coupled systems is theoretically examined, which indeed +supports the experimental outcome. +Here, we highlight two samples, where ~500 nm thick +(0001) ZnO epitaxial films were deposited using pulsed laser +deposition (PLD) technique on two different c-oriented p- +type GaN/sapphire templates with room temperature hole +concentrations of nh~1×1017 (sample D1) and 1×1016cm−3 +(sample D2) in the GaN layer. ZnO layers are found to be +unintentionally n-type doped with electron concentration of +nn~ 1 × 1019 cm−3 at room temperature. Details of the +growth, structural and morphological properties of these films +can be found in the supplementary information S1. +Photoluminescence (PL) and electroluminescence (EL) +measurements at different temperatures ranging from 10 to +300K were carried out in a closed cycle helium cryostat using +a 550 mm focal length monochromator equipped with a UV- +enhanced CCD detector. He–Cd laser of wavelength 325 nm +was used as the excitation source for PL. Ni/Au (30nm/60nm) +and Ti/Au (20nm/100nm) metal contacts were deposited on +p-GaN and n-ZnO sides, respectively. Ni/Au contacts were +rapid thermally annealed at 435ºC for sample D1 and at 400ºC +for sample D2, in N2 atmosphere for 5 minutes to get ohmic +behaviour. Device structure is shown schematically in the +inset of Fig.1(a). Current-voltage profiles recorded between +the contacts on different sides were found to exhibit rectifying +characteristics for both the samples down to the lowest +temperature (10K) as discussed in the supplementary S2. +Fig.1(a) presents the EL spectra recorded for device D1 at +10K at different applied forward voltages. At sufficiently low +voltages, the spectra are featured by two peaks at 3.21 and +3.12 eV. Intensity of both the peaks found by gaussian +deconvolution of the combined feature (discussed in +supplementary S3), increases with the applied bias. The lower +energy peak does not show any shift with 𝑉𝑏, while the higher +energy peak red-shifts as the bias voltage is increased as can +be seen in Fig.1(b). Fig.1(c) shows the peak position variation +of the 3.21 eV feature with 𝑉𝑏. Interestingly, the peak position +shifts at a much slower rate for 𝑉𝑏 < 8V, beyond which it +decreases at a much faster pace. This low voltage regime can +be termed as regime-(I). As 𝑉𝑏 is increased further, a doublet +structure starts to appear at 3.10 and 3.15 eV as can be +observed in Fig.1(a). Beyond a threshold voltage of around +10V, the intensity of this doublet feature rapidly increases and +fully overwhelms the 3.21 and 3.12 eV features, which marks +the beginning of regime-(II). In this regime, a weaker feature +also appears around 2.4 eV. For 𝑉𝑏 >12.5 V, intensity of the +~3eV band decreases sharply, while its shape changes +significantly with increasing bias. In this regime-(III), 3.1 eV +doublet structure disappears. Instead, a peak appears at ~3.27 +eV with a low energy hump at 2.9 eV. Interestingly, 2.4 eV +feature continues to grow without showing any peak-shift. +Fig.1(d) shows the 10K EL spectra recorded at different +applied forward voltages for the sample D2. Evidently, all +spectra are featured by three peaks. Two higher energy humps +can be seen at 3.23 and 3.14 eV, while the most dominant +peak appears at ~2.78 eV. The 2.78 eV feature shows a clear +blue shift with the increasing bias, which may suggest its +donor-acceptor-pair (DAP) origin. Unlike in device D1, none +of the 3.23 and 3.14 eV peaks show any shift with 𝑉𝑏 in this +sample. Note that comparisons between 10K EL spectra and +PL spectra recorded separately on ZnO film and the GaN +template for both the samples does not find any match of the +positions of the EL peaks appearing above 3 eV with any of +the near band edge features of either GaN or ZnO. This +strongly suggests that these EL transitions must be originating +from the interfacial region (see supplementary S3). + +Fig.1: (a) EL spectra for the device D1 recorded at 10K under +different applied forward biases. Inset: schematic view of the +devices used in this study. (b) Highlights the bias dependent +spectral change in regime-(I) for the same device. (c) Position +of the 3.21 eV feature as a function of the applied bias 𝑉𝑏 in +regime-(I). (d) EL spectra recorded at 10K under different +applied biases for the device D2. +Note that our ZnO layers are deposited without any oxygen +treatment of Ga-polar (0001) GaN surface before the PLD +growth. Such a pre-treatment has been found to be necessary +to grow O-polar (0001̅)ZnO film on c-GaN substrates29,30. +ZnO epitaxial layers are thus expected to be Zn-polar. As +mentioned earlier that both GaN and ZnO have spontaneous +polarization 𝑃𝑠𝑝 = −0.029 and −0.05 C/m2, respectively31,32 +along [0001] direction. As a result, accumulation of a net +positive charge of 0.021 C/m2 is expected at the interface as +shown schematically in Fig.2(a). Since XRD study does not +evidence biaxial strain in these ZnO films (see supplementary +S1), piezoelectric polarization at the interface can be +excluded in our samples. Here, the energy band diagrams +across the junction between Zn-polar n-ZnO grown on Ga- +polar p-GaN are obtained by solving poisson equation with +appropriate boundary and interface conditions considering +2.0 +2.5 +3.0 +3.5 +3.0 +3.1 +3.2 +3.3 +5 +6 +7 +8 +9 +10 +3.19 +3.21 +2.0 +2.4 +2.8 +3.2 +Ni/Au +Ti/Au +n-ZnO +p-GaN +Sapphire +NiO +EL Intensity (a.u.) +ħw (eV) +(a) +T=10K +(b) +D1 +8.8V +5.5V +12.1V +14.6V +D1 +D1 +D2 +EL Intensity (a.u.) +ħw (eV) +9.5V +9.3V +8.4V +7.8V +7.5V +7.1V +6.5V +5.5V +T=10K +Etx (eV) +Vb (volt) +(c) +(d) +EL Intensity (a.u.) +ħw (eV) + 12V + 9.1V + 6.5V + 5.3V + 4.0V +T=10K + +3 + +the effects of spontaneous polarization and band off-set at the +interface. More details of the calculation and the parameters +used for the same are provided in supplementary S4. +Fig.2(a) presents the band diagram calculated at 10K for +bias voltages 𝑉𝑗 = 0 and 2V at the junction in case of the +ZnO/GaN system with the acceptor concentration of 𝑁𝑎𝐺 = +2 × 1018 cm-3 in the GaN side and donor concentration of +𝑁𝑑𝑍 = 1 × 1019 cm-3 in the ZnO side. Formation of the +polarization induced triangular potential well adjacent to the +junction for the electrons in the ZnO side and a depletion +region in the GaN side is evident in both the cases. The +potential well can quantum mechanically confine electrons as +the solution of the Schrodinger equation returns eigen states +below the fermi level even up to 𝑉𝑗~3V. In Fig. 2(b), the +conduction band diagrams in a close vicinity of the junction +are compared for 𝑉𝑗 = 0 and 2V. Note that for better +comparison of the shape of the potential wells in the two +cases, the band energy in the ZnO side at the junction has been +treated as the reference in the figure. Ground state +wavefunctions and energies (represented by the position of +the base of the respective wavefunctions) are also shown in +the figure for the two bias voltages. +Beyond a certain forward bias voltage, hole concentration +near to the interface starts to build up. If the temperature is +sufficiently low, the valence band off-set can restrict the holes +to jump over to the ZnO side. In this scenario, ground state +electrons confined in the triangular potential well in the ZnO +side can be coupled with the holes to form the indirect +excitons (IDX), in which electron part stays in a parallel plane +that is 𝑑𝑥 distance away from the interface and the hole part +is located in the GaN side adjacent to the junction. This is +shown schematically in the inset of Fig.2(a). One can +consider 𝑑𝑥 = < 𝑥 > , the expectation value of position of the +ground state electron in the triangular potential well. As the +forward bias is increased, the well narrows leading to an +upshift of the ground state energy as can be seen in Fig.2(b). +This results in the reduction of 𝑑𝑥, which is also evident from +the figure. Binding energy 𝐸𝑏𝑥 of such excitons can thus be +controlled through applied forward bias. 𝐸𝑏𝑥 can be obtained +by solving the Schrodinger equation in cylindrical coordinate +considering electron-hole interaction potential as 𝑉(𝑟) = +− 𝛼𝑒2 4𝜋𝜖𝑜𝜅𝑎𝑣√𝜌2 + 𝑑𝑥2 +⁄ +, +where +𝜅𝑎𝑣 = (1 2 +⁄ )(𝜅𝐺𝑎𝑁 + +𝜅𝑍𝑛𝑂) the average of the dielectric constants of GaN and ZnO, +𝜖𝑜 the vacuum permittivity and 𝛼 is a constant introduced to +take into account the overestimation of dielectric screening +effect in GaN and ZnO33. +More details of the calculation can be found in +supplementary S5. Transition energy of such excitons can be +expressed +as +𝐸𝑡𝑥 = 𝐸𝑐1 +𝑍𝑛𝑂(0) − 𝐸𝑣 +𝐺𝑎𝑁(0) − 𝐸𝑏𝑥, +where +𝐸𝑐1 +𝑍𝑛𝑂(0) and 𝐸𝑣 +𝐺𝑎𝑁(0) represent the ground state energy of +the potential well in the ZnO side and the energy at valence +band maximum in the GaN just at the junction. In Fig. 2(c), +calculated 𝐸𝑏𝑥 and 𝐸𝑡𝑥 are plotted as functions of the applied +bias at the junction 𝑉𝑗. As expected, 𝐸𝑏𝑥 increases with 𝑉𝑗. +Note that 𝐸𝑡𝑥 versus 𝑉𝑗 profile shows quite a similar +behaviour as that is experimentally observed for the sample +D1 in Fig. 1(c). 𝛼 is taken to be 2 in these calculations36. Note +that the applied bias across the two contact pads 𝑉𝑏, which is +plotted along x-axis in Fig. 1(c), should not be the same as the +junction bias 𝑉𝑗 plotted in Fig. 2(c). In reality, 𝑉𝑗 is expected +to be less than 𝑉𝑏. Nevertheless, a similarity in the shape of +the two plots strongly support the assignment of 3.21 eV +transition observed in sample D1 to IDX. + +Fig.2: (a) Band diagrams calculated for two different applied +biases. Spontaneous polarization directions in GaN and ZnO +are schematically shown at the junction. Inset: schematic +representation of the indirect exciton IDX. (b) Conduction +band profile at the junction in an expanded scale. (c) +Transition energy and binding energy of IDX as a function of +applied bias at the junction. + +Fig.3: Band diagrams for different acceptor concentrations +𝑁𝑎𝐺 in GaN side. Insets compare the conduction and valence +band profiles at the junction. +-45 +-30 +-15 +0 +15 +-2 +2 +0 +10 +0.0 +0.8 ++ +- +Energy (eV) +x (nm) +GaN +ZnO +Vj = 0V +(a) +(b) +Vj = 2V +(c) +qVj ++ +- +E (eV) +x (nm) +0.16 +0.28 +Ebx (eV) +0 +1 +2 +3 +3.19 +3.23 +Vj (volt) +Etx (eV) +-1 +1 +-0.1 +0.1 +0 +1 +2 +3 +3.19 +3.23 +-200 +-100 +0 +-2 +2 +-1 +1 +-0.1 +0.1 +-3 +9 +0.0 +0.8 +VB +Energy (eV) +x (nm) +GaN +ZnO +CB +NaG = 1´1017cm-3 +NaG = 2´1018cm-3 +E(eV) +x (nm) +E(eV) +x (nm) + +4 + +In Fig.3, band diagrams calculated at 10K for 𝑉𝑗 = 0 in +case of the acceptor density 𝑁𝑎𝐺 = 1 × 1017 and 2 × 1018 +cm-3 in the GaN side, are compared. Clearly, even at zero bias +condition, the potential well (barrier) for the electrons (holes) +is significantly narrower in case of lower 𝑁𝑎𝐺 + (see insets of +the figure), which reduces the chance of IDX formation. This +can explain why no EL peak showing red-shift with applied +bias could be detected in sample D2, where 𝑁𝑎𝐺 is indeed of +the order of 1017 cm-3. +We believe that the 3.23 eV EL peak in sample D2 [see +Fig. 1(d)] is arising from transition between the shallow donor +states in ZnO to valence band edge in GaN (𝐸𝑑𝑍𝑣𝐺). While, +3.14 eV EL peak can be attributed to conduction band (or +shallow donors) to Mg acceptor states in GaN (𝐸𝑐𝐺𝑎𝐺) side of +the junction34. The most dominant EL feature appearing at +2.78 eV in this sample can be assigned to blue luminescence +(BL) band often reported in low temperature PL of Mg doped +GaN (see supplementary S3). Assignment of different +transitions are schematically shown in Fig. 4(a). In sample D1 +with 𝑁𝑎𝐺~2 × 1018 cm-3, IDX formation is theoretically +sustainable and hence 3.21 eV IDX-feature can be observed +at low temperatures. In this sample, IDX transition must be +dominating over 𝐸𝑑𝑍𝑣𝐺 transition in regime-(I). But, EcGaG +peak is visible [Fig. 1(a)]. As the forward bias increases, the +triangular potential well (barrier) in the conduction (valence) +band of the ZnO side narrows, which increases the binding +energy of the IDX, but at the same time reduces their chance +of formation by enhancing the leakage of the electrons (holes) +into GaN (ZnO). Above a certain threshold bias, IDX +formation becomes unsustainable and various other +transitions start to dominate mainly in GaN side (as holes are +less mobile than electrons). We believe that the rapid increase +of 3.10 eV peak beyond 10V of applied bias is the point when +electrons and holes escapes into GaN and ZnO sides in +sample D1 [see Fig. 1(a)]. Since sample D2 is always in +regime-(II), sudden rise of any EL peak as a function of the +bias could not be seen there. Increase of the applied bias in +regime-(II) results in more electrons leaking into GaN side +that leads to the enhancement of BL and YL transitions as +observed in both the samples [see Fig. 1(a) and (d)]. At +sufficiently high bias [regime-(III)], an EL peak emerges at +3.27 eV in sample D1. This can be assigned to EdZvG. In this +regime, EL yield reduces with increasing bias, which can be +attributed to Auger recombination process35. +Fig. 4(b) shows the evaluation of the EL spectra recorded +at a bias of 8.8V with the increase of temperature in sample +D1. The two peaks at 3.12 (𝐸𝑐𝐺𝑎𝐺) and 3.21 eV (IDX) are +dominant only below ~60K. Above 60K, 3.10 eV doublet +feature rapidly increases marking the onset of thermal escape +of the electrons from the potential well. This leads to the +sudden influx of electrons in the GaN side. However, for 𝑇 > +100K, overall EL yield starts to drop, which can be attributed +to nonradiative recombination processes. At higher +temperatures, transitions can be observed even in the ZnO +side indicating thermal escape of the holes over the junction +barrier into the ZnO side (see supplementary S3). Fig.4(c) +presents the EL spectra recorded below 60K. Inset shows the +variation of the integrated intensity of the 𝐸𝑐𝐺𝑎𝐺 and IDX +features as functions of temperature. It is worth noticing that +both the peaks initially increase as the hole population at the +junction rises with temperature. At higher temperatures, the +intensity of 𝐸𝑐𝐺𝑎𝐺 peak reaches a plateau, while that of the +IDX decreases with increasing temperature. This can be +attributed to the dissociation of IDX due to the thermal escape +of the holes over the junction barrier. In Fig.4(d), IDX +intensities (𝐼𝐼𝐷𝑋) recorded for different bias voltages are +plotted with the inverse of temperature (1/𝑇). Binding energy +(𝐸𝑏𝑥) of the IDX, which has been obtained by fitting the data +using equation 𝐼𝐼𝐷𝑋 = 𝐼𝑜 [1 + 𝐶 exp (− 𝐸𝑏𝑥 𝑘𝐵𝑇) +⁄ +] +⁄ + with 𝐼𝑜 +and 𝐶 are temperature independent constants, are plotted +versus 𝑉𝑏 in the inset. Evidently, 𝐸𝑏𝑥 increases with the +applied forward bias and can even cross the excitonic binding +energy value of ~60 meV for bulk ZnO37. Note that our +theoretical calculations also predict large binding energy +values, which increase with bias, for these excitons [see Fig. +2(c)]37. + +Fig.4: (a) Schematic depiction of different transitions taking +place at the interface and inside the GaN layer. (b) EL spectra +recorded for 𝑉𝑏 = 8.8V at different temperatures in sample +D1. (c) EL spectra recorded at different temperatures below +60K for 𝑉𝑏 =8.8V. Inset: integrated intensity of the EcGaG and +IDX features versus temperature. (d) IDX intensity versus +inverse of temperature for different bias voltages. Inset: +binding energy (𝐸𝑏𝑥) of the excitons as a function of the +applied bias. +Electroluminescence properties of c-oriented n-ZnO/p- +GaN heterojunctions are studied as functions of applied bias +and temperature. The study evidences the formation of +2.0 2.4 2.8 3.2 3.6 +0 +300 +3.0 +3.1 +3.2 +3.3 0.016 +0.020 +0.024 +50 +250 +5 +9 +55 +85 +10 +60 +T(K) +EL Intensity (a.u.) +ћw (eV) +(b) +(a) +EL Intensity (a.u.) +ħw (eV) +(c) +10 +300 +150 +100 +200 +50 +10K +20K +30K +57K +50K +46K +44K +40K +(d) +IIDX (a.u.) +1/T (K-1) + 5.5V + 7.1V + 8.8V +Ebx(meV) +Vb (V) +I.A (a.u) +T (K) +IDX +cGaG +GaN +ZnO +EIDX +EdZvG +EcGaG +Ebx + +5 + +indirect excitons (IDX) at the interface, when the temperature +is sufficiently low. The corresponding EL peak shows a +systematic redshift with increasing forward bias. Binding +energy of these electron-hole coupled entities is found to be +quite high. The value can be enhanced to even more than that +of ZnO by increasing the applied bias up to a certain limit. +Beyond that point, formation of these excitons becomes +unsustainable, which can be attributed to the leakage of +electrons (holes) into GaN (ZnO) side. Band diagram and the +binding energy of these indirect excitons are also theoretically +calculated. Theoretical results are found to represent the +experimental data very well. +The authors acknowledge financial support from the +Department of Science and Technology (DST), Government +of India, under Grant No: CRG/2018/001343. They would +also like to acknowledge the use of various facilities under +the Industrial Research and Consultancy Centre (IRCC), +Sophisticated Analytical Instrument Facility (SAIF) and the +Centre for Excellence in Nanoelectronics (CEN), IIT +Bombay. +References +1 R.K. Saroj, S. Deb, and S. Dhar, Semicond. Sci. Technol. 33, 125012 +(2018). +2 F. Rahman, Opt. 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Correction factor 𝛼 should thus be ~2 in +case of bulk. +37 Values of 𝐸𝑏𝑥 are likely to be underestimated here as these are extracted +from the fitting of data up to only 60K. Data beyond 60K is not possible as +IDX feature is overwhelmed by other transitions due to the thermal escape +of the carriers beyond that temperature. The actual values of 𝐸𝑏𝑥 thus could +be even larger than the values shown in the inset. + + +1 + + + +Supplementary Material +Spatially indirect interfacial excitons in n-ZnO/p- +GaN heterostructures +Simran, Subhabrata Dhar +Department of Physics, IIT Bombay, Mumbai-Maharashtra-400076, India. +E-mail: dhar@phy.iitb.ac.in + + + + + + + + + + + + + + + + + + + +2 + +S1. Growth, structural and morphological properties of the ZnO layer +Growth Process: ZnO films were deposited on (0001)-GaN/sapphire substrates using a high vacuum pulsed laser deposition +(PLD) system. More details about the ZnO pellet making and growth process can be found elsewhere1. A KrF excimer laser with a +wavelength of 248 nm and a pulse duration of 30 ns was used to ablate the ZnO pellet target. The laser beam was focused onto the +rotating target, which was kept at a distance of 5 cm from the substrate. Pulse energy and frequency of the laser were set at optimized +values of 1.5 J cm−2 and 5 Hz, respectively. The base pressure of the chamber was below 5 × 10−6 mbar. Prior to the deposition, +the substrate was cleaned by ultrasonication in trichloroethylene, acetone, and methanol for 5 min each and then immersed in dilute +hydrofluoric acid (1:10 ratio) for 1 min before being rinsed in methanol and dried in a N2 flow. Here, we have highlighted two +samples as sample D1 and D2 which were deposited at a growth temperature of 500ºC and at oxygen chamber pressures of 15 and +10 mbar on the GaN templates with 𝑛ℎ~1× 1017 and 1 × 1016 cm−3, respectively. Growth time was adjusted to 12500 pulses, +which corresponds to 41.66 min for both the samples. After the growth, samples were cooled naturally to 100°C. Throughout the +cooling stage, oxygen pressure was maintained at the level that was used during growth. The sample was removed from the chamber +at room temperature. +30 +40 +50 +60 +70 +80 +30 +40 +50 +60 +70 +72 +73 +Intensity (a.u.) +2q (deg.) +Intensity (a.u.) +2qc/f (deg.) +ZnO (1010) +GaN (1010) +ZnO (2020) +GaN (2020) +(a) +(b) +ZnO (0002) +ZnO (0004) +(0006) +(0224) +ZnO(1120) +I (a.u.) +2q (deg.) +(0004) +ZnO +GaN +2µm +100 nm +0 + =15mbar +ZnO +GaN +500 +nm + +Fig.S1 High-resolution x-ray diffraction measurements for the samples have been carried out in a Rigaku Smart Lab diffractometer +with Cu Kα x-ray radiation. (a) ω-2θ profile recorded for the sample D1, which is featured by (0002) peak and its higher order +(0004) reflection for the wurtzite phase of ZnO and GaN (template). It should be noted that these peaks are expected to appear at +almost the same position for ZnO and GaN as the lattice constants of the two materials are very similar. The (0006) peak associated +with the sapphire substrate is also visible in the scan. Inset shows the (0004) peak in the expanded scale, where the ZnO and GaN +features can be clearly distinguished. (b) 2θχ − ϕ profile recorded for the same sample. It is featured by (101̅0) and (202̅0) +reflections from both ZnO and GaN. The (101̅0) peak for ZnO almost coincides with that of GaN. However, at the (202̅0) reflection, +ZnO and GaN features can be clearly distinguished. All these results show the epitaxial growth of (0001) ZnO film on top of (0001) +GaN template. 𝑎 and 𝑐 lattice parameters are found to be 5.204 and 3.263Å from the XRD peak positions. These values match quite +well with those of bulk ZnO2–4. This suggests that the grown ZnO layer is unstrained. Note that similar results are obtained for all +the samples investigated here. (c) Atomic force micrograph recorded for the ZnO surface of sample D1. A smooth and continuous +deposition is quite evident. The rms roughness is found to be ~19 nm for both the samples. Inset: cross-sectional SEM image for +sample D1. Thickness of the ZnO film is found to be ~500 nm for both the samples investigated here. + + + + + +3 + +S2. Current-voltage characteristics + +Fig.S2 (a) Schematic depiction of the n-ZnO/p-GaN heterojunction device. Current-voltage (I-V) profiles recorded at 300K between +(b) Ni/Au contacts on p-GaN side and (c) Ti/Au contacts on n-ZnO side for both the devices. While on ZnO, contacts are showing +a clear ohmic nature. On GaN side, the profiles are almost linear. (d) I-V profiles recorded between the contacts on GaN and ZnO +sides at 10 and 300K for device D1. Both the profiles display rectifying behaviour implying the formation of a depletion region at +the interface. + + + + + + + + + + + + + + +-10 -5 +0 +5 +10 +0 +1 +2 +3 +-10 -5 +0 +5 10 +0 +1 +2 +3 +p-GaN +Sapphire +n-ZnO +Ti/Au +Ni/Au +Current (mA) +Voltage (V) + 300K + 10K +-6 -4 -2 0 2 4 6 +-0.08 +0.08 +Voltage (V) +Current (mA) +(b) +(a) +(d) +(c) +p-p-GaN +p-n-D1 +n-n-ZnO +-0.4 +-0.2 +0.0 +0.2 +-2 +-1 +0 +1 +2 +-2 +0 +2 + D1 + D2 +Voltage (V) + D1 + D2 +-0.4 +-0.2 +0.0 +0.2 +0.4 + +4 + +S3. Comparison of electroluminescence (EL) and photoluminescence (PL) spectra + +Fig.S3.1: (a) EL spectra recorded for the device D1 at 300K at different applied forward voltages. Inset shows the 2.85 eV EL peaks +in expanded scales. (b) Compares the EL spectrum obtained at 12.5 V of forward bias with the room-temperature PL spectra recorded +separately on ZnO and GaN. EL spectra shown in (a) are featured by four peaks appearing at 2.21 eV, 2.85 eV, 3.27 eV and 3.35 +eV. It is quite evident from (a) and (b) that the peaks observed at 3.27 eV and 3.35 eV in EL are resulting from near band-edge +emissions in ZnO and GaN, respectively. While the most dominant 2.21 eV EL feature matches very well with the defect related +yellow luminescence (YL) feature in GaN PL and hence can be attributed to YL transition in GaN. 2.85 eV EL feature, which +appears at a slightly red shifted position as compared to the blue luminescence (BL) PL peak at 2.9 eV in GaN. Note that the 2.85 +eV EL peak shows a blue shift with the increase of the forward bias as shown in the inset of (a). These findings may suggest that +the origin of the 2.85 eV EL peak is the donor-acceptor pair (DAP) recombination from ZnO donor to GaN acceptor states at the +junction (c) EL spectra recorded for 8.8V applied forward bias at 40K temperature for same device. EL feature can be deconvoluted +with five gaussian functions. Integrated intensity and peak positions for the two peaks (~3.21 eV and ~3.12 eV) could be obtained +from this fitting. + +Fig.S3.2: (a) PL spectra of ZnO layer and GaN substrate recorded at 10K are compared with the 10K EL spectra for sample D1 at +bias voltages belonging to three regimes; 8.8V (regime- I), 12.1V (regime-II) and 14.6V (regime-III). Note that none of the EL +peaks appearing above 3 eV band matches with any of the near band edge features of either GaN or ZnO in any of the regimes, +which strongly suggests that these EL transitions must be originating from the interfacial region. (b) 10K EL profile for sample D2 +recorded under 𝑉𝑏 = 9V for the device and PL spectra recorded separately on the ZnO film and GaN substrate at 10K. The most +dominant EL feature appearing at 2.78 eV in this sample can be assigned to broad blue luminescence (BL) band observed in GaN +PL (also often reported in low temperature PL of Mg doped GaN). However, one must note the line-shape of the EL feature is +seemingly quite different from that of the PL. Specially, the 3.23 and 3.14 eV EL features cannot be seen in GaN PL. This may +imply that these EL features must also be originating from the interface like in case of device D1. +-10 -5 +0 +5 +10 +0 +1 +2 +3 +2.0 +2.5 +3.0 +3.5 +2.0 +2.5 +3.0 +3.5 +0.01 +0.1 +1 +2.8 +2.6 2.8 3.0 3.2 3.4 +p-GaN +Sapphire +n-ZnO +Ti/Au +Ni/Au +EL Intensity (a.u.) +ħw (eV) +T=300K +(b) +9.7 +10.2 +12.5 +(a) +(c) T=40K +T=300K +I +ħw (eV) +EL@12.5V +PL@GaN +PL@ZnO +ħw (eV) +6.6 +8.0 +EL Intensity (a.u.) +ħw (eV) +3.0 +3.2 +3.4 +0 +1 +2.0 +2.5 +3.0 +3.5 +Intensity +ħw (eV) +EL@8.8V +EL@12.1V +EL@14.6V +PL@GaN +PL@ZnO +Intensity (a.u.) +ħw (eV) +EL@9.1V +PL@GaN +PL@ZnO + +(a) +(b) +D2 +D1 + +5 + +S4. Calculation of the band diagram: + +Fig.S4: Schematic representation of polarization charge accumulation at the interface of (0001) ZnO layer grown on (0001) GaN. + +Band diagrams can be obtained by solving Poisson equation + + +where 𝑞 the electronic charge, 𝜖 the dielectric permittivity of the material, 𝐸(𝑥) the conduction (valence) band minimum +(maximum) as a function of position 𝑥. +In GaN side, 𝜌 = 𝑞[𝑝(𝑥) − 𝑁𝑎𝐺 +− ], where 𝑝(𝑥) = 𝑁𝑣𝐺𝐹1/2(𝜂) the hole concentration as a function of position, 𝑁𝑣𝐺 = +2(𝑚𝑝𝐺𝑘𝐵𝑇/2𝜋ℏ2)3/2 the effective density of state of the valence band, 𝑚𝑝𝐺 the hole effective mass, 𝜂 = (𝐸𝑣𝐺(𝑥) − 𝐸𝐹𝐺)/𝑘𝐵𝑇, +𝐸𝑣𝐺(𝑥) and 𝐸𝐹𝐺 stand for valence band maximum and fermi level in GaN side, 𝑁𝑎𝐺 +− = 𝑁𝑎𝐺/(1 + 𝑔𝑣𝑎 exp(𝜂) exp(∆𝐸𝑎𝐺/𝑘𝐵𝑇)) the +ionized acceptor concentration and ∆𝐸𝑎𝐺 the activation energy of the shallow acceptors and 𝑔𝑣𝑎 = 2. In the neutral region, electric +field should be zero and hence (𝑑𝐸𝑣𝐺/𝑑𝑥) (𝑛𝑒𝑢𝑡𝑟𝑎𝑙) = 0. Conduction band minimum in GaN side as a function of position can be +expressed as 𝐸𝑐𝐺(𝑥) = 𝐸𝑣𝐺(𝑥) + 𝐸𝑔𝐺, where 𝐸𝑔𝐺 the band gap of GaN. +In ZnO side, at the junction, 𝐸𝑐𝑍(𝑗) = 𝐸𝑐𝐺(𝑗) − ∆𝐸𝑐 , where 𝑗 stands for the junction point, 𝐸𝑐𝑍 and ∆𝐸𝑐 represent conduction +band minimum in ZnO and conduction band off-set at the interface, respectively. 𝜌 = 𝑞[𝑛(𝑥) − 𝑁𝑑𝑍 ++ ], where 𝑛(𝑥) = 𝑁𝑐𝑍𝐹1/2(𝜂) the +electron concentration as a function of position, 𝑁𝑐𝐺 = 2(𝑚𝑛𝑧𝑘𝐵𝑇/2𝜋ℏ2)3/2 the effective density of state of the conduction band, +𝑚𝑛𝑍 the electron effective mass in ZnO, 𝜂 = (𝐸𝐹𝑍(𝑥) − 𝐸𝑐𝑍)/𝑘𝐵𝑇, 𝐸𝑐𝑍(𝑥) and 𝐸𝐹𝑍 stand for conduction band minimum and fermi +level in ZnO side, 𝑁𝑑𝑍 ++ = 𝑁𝑑𝑍/(1 + 𝑔𝑐𝑑 exp(𝜂) exp(∆𝐸𝑑𝑍/𝑘𝐵𝑇)) the ionized donor concentration and ∆𝐸𝑑𝑍 the activation energy +of the shallow donors. 𝑔𝑐𝑑 = 2. Furthermore, the condition 𝐷𝑗𝑍 = (𝑃𝐺 − 𝑃𝑍) + 𝐷𝑗𝐺 should be satisfied at the interface, where 𝐷𝑗𝐺 +and 𝐷𝑗𝑍 represent the displacement field at the junction in the GaN and ZnO sides, respectively. 𝑃𝐺 and 𝑃𝑍 stand for the spontaneous +polarization, respectively, in GaN and ZnO. In the neutral region, electric field should be zero and hence (𝑑𝐸𝑐𝑍/𝑑𝑥) (𝑛𝑒𝑢𝑡𝑟𝑎𝑙) = 0. +Valence band maximum in the ZnO side as a function of position can be expressed as 𝐸𝑣𝑍(𝑥) = 𝐸𝑐𝑍(𝑥) − 𝐸𝑔𝑍, where 𝐸𝑔𝑍 the band +gap of ZnO. Note that under an applied voltage of 𝑉𝑗 the condition 𝑞𝑉𝑗 = 𝐸𝐹𝑍 − 𝐸𝐹𝐺 is satisfied. +Equation (S4.1) is solved numerically satisfying above mentioned boundary/interfacial conditions. Parameters used in these +calculations are listed in Table S4.1. Note that the temperature dependent variation of band gap in the two materials has been +considered as 𝐸𝑔(𝑇) = 𝐸𝑔(0) − 𝛾𝑇2/(𝑇 + 𝛽). + + + + + + +𝑑2𝑉 +𝑑𝑥2 = − 𝜌 +𝜖 +⇒ +𝑑2𝐸(𝑥) +𝑑𝑥2 += +𝑞𝜌 +𝜖 (S4.1) +p-GaN +n-ZnO + + + + + +[0001] +[0001] +𝑃𝐺 = −0.029 +𝑃𝑍 = −0.05 +Polarization induced +charge accumulation + +6 + +Table S4.1: Material parameters used for the calculation + +GaN +ZnO +Band gap at 0K (𝐸𝑔) +3.47 eV [A] +3.42 eV [E] +Temp. coefficients of 𝐸𝑔 (𝛾, 𝛽) +7.2× 10−4 eVK-1, 600K [B] +5.3× 10−4 eVK-1, 330K [F] +Hole effective mass (𝑚𝑝) +1.4 𝑚𝑜 [A] +0.59 𝑚𝑜 [E] +Electron effective mass (𝑚𝑛) +0.2 𝑚𝑜 [A] +0.24 𝑚𝑜 [E] +Dielectric constant (𝜅) +9.5 [C] +8.5 [E] +Spontaneous polarization (𝑃) +−0.029 C/m2 [D] +−0.05 C/m2 [G] +Donor concentration (𝑁𝑑) +1 × 1013cm−3 +1 × 1019cm−3 +Acceptor concentration (𝑁𝑎) +1 × 1017cm−3 and 2 × 1018cm−3 +1 × 1013cm−3 +Donor activation energy (∆𝐸𝑑) +15 meV +30 meV +Acceptor activation energy (∆𝐸𝑎) +140 meV [A] +180 meV +Conduction band off-set (∆𝐸𝑐) +200 meV + +[A] Bougrov V. et al., in Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe., John Wiley & Sons, +Inc., New York, 2001, 1-30. +[B] M. Ilegems et al., Journal of Applied Physics 43, 3797 (1972). +[C] Barker et al., Infrared Lattice Vibrations and Free-Electron Dispersion in GaN, Phys. Rev. B 7, 743 (1973). +[D] J. Lähnemann et al., Phys. Rev. B 86, 081302 (2012). +[E] D.P. Norton et al., Mater. Today 7, 34 (2004). +[F] R.C. Rai et al., Journal of Applied Physics 111, 073511 (2012). +[G] A. Dal Corso et al., Phys. Rev. B 50, 10715 (1994). + + + + + + + + + +7 + +S5. Calculation of binding energy of the indirect excitons (IDX): +Binding energy 𝐸𝑏𝑥 of such excitons can be obtained by solving the Schrodinger equation in cylindrical coordinate5. The radial +part of the wavefunction 𝑅(𝜌) satisfies +𝜌2 𝑑2𝑅 +𝑑𝜌2 + 𝜌 𝑑𝑅 +𝑑𝜌 + [2𝜇𝜌2 +ℏ2 {𝐸𝑛𝑙 − 𝑉(𝑟)} − 𝑙2] 𝑅(𝜌) = 0 (S5.1) +where, 𝜌 represents the cylindrical coordinate for the relative position of the electron and hole. 𝐸𝑛𝑙 the eigen energy for the +principle quantum number 𝑛 and angular momentum quantum number 𝑙. 𝜇 = 𝑚𝑒 + 𝑚ℎ 𝑚𝑒𝑚ℎ +⁄ + the reduced mass of the exciton. +Electron hole interaction potential can be considered as 𝑉(𝑟) = − 𝛼𝑒2 4𝜋𝜖𝑜𝜅𝑎𝑣√𝜌2 + 𝑑𝑥2 +⁄ +, where 𝜅𝑎𝑣 = (1 2 +⁄ )(𝜅𝐺𝑎𝑁 + 𝜅𝑍𝑛𝑂) the +average of the dielectric constants of GaN and ZnO, 𝜖𝑜 the vacuum permittivity, 𝑑𝑥 is the average electron-hole separation along +the vertical direction and 𝛼 is a constant introduced to take into account the overestimation of dielectric screening effect in GaN and +ZnO. While the azimuthal part of the wavefunction satisfies 𝛷(𝜑) = 𝑒−𝑖𝑙𝜑 √2𝜋 +⁄ +, where 𝑙 = −𝑛 + 1, −𝑛 + 2, … . ,0, … . , 𝑛 − 2, 𝑛 − +1 for the principle quantum number 𝑛. Equation (1) can be solved using Mathematica’s ‘NDEigensystem’ function to obtain the +binding energy of the excitons 𝐸𝑏𝑥. Note that 𝐸𝑏𝑥 = 𝐸𝑛𝑙, when 𝑛 = 0 and 𝑙 = 0. + + + + + + + + + + + + + + + + + + + + + + +8 + +References +1 Simran, S.K. Yadav, P. Chakrabarti, and S. Dhar, J. Appl. Phys. 131, 015302 (2022). +2 F. Rahman, Opt. Eng. 58, 1 (2019). +3 Ü. Özgür, Y.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Doğan, V. Avrutin, S.-J. Cho, and H. Morkoç, J. Appl. Phys. 98, +041301 (2005). +4 D.P. Norton, Y.W. Heo, M.P. Ivill, K. Ip, S.J. Pearton, M.F. Chisholm, and T. Steiner, Mater. Today 7, 34 (2004). +5 K.W. Lau, Calvin, Z. Gong, H. Yu, and W. Yao, Phys. Rev. B 98, 115427 (2018). + + + + + + + + + + + diff --git a/xtE2T4oBgHgl3EQfhQc5/content/tmp_files/load_file.txt b/xtE2T4oBgHgl3EQfhQc5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c84e7d8d87615d88bddc9fb58b601867d4bb509 --- /dev/null +++ b/xtE2T4oBgHgl3EQfhQc5/content/tmp_files/load_file.txt @@ -0,0 +1,1002 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf,len=1001 +page_content='1 Spatially indirect interfacial excitons in n-ZnO/p- GaN heterostructures Simran, Subhabrata Dhar\uf02a Department of Physics, IIT Bombay, Mumbai-Maharashtra-400076, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' \uf02aE-mail: dhar@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='in Abstract Electroluminescence properties of epitaxially grown n-ZnO/p-GaN pn-heterojunctions are investigated as functions of applied bias and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The study reveals the existence of indirect interfacial excitons at sufficiently low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Electroluminescence feature associated with these excitons redshifts with increasing forward bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' It has been found that the binding energy of these entities can be controlled through applied forward bias and can even be made higher than that of the excitons in ZnO bulk (60 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' However, formation of these excitons becomes unsustainable when either the applied bias or the temperature crosses a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This has been explained in terms of leakage and thermal escape of electrons (holes) into GaN (ZnO) side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Calculations for the band diagram and the binding energy of these spatially indirect electron-hole coulomb-coupled entities are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Theoretical results are found to explain the experimental findings quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ZnO has direct band gap (Eg) of \uf07e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='37 eV and large exciton binding energy of \uf07e60meV, which make the semiconductor an attractive candidate for optoelectronis1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' However, in spite of intense research for the past several years, controllable and reproducible p-type doping in ZnO is yet to be accomplished3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Heterojunctions of n-ZnO with other p-type layers can be an alternative to exploit all the merits of ZnO while bypassing the p-type doping challenge1,5–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In this regard, GaN, which has a very similar lattice structure as ZnO with lattice mismatch <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8% along c-direction10 and where controllable p-type doping is possible11–13, can arguably be a wonderful choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Electroluminescence (EL) from n-ZnO/p- GaN heterostructures grown by different techniques has indeed been reported by several groups14–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' However, a detailed and systematic study to understand EL in this system is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ZnO and GaN have type-II band alignment19–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The two materials also have significantly different spontaneous polarizations along the c-direction, which results in a net positive charge accumulation at the interface, when a (0001) ZnO film is grown on (0001) GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This polarization field may cause quantum confinement of the conduction band electrons in the ZnO side of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The combination of the band alignment and the polarization charge accumulation at the interface can lead to a unique type of bound indirect excitons formation at the interface, whose electron part is quantum confined along c-direction in the ZnO side, while the hole part stays at the GaN side of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Like the interlayer excitons observed in homo-/hetero-bilayers of transition metal dichalcogenides (TMDs)22–26, these spatially indirect excitons should also have planer character with a finite c-directional dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Here, the added advantage is the possibility to bring the plane of electron motion sufficiently close to that of the hole through applied forward bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Certain important properties of these excitons, such as the transition energy, binding energy and g-factor can thus be modified through applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Due to the spatial separation between the electron and hole, these excitons are expected to show long lifetime, which could be useful for the development of exciton based logic circuits in the future23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that interlayer excitons in TMD heterostructures are found to show lifetime exceeding nanoseconds24,25 and electrically tuneable resonance26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' It should be further mentioned that Bose-Einstein condensation of these entities is theoretically possible as they possess finite perpendicular dipole moment 27,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' A search for such excitons in ZnO/GaN heterojunction could thus be highly exciting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Here, we investigate the behaviour of electroluminescence (EL) as a function of applied bias at different temperatures in pn-heterojunctions consisting of c-oriented n-ZnO layer grown epitaxially on c-GaN/sapphire template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In certain samples, an EL peak is found at low temperatures, which shows a redshift with increasing forward bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Our study assigns this peak to spatially indirect interfacial excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Binding energy of these entities can be significantly increased by changing the applied bias and can even be made larger than that of ZnO bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Formation of such indirect electron-hole 2 coupled systems is theoretically examined, which indeed supports the experimental outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Here, we highlight two samples, where ~500 nm thick (0001) ZnO epitaxial films were deposited using pulsed laser deposition (PLD) technique on two different c-oriented p- type GaN/sapphire templates with room temperature hole concentrations of nh~1×1017 (sample D1) and 1×1016cm−3 (sample D2) in the GaN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ZnO layers are found to be unintentionally n-type doped with electron concentration of nn~ 1 × 1019 cm−3 at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Details of the growth, structural and morphological properties of these films can be found in the supplementary information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Photoluminescence (PL) and electroluminescence (EL) measurements at different temperatures ranging from 10 to 300K were carried out in a closed cycle helium cryostat using a 550 mm focal length monochromator equipped with a UV- enhanced CCD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' He–Cd laser of wavelength 325 nm was used as the excitation source for PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ni/Au (30nm/60nm) and Ti/Au (20nm/100nm) metal contacts were deposited on p-GaN and n-ZnO sides, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ni/Au contacts were rapid thermally annealed at 435ºC for sample D1 and at 400ºC for sample D2, in N2 atmosphere for 5 minutes to get ohmic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Device structure is shown schematically in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Current-voltage profiles recorded between the contacts on different sides were found to exhibit rectifying characteristics for both the samples down to the lowest temperature (10K) as discussed in the supplementary S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1(a) presents the EL spectra recorded for device D1 at 10K at different applied forward voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' At sufficiently low voltages, the spectra are featured by two peaks at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='12 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Intensity of both the peaks found by gaussian deconvolution of the combined feature (discussed in supplementary S3), increases with the applied bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The lower energy peak does not show any shift with 𝑉𝑏, while the higher energy peak red-shifts as the bias voltage is increased as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1(c) shows the peak position variation of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV feature with 𝑉𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Interestingly, the peak position shifts at a much slower rate for 𝑉𝑏 < 8V, beyond which it decreases at a much faster pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This low voltage regime can be termed as regime-(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' As 𝑉𝑏 is increased further, a doublet structure starts to appear at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='15 eV as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Beyond a threshold voltage of around 10V, the intensity of this doublet feature rapidly increases and fully overwhelms the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='12 eV features, which marks the beginning of regime-(II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In this regime, a weaker feature also appears around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' For 𝑉𝑏 >12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 V, intensity of the ~3eV band decreases sharply, while its shape changes significantly with increasing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In this regime-(III), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 eV doublet structure disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Instead, a peak appears at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='27 eV with a low energy hump at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='9 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Interestingly, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 eV feature continues to grow without showing any peak-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1(d) shows the 10K EL spectra recorded at different applied forward voltages for the sample D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Evidently, all spectra are featured by three peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Two higher energy humps can be seen at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='23 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='14 eV, while the most dominant peak appears at ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='78 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='78 eV feature shows a clear blue shift with the increasing bias, which may suggest its donor-acceptor-pair (DAP) origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Unlike in device D1, none of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='23 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='14 eV peaks show any shift with 𝑉𝑏 in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that comparisons between 10K EL spectra and PL spectra recorded separately on ZnO film and the GaN template for both the samples does not find any match of the positions of the EL peaks appearing above 3 eV with any of the near band edge features of either GaN or ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This strongly suggests that these EL transitions must be originating from the interfacial region (see supplementary S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1: (a) EL spectra for the device D1 recorded at 10K under different applied forward biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset: schematic view of the devices used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (b) Highlights the bias dependent spectral change in regime-(I) for the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (c) Position of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV feature as a function of the applied bias 𝑉𝑏 in regime-(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (d) EL spectra recorded at 10K under different applied biases for the device D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that our ZnO layers are deposited without any oxygen treatment of Ga-polar (0001) GaN surface before the PLD growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Such a pre-treatment has been found to be necessary to grow O-polar (0001̅)ZnO film on c-GaN substrates29,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ZnO epitaxial layers are thus expected to be Zn-polar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' As mentioned earlier that both GaN and ZnO have spontaneous polarization 𝑃𝑠𝑝 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='029 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='05 C/m2, respectively31,32 along [0001] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' As a result, accumulation of a net positive charge of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='021 C/m2 is expected at the interface as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Since XRD study does not evidence biaxial strain in these ZnO films (see supplementary S1), piezoelectric polarization at the interface can be excluded in our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Here, the energy band diagrams across the junction between Zn-polar n-ZnO grown on Ga- polar p-GaN are obtained by solving poisson equation with appropriate boundary and interface conditions considering 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3 5 6 7 8 9 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 Ni/Au Ti/Au n-ZnO p-GaN Sapphire NiO EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) (a) T=10K (b) D1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='6V D1 D1 D2 EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3V 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4V 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V T=10K Etx (eV) Vb (volt) (c) (d) EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) 12V 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3V 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0V T=10K 3 the effects of spontaneous polarization and band off-set at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' More details of the calculation and the parameters used for the same are provided in supplementary S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2(a) presents the band diagram calculated at 10K for bias voltages 𝑉𝑗 = 0 and 2V at the junction in case of the ZnO/GaN system with the acceptor concentration of 𝑁𝑎𝐺 = 2 × 1018 cm-3 in the GaN side and donor concentration of 𝑁𝑑𝑍 = 1 × 1019 cm-3 in the ZnO side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Formation of the polarization induced triangular potential well adjacent to the junction for the electrons in the ZnO side and a depletion region in the GaN side is evident in both the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The potential well can quantum mechanically confine electrons as the solution of the Schrodinger equation returns eigen states below the fermi level even up to 𝑉𝑗~3V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 2(b), the conduction band diagrams in a close vicinity of the junction are compared for 𝑉𝑗 = 0 and 2V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that for better comparison of the shape of the potential wells in the two cases, the band energy in the ZnO side at the junction has been treated as the reference in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ground state wavefunctions and energies (represented by the position of the base of the respective wavefunctions) are also shown in the figure for the two bias voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Beyond a certain forward bias voltage, hole concentration near to the interface starts to build up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' If the temperature is sufficiently low, the valence band off-set can restrict the holes to jump over to the ZnO side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In this scenario, ground state electrons confined in the triangular potential well in the ZnO side can be coupled with the holes to form the indirect excitons (IDX), in which electron part stays in a parallel plane that is 𝑑𝑥 distance away from the interface and the hole part is located in the GaN side adjacent to the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This is shown schematically in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' One can consider 𝑑𝑥 = < 𝑥 > , the expectation value of position of the ground state electron in the triangular potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' As the forward bias is increased, the well narrows leading to an upshift of the ground state energy as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This results in the reduction of 𝑑𝑥, which is also evident from the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Binding energy 𝐸𝑏𝑥 of such excitons can thus be controlled through applied forward bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝐸𝑏𝑥 can be obtained by solving the Schrodinger equation in cylindrical coordinate considering electron-hole interaction potential as 𝑉(𝑟) = − 𝛼𝑒2 4𝜋𝜖𝑜𝜅𝑎𝑣√𝜌2 + 𝑑𝑥2 ⁄ , where 𝜅𝑎𝑣 = (1 2 ⁄ )(𝜅𝐺𝑎𝑁 + 𝜅𝑍𝑛𝑂) the average of the dielectric constants of GaN and ZnO, 𝜖𝑜 the vacuum permittivity and 𝛼 is a constant introduced to take into account the overestimation of dielectric screening effect in GaN and ZnO33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' More details of the calculation can be found in supplementary S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Transition energy of such excitons can be expressed as 𝐸𝑡𝑥 = 𝐸𝑐1 𝑍𝑛𝑂(0) − 𝐸𝑣 𝐺𝑎𝑁(0) − 𝐸𝑏𝑥, where 𝐸𝑐1 𝑍𝑛𝑂(0) and 𝐸𝑣 𝐺𝑎𝑁(0) represent the ground state energy of the potential well in the ZnO side and the energy at valence band maximum in the GaN just at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 2(c), calculated 𝐸𝑏𝑥 and 𝐸𝑡𝑥 are plotted as functions of the applied bias at the junction 𝑉𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' As expected, 𝐸𝑏𝑥 increases with 𝑉𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that 𝐸𝑡𝑥 versus 𝑉𝑗 profile shows quite a similar behaviour as that is experimentally observed for the sample D1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝛼 is taken to be 2 in these calculations36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that the applied bias across the two contact pads 𝑉𝑏, which is plotted along x-axis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1(c), should not be the same as the junction bias 𝑉𝑗 plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In reality, 𝑉𝑗 is expected to be less than 𝑉𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Nevertheless, a similarity in the shape of the two plots strongly support the assignment of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV transition observed in sample D1 to IDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2: (a) Band diagrams calculated for two different applied biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Spontaneous polarization directions in GaN and ZnO are schematically shown at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset: schematic representation of the indirect exciton IDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (b) Conduction band profile at the junction in an expanded scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (c) Transition energy and binding energy of IDX as a function of applied bias at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3: Band diagrams for different acceptor concentrations 𝑁𝑎𝐺 in GaN side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Insets compare the conduction and valence band profiles at the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' -45 -30 -15 0 15 -2 2 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8 + - Energy (eV) x (nm) GaN ZnO Vj = 0V (a) (b) Vj = 2V (c) qVj + - E (eV) x (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='28 Ebx (eV) 0 1 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='23 Vj (volt) Etx (eV) -1 1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 0 1 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='23 -200 -100 0 -2 2 -1 1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 -3 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8 VB Energy (eV) x (nm) GaN ZnO CB NaG = 1´1017cm-3 NaG = 2´1018cm-3 E(eV) x (nm) E(eV) x (nm) 4 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3, band diagrams calculated at 10K for 𝑉𝑗 = 0 in case of the acceptor density 𝑁𝑎𝐺 = 1 × 1017 and 2 × 1018 cm-3 in the GaN side, are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Clearly, even at zero bias condition, the potential well (barrier) for the electrons (holes) is significantly narrower in case of lower 𝑁𝑎𝐺 (see insets of the figure), which reduces the chance of IDX formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This can explain why no EL peak showing red-shift with applied bias could be detected in sample D2, where 𝑁𝑎𝐺 is indeed of the order of 1017 cm-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' We believe that the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='23 eV EL peak in sample D2 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1(d)] is arising from transition between the shallow donor states in ZnO to valence band edge in GaN (𝐸𝑑𝑍𝑣𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' While, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='14 eV EL peak can be attributed to conduction band (or shallow donors) to Mg acceptor states in GaN (𝐸𝑐𝐺𝑎𝐺) side of the junction34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The most dominant EL feature appearing at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='78 eV in this sample can be assigned to blue luminescence (BL) band often reported in low temperature PL of Mg doped GaN (see supplementary S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Assignment of different transitions are schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In sample D1 with 𝑁𝑎𝐺~2 × 1018 cm-3, IDX formation is theoretically sustainable and hence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV IDX-feature can be observed at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In this sample, IDX transition must be dominating over 𝐸𝑑𝑍𝑣𝐺 transition in regime-(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' But, EcGaG peak is visible [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' As the forward bias increases, the triangular potential well (barrier) in the conduction (valence) band of the ZnO side narrows, which increases the binding energy of the IDX, but at the same time reduces their chance of formation by enhancing the leakage of the electrons (holes) into GaN (ZnO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Above a certain threshold bias, IDX formation becomes unsustainable and various other transitions start to dominate mainly in GaN side (as holes are less mobile than electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' We believe that the rapid increase of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='10 eV peak beyond 10V of applied bias is the point when electrons and holes escapes into GaN and ZnO sides in sample D1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Since sample D2 is always in regime-(II), sudden rise of any EL peak as a function of the bias could not be seen there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Increase of the applied bias in regime-(II) results in more electrons leaking into GaN side that leads to the enhancement of BL and YL transitions as observed in both the samples [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1(a) and (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' At sufficiently high bias [regime-(III)], an EL peak emerges at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='27 eV in sample D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This can be assigned to EdZvG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In this regime, EL yield reduces with increasing bias, which can be attributed to Auger recombination process35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 4(b) shows the evaluation of the EL spectra recorded at a bias of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V with the increase of temperature in sample D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The two peaks at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='12 (𝐸𝑐𝐺𝑎𝐺) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV (IDX) are dominant only below ~60K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Above 60K, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='10 eV doublet feature rapidly increases marking the onset of thermal escape of the electrons from the potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This leads to the sudden influx of electrons in the GaN side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' However, for 𝑇 > 100K, overall EL yield starts to drop, which can be attributed to nonradiative recombination processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' At higher temperatures, transitions can be observed even in the ZnO side indicating thermal escape of the holes over the junction barrier into the ZnO side (see supplementary S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4(c) presents the EL spectra recorded below 60K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset shows the variation of the integrated intensity of the 𝐸𝑐𝐺𝑎𝐺 and IDX features as functions of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' It is worth noticing that both the peaks initially increase as the hole population at the junction rises with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' At higher temperatures, the intensity of 𝐸𝑐𝐺𝑎𝐺 peak reaches a plateau, while that of the IDX decreases with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This can be attributed to the dissociation of IDX due to the thermal escape of the holes over the junction barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4(d), IDX intensities (𝐼𝐼𝐷𝑋) recorded for different bias voltages are plotted with the inverse of temperature (1/𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Binding energy (𝐸𝑏𝑥) of the IDX, which has been obtained by fitting the data using equation 𝐼𝐼𝐷𝑋 = 𝐼𝑜 [1 + 𝐶 exp (− 𝐸𝑏𝑥 𝑘𝐵𝑇) ⁄ ] ⁄ with 𝐼𝑜 and 𝐶 are temperature independent constants, are plotted versus 𝑉𝑏 in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Evidently, 𝐸𝑏𝑥 increases with the applied forward bias and can even cross the excitonic binding energy value of ~60 meV for bulk ZnO37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that our theoretical calculations also predict large binding energy values, which increase with bias, for these excitons [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 2(c)]37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4: (a) Schematic depiction of different transitions taking place at the interface and inside the GaN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (b) EL spectra recorded for 𝑉𝑏 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V at different temperatures in sample D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (c) EL spectra recorded at different temperatures below 60K for 𝑉𝑏 =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset: integrated intensity of the EcGaG and IDX features versus temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (d) IDX intensity versus inverse of temperature for different bias voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset: binding energy (𝐸𝑏𝑥) of the excitons as a function of the applied bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Electroluminescence properties of c-oriented n-ZnO/p- GaN heterojunctions are studied as functions of applied bias and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The study evidences the formation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='6 0 300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='024 50 250 5 9 55 85 10 60 T(K) EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ћw (eV) (b) (a) EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) (c) 10 300 150 100 200 50 10K 20K 30K 57K 50K 46K 44K 40K (d) IIDX (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') 1/T (K-1) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V Ebx(meV) Vb (V) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='A (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u) T (K) IDX cGaG GaN ZnO EIDX EdZvG EcGaG Ebx 5 indirect excitons (IDX) at the interface, when the temperature is sufficiently low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The corresponding EL peak shows a systematic redshift with increasing forward bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Binding energy of these electron-hole coupled entities is found to be quite high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The value can be enhanced to even more than that of ZnO by increasing the applied bias up to a certain limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Beyond that point, formation of these excitons becomes unsustainable, which can be attributed to the leakage of electrons (holes) into GaN (ZnO) side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Band diagram and the binding energy of these indirect excitons are also theoretically calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Theoretical results are found to represent the experimental data very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The authors acknowledge financial support from the Department of Science and Technology (DST), Government of India, under Grant No: CRG/2018/001343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' They would also like to acknowledge the use of various facilities under the Industrial Research and Consultancy Centre (IRCC), Sophisticated Analytical Instrument Facility (SAIF) and the Centre for Excellence in Nanoelectronics (CEN), IIT Bombay.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lin, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chen, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 95, 131117 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Saroj, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Barick, and S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Saroj and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Dhar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 120, 075701 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 10 Ü.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Özgür, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Alivov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Teke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Reshchikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Doğan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Avrutin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Cho, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Morkoç, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 98, 041301 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 11 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Yang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Tseng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Tseng, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Growth 439, 87 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Namkoong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Trybus, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Moseley, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Doolittle, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Look, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 93, 172112 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 13 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Yohannes and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Kuo, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Semicond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 29, 288 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Alivov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Van Nostrand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Look, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chukichev, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ataev, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 83, 2943 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 15 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-X.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Huang, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 91, 091107 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 18 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Zhang, R.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 46, 065101 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 19 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Hong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Hanada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Makino, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ko, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Yao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Tanaka, H.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Bailey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' McConville, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Martel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Moreno, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Frayssinet, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' High, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Novitskaya, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Butov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Hanson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Gossard, Science (80-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 321, 229 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 24 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rivera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 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Mandrus, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Yao, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Xu, Science (80-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 351, 688 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Bechstedt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Trampert, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Geelhaar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' B 86, 081302 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Dvorak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Wei, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 110, 016402 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 34 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Reshchikov and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Morkoç, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 97, 061301 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 35 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Wei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Huo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Wei, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Duan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Zeng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Li, IEEE Photonics J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 6, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 36 Hydrogenic model underestimates the binding energy of exciton by almost a factor of half as compared to the experimental value in ZnO, when the dielectric screening of the coulomb field is taken care of simply via dielectric constant of the material (κ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Correction factor 𝛼 should thus be ~2 in case of bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 37 Values of 𝐸𝑏𝑥 are likely to be underestimated here as these are extracted from the fitting of data up to only 60K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Data beyond 60K is not possible as IDX feature is overwhelmed by other transitions due to the thermal escape of the carriers beyond that temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The actual values of 𝐸𝑏𝑥 thus could be even larger than the values shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 1 Supplementary Material Spatially indirect interfacial excitons in n-ZnO/p- GaN heterostructures Simran, Subhabrata Dhar\uf02a Department of Physics, IIT Bombay, Mumbai-Maharashtra-400076, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' \uf02aE-mail: dhar@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='in 2 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Growth, structural and morphological properties of the ZnO layer Growth Process: ZnO films were deposited on (0001)-GaN/sapphire substrates using a high vacuum pulsed laser deposition (PLD) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' More details about the ZnO pellet making and growth process can be found elsewhere1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' A KrF excimer laser with a wavelength of 248 nm and a pulse duration of 30 ns was used to ablate the ZnO pellet target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The laser beam was focused onto the rotating target, which was kept at a distance of 5 cm from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Pulse energy and frequency of the laser were set at optimized values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 J cm−2 and 5 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The base pressure of the chamber was below 5 × 10−6 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Prior to the deposition, the substrate was cleaned by ultrasonication in trichloroethylene, acetone, and methanol for 5 min each and then immersed in dilute hydrofluoric acid (1:10 ratio) for 1 min before being rinsed in methanol and dried in a N2 flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Here, we have highlighted two samples as sample D1 and D2 which were deposited at a growth temperature of 500ºC and at oxygen chamber pressures of 15 and 10 mbar on the GaN templates with 𝑛ℎ~1× 1017 and 1 × 1016 cm−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Growth time was adjusted to 12500 pulses, which corresponds to 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='66 min for both the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' After the growth, samples were cooled naturally to 100°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Throughout the cooling stage, oxygen pressure was maintained at the level that was used during growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The sample was removed from the chamber at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 30 40 50 60 70 80 30 40 50 60 70 72 73 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') 2q (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') 2qc/f (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ZnO (1010) GaN (1010) ZnO (2020) GaN (2020) (a) (b) ZnO (0002) ZnO (0004) \uf02a(0006) \uf02a(0224) ZnO(1120) I (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') 2q (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') (0004) ZnO GaN 2µm 100 nm 0 =15mbar ZnO GaN 500 nm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='S1 High-resolution x-ray diffraction measurements for the samples have been carried out in a Rigaku Smart Lab diffractometer with Cu Kα x-ray radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (a) ω-2θ profile recorded for the sample D1, which is featured by (0002) peak and its higher order (0004) reflection for the wurtzite phase of ZnO and GaN (template).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' It should be noted that these peaks are expected to appear at almost the same position for ZnO and GaN as the lattice constants of the two materials are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The (0006) peak associated with the sapphire substrate is also visible in the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset shows the (0004) peak in the expanded scale, where the ZnO and GaN features can be clearly distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (b) 2θχ − ϕ profile recorded for the same sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' It is featured by (101̅0) and (202̅0) reflections from both ZnO and GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The (101̅0) peak for ZnO almost coincides with that of GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' However, at the (202̅0) reflection, ZnO and GaN features can be clearly distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' All these results show the epitaxial growth of (0001) ZnO film on top of (0001) GaN template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝑎 and 𝑐 lattice parameters are found to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='204 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='263Å from the XRD peak positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' These values match quite well with those of bulk ZnO2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This suggests that the grown ZnO layer is unstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that similar results are obtained for all the samples investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (c) Atomic force micrograph recorded for the ZnO surface of sample D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' A smooth and continuous deposition is quite evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The rms roughness is found to be ~19 nm for both the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset: cross-sectional SEM image for sample D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Thickness of the ZnO film is found to be ~500 nm for both the samples investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 3 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Current voltage characteristics Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='S2 (a) Schematic depiction of the n-ZnO/p-GaN heterojunction device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Current-voltage (I-V) profiles recorded at 300K between (b) Ni/Au contacts on p-GaN side and (c) Ti/Au contacts on n-ZnO side for both the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' While on ZnO, contacts are showing a clear ohmic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' On GaN side, the profiles are almost linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (d) I-V profiles recorded between the contacts on GaN and ZnO sides at 10 and 300K for device D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Both the profiles display rectifying behaviour implying the formation of a depletion region at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 10 5 0 5 10 0 1 2 3 10 5 0 5 10 0 1 2 3 p GaN Sapphire n ZnO Ti/Au Ni/Au Current (mA) Voltage (V) 300K 10K 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='08 Voltage (V) Current (mA) (b) (a) (d) (c) p p GaN p n D1 n n ZnO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 2 1 0 1 2 2 0 2 D1 D2 Voltage (V) D1 D2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 4 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Comparison of electroluminescence (EL) and photoluminescence (PL) spectra Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1: (a) EL spectra recorded for the device D1 at 300K at different applied forward voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Inset shows the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='85 eV EL peaks in expanded scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (b) Compares the EL spectrum obtained at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 V of forward bias with the room-temperature PL spectra recorded separately on ZnO and GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' EL spectra shown in (a) are featured by four peaks appearing at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='85 eV, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='27 eV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='35 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' It is quite evident from (a) and (b) that the peaks observed at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='27 eV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='35 eV in EL are resulting from near band-edge emissions in ZnO and GaN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' While the most dominant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV EL feature matches very well with the defect related yellow luminescence (YL) feature in GaN PL and hence can be attributed to YL transition in GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='85 eV EL feature, which appears at a slightly red shifted position as compared to the blue luminescence (BL) PL peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='9 eV in GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='85 eV EL peak shows a blue shift with the increase of the forward bias as shown in the inset of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' These findings may suggest that the origin of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='85 eV EL peak is the donor-acceptor pair (DAP) recombination from ZnO donor to GaN acceptor states at the junction (c) EL spectra recorded for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V applied forward bias at 40K temperature for same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' EL feature can be deconvoluted with five gaussian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Integrated intensity and peak positions for the two peaks (~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='21 eV and ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='12 eV) could be obtained from this fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2: (a) PL spectra of ZnO layer and GaN substrate recorded at 10K are compared with the 10K EL spectra for sample D1 at bias voltages belonging to three regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V (regime- I), 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V (regime-II) and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='6V (regime-III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that none of the EL peaks appearing above 3 eV band matches with any of the near band edge features of either GaN or ZnO in any of the regimes, which strongly suggests that these EL transitions must be originating from the interfacial region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' (b) 10K EL profile for sample D2 recorded under 𝑉𝑏 = 9V for the device and PL spectra recorded separately on the ZnO film and GaN substrate at 10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The most dominant EL feature appearing at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='78 eV in this sample can be assigned to broad blue luminescence (BL) band observed in GaN PL (also often reported in low temperature PL of Mg doped GaN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' However, one must note the line-shape of the EL feature is seemingly quite different from that of the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Specially, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='23 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='14 eV EL features cannot be seen in GaN PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' This may imply that these EL features must also be originating from the interface like in case of device D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' -10 -5 0 5 10 0 1 2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 p-GaN Sapphire n-ZnO Ti/Au Ni/Au EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) T=300K (b) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 (a) (c) T=40K T=300K I ħw (eV) EL@12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5V PL@GaN PL@ZnO ħw (eV) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 EL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 0 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 Intensity ħw (eV) EL@8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='8V EL@12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V EL@14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='6V PL@GaN PL@ZnO Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=') ħw (eV) EL@9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1V PL@GaN PL@ZnO (a) (b) D2 D1 5 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Calculation of the band diagram: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='S4: Schematic representation of polarization charge accumulation at the interface of (0001) ZnO layer grown on (0001) GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Band diagrams can be obtained by solving Poisson equation where 𝑞 the electronic charge, 𝜖 the dielectric permittivity of the material, 𝐸(𝑥) the conduction (valence) band minimum (maximum) as a function of position 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In GaN side, 𝜌 = 𝑞[𝑝(𝑥) − 𝑁𝑎𝐺 − ], where 𝑝(𝑥) = 𝑁𝑣𝐺𝐹1/2(𝜂) the hole concentration as a function of position, 𝑁𝑣𝐺 = 2(𝑚𝑝𝐺𝑘𝐵𝑇/2𝜋ℏ2)3/2 the effective density of state of the valence band, 𝑚𝑝𝐺 the hole effective mass, 𝜂 = (𝐸𝑣𝐺(𝑥) − 𝐸𝐹𝐺)/𝑘𝐵𝑇, 𝐸𝑣𝐺(𝑥) and 𝐸𝐹𝐺 stand for valence band maximum and fermi level in GaN side, 𝑁𝑎𝐺 − = 𝑁𝑎𝐺/(1 + 𝑔𝑣𝑎 exp(𝜂) exp(∆𝐸𝑎𝐺/𝑘𝐵𝑇)) the ionized acceptor concentration and ∆𝐸𝑎𝐺 the activation energy of the shallow acceptors and 𝑔𝑣𝑎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In the neutral region, electric field should be zero and hence (𝑑𝐸𝑣𝐺/𝑑𝑥) (𝑛𝑒𝑢𝑡𝑟𝑎𝑙) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Conduction band minimum in GaN side as a function of position can be expressed as 𝐸𝑐𝐺(𝑥) = 𝐸𝑣𝐺(𝑥) + 𝐸𝑔𝐺, where 𝐸𝑔𝐺 the band gap of GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In ZnO side, at the junction, 𝐸𝑐𝑍(𝑗) = 𝐸𝑐𝐺(𝑗) − ∆𝐸𝑐 , where 𝑗 stands for the junction point, 𝐸𝑐𝑍 and ∆𝐸𝑐 represent conduction band minimum in ZnO and conduction band off-set at the interface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝜌 = 𝑞[𝑛(𝑥) − 𝑁𝑑𝑍 + ], where 𝑛(𝑥) = 𝑁𝑐𝑍𝐹1/2(𝜂) the electron concentration as a function of position, 𝑁𝑐𝐺 = 2(𝑚𝑛𝑧𝑘𝐵𝑇/2𝜋ℏ2)3/2 the effective density of state of the conduction band, 𝑚𝑛𝑍 the electron effective mass in ZnO, 𝜂 = (𝐸𝐹𝑍(𝑥) − 𝐸𝑐𝑍)/𝑘𝐵𝑇, 𝐸𝑐𝑍(𝑥) and 𝐸𝐹𝑍 stand for conduction band minimum and fermi level in ZnO side, 𝑁𝑑𝑍 + = 𝑁𝑑𝑍/(1 + 𝑔𝑐𝑑 exp(𝜂) exp(∆𝐸𝑑𝑍/𝑘𝐵𝑇)) the ionized donor concentration and ∆𝐸𝑑𝑍 the activation energy of the shallow donors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝑔𝑐𝑑 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Furthermore, the condition 𝐷𝑗𝑍 = (𝑃𝐺 − 𝑃𝑍) + 𝐷𝑗𝐺 should be satisfied at the interface, where 𝐷𝑗𝐺 and 𝐷𝑗𝑍 represent the displacement field at the junction in the GaN and ZnO sides, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝑃𝐺 and 𝑃𝑍 stand for the spontaneous polarization, respectively, in GaN and ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' In the neutral region, electric field should be zero and hence (𝑑𝐸𝑐𝑍/𝑑𝑥) (𝑛𝑒𝑢𝑡𝑟𝑎𝑙) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Valence band maximum in the ZnO side as a function of position can be expressed as 𝐸𝑣𝑍(𝑥) = 𝐸𝑐𝑍(𝑥) − 𝐸𝑔𝑍, where 𝐸𝑔𝑍 the band gap of ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that under an applied voltage of 𝑉𝑗 the condition 𝑞𝑉𝑗 = 𝐸𝐹𝑍 − 𝐸𝐹𝐺 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Equation (S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1) is solved numerically satisfying above mentioned boundary/interfacial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Parameters used in these calculations are listed in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that the temperature dependent variation of band gap in the two materials has been considered as 𝐸𝑔(𝑇) = 𝐸𝑔(0) − 𝛾𝑇2/(𝑇 + 𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝑑2𝑉 𝑑𝑥2 = − 𝜌 𝜖 ⇒ 𝑑2𝐸(𝑥) 𝑑𝑥2 = 𝑞𝜌 𝜖 (S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1) p-GaN n-ZnO \uf0c5 \uf0c5 \uf0c5 \uf0c5 \uf0c5 [0001] [0001] 𝑃𝐺 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='029 𝑃𝑍 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='05 Polarization induced charge accumulation 6 Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1: Material parameters used for the calculation GaN ZnO Band gap at 0K (𝐸𝑔) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='47 eV [A] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='42 eV [E] Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' coefficients of 𝐸𝑔 (𝛾, 𝛽) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2× 10−4 eVK-1, 600K [B] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='3× 10−4 eVK-1, 330K [F] Hole effective mass (𝑚𝑝) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='4 𝑚𝑜 [A] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='59 𝑚𝑜 [E] Electron effective mass (𝑚𝑛) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='2 𝑚𝑜 [A] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='24 𝑚𝑜 [E] Dielectric constant (𝜅) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 [C] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='5 [E] Spontaneous polarization (𝑃) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='029 C/m2 [D] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='05 C/m2 [G] Donor concentration (𝑁𝑑) 1 × 1013cm−3 1 × 1019cm−3 Acceptor concentration (𝑁𝑎) 1 × 1017cm−3 and 2 × 1018cm−3 1 × 1013cm−3 Donor activation energy (∆𝐸𝑑) 15 meV 30 meV Acceptor activation energy (∆𝐸𝑎) 140 meV [A] 180 meV Conduction band off-set (∆𝐸𝑐) 200 meV [A] Bougrov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' et 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Today 7, 34 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' [F] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=', Journal of Applied Physics 111, 073511 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' [G] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Dal Corso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' B 50, 10715 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 7 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Calculation of binding energy of the indirect excitons (IDX): Binding energy 𝐸𝑏𝑥 of such excitons can be obtained by solving the Schrodinger equation in cylindrical coordinate5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' The radial part of the wavefunction 𝑅(𝜌) satisfies 𝜌2 𝑑2𝑅 𝑑𝜌2 + 𝜌 𝑑𝑅 𝑑𝜌 + [2𝜇𝜌2 ℏ2 {𝐸𝑛𝑙 − 𝑉(𝑟)} − 𝑙2] 𝑅(𝜌) = 0 (S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='1) where, 𝜌 represents the cylindrical coordinate for the relative position of the electron and hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝐸𝑛𝑙 the eigen energy for the principle quantum number 𝑛 and angular momentum quantum number 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 𝜇 = 𝑚𝑒 + 𝑚ℎ 𝑚𝑒𝑚ℎ ⁄ the reduced mass of the exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Electron hole interaction potential can be considered as 𝑉(𝑟) = − 𝛼𝑒2 4𝜋𝜖𝑜𝜅𝑎𝑣√𝜌2 + 𝑑𝑥2 ⁄ , where 𝜅𝑎𝑣 = (1 2 ⁄ )(𝜅𝐺𝑎𝑁 + 𝜅𝑍𝑛𝑂) the average of the dielectric constants of GaN and ZnO, 𝜖𝑜 the vacuum permittivity, 𝑑𝑥 is the average electron-hole separation along the vertical direction and 𝛼 is a constant introduced to take into account the overestimation of dielectric screening effect in GaN and ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' While the azimuthal part of the wavefunction satisfies 𝛷(𝜑) = 𝑒−𝑖𝑙𝜑 √2𝜋 ⁄ , where 𝑙 = −𝑛 + 1, −𝑛 + 2, … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' ,0, … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' , 𝑛 − 2, 𝑛 − 1 for the principle quantum number 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Equation (1) can be solved using Mathematica’s ‘NDEigensystem’ function to obtain the binding energy of the excitons 𝐸𝑏𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Note that 𝐸𝑏𝑥 = 𝐸𝑛𝑙, when 𝑛 = 0 and 𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 8 References 1 Simran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rahman, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 58, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 3 Ü.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Özgür, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Alivov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Teke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Reshchikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Doğan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Avrutin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Cho, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Morkoç, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 98, 041301 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Norton, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Heo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ivill, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Ip, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Pearton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Chisholm, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Steiner, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Today 7, 34 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Lau, Calvin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Gong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Yu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Yao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} +page_content=' B 98, 115427 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE2T4oBgHgl3EQfhQc5/content/2301.03945v1.pdf'} diff 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